polymers 0.3.7

"""Module to test the local module.

"""
import unittest
import numpy as np
from polymers import physics
from ..test import Parameters
from ....test import integrate

parameters = Parameters()
FJC = physics.single_chain.fjc.thermodynamics.modified_canonical.FJC


class Base(unittest.TestCase):
    """Class for basic tests.

    """
    def test_init(self):
        """Function to test instantiation.

        """
        for _ in range(parameters.number_of_loops):
            _ = FJC(
                parameters.number_of_links_minimum,
                parameters.link_length_reference,
                parameters.hinge_mass_reference
            )

    def test_number_of_links(self):
        """Function to test the number of links during instantiation.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            self.assertEqual(
                number_of_links,
                FJC(
                    number_of_links,
                    parameters.link_length_reference,
                    parameters.hinge_mass_reference
                ).number_of_links
            )

    def test_link_length(self):
        """Function to test the link length during instantiation.

        """
        for _ in range(parameters.number_of_loops):
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            self.assertEqual(
                link_length,
                FJC(
                    parameters.number_of_links_minimum,
                    link_length,
                    parameters.hinge_mass_reference
                ).link_length
            )

    def test_hinge_mass(self):
        """Function to test the hinge mass during instantiation.

        """
        for _ in range(parameters.number_of_loops):
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            self.assertEqual(
                hinge_mass,
                FJC(
                    parameters.number_of_links_minimum,
                    parameters.link_length_reference,
                    hinge_mass
                ).hinge_mass
            )

    def test_all_parameters(self):
        """Function to test all parameters during instantiation.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            self.assertEqual(
                number_of_links,
                model.number_of_links
            )
            self.assertEqual(
                link_length,
                model.link_length
            )
            self.assertEqual(
                hinge_mass,
                model.hinge_mass
            )


class Nondimensional(unittest.TestCase):
    """Class for nondimensionalization tests.

    """
    def test_end_to_end_length(self):
        """Function to test the nondimensionalization
        of the end-to-end length.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            nondimensional_end_to_end_length = \
                model.nondimensional_end_to_end_length(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness
                )
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            end_to_end_length = \
                model.end_to_end_length(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            residual_abs = \
                end_to_end_length \
                / link_length \
                - nondimensional_end_to_end_length
            residual_rel = \
                residual_abs / \
                nondimensional_end_to_end_length
            self.assertLessEqual(
                np.abs(residual_abs),
                parameters.abs_tol
            )
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_end_to_end_length_per_link(self):
        """Function to test the nondimensionalization
        of the end-to-end length per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            nondimensional_end_to_end_length_per_link = \
                model.nondimensional_end_to_end_length_per_link(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness
                )
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            end_to_end_length_per_link = \
                model.end_to_end_length_per_link(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            residual_abs = \
                end_to_end_length_per_link \
                / link_length \
                - nondimensional_end_to_end_length_per_link
            residual_rel = \
                residual_abs / \
                nondimensional_end_to_end_length_per_link
            self.assertLessEqual(
                np.abs(residual_abs),
                parameters.abs_tol
            )
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_force(self):
        """Function to test the nondimensionalization
        of the force.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_small * \
                (0.5 + (0.5 - np.random.rand()))
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            nondimensional_force = \
                model.nondimensional_force(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness
                )
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            force = \
                model.force(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            residual_abs = \
                force \
                / parameters.boltzmann_constant/temperature*link_length \
                - nondimensional_force
            residual_rel = \
                residual_abs / \
                nondimensional_force
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_helmholtz_free_energy(self):
        """Function to test the nondimensionalization
        of the Helmholtz free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_small * \
                (0.5 + (0.5 - np.random.rand()))
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            nondimensional_helmholtz_free_energy = \
                model.nondimensional_helmholtz_free_energy(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness,
                    temperature
                )
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            helmholtz_free_energy = \
                model.helmholtz_free_energy(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            residual_abs = \
                helmholtz_free_energy \
                / parameters.boltzmann_constant/temperature \
                - nondimensional_helmholtz_free_energy
            residual_rel = \
                residual_abs / \
                nondimensional_helmholtz_free_energy
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_helmholtz_free_energy_per_link(self):
        """Function to test the nondimensionalization
        of the Helmholtz free energy per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_small * \
                (0.5 + (0.5 - np.random.rand()))
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            nondimensional_helmholtz_free_energy_per_link = \
                model.nondimensional_helmholtz_free_energy_per_link(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness,
                    temperature
                )
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            helmholtz_free_energy_per_link = \
                model.helmholtz_free_energy_per_link(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            residual_abs = \
                helmholtz_free_energy_per_link \
                / parameters.boltzmann_constant/temperature \
                - nondimensional_helmholtz_free_energy_per_link
            residual_rel = \
                residual_abs / \
                nondimensional_helmholtz_free_energy_per_link
            self.assertLessEqual(
                np.abs(residual_abs),
                parameters.abs_tol
            )
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_relative_helmholtz_free_energy(self):
        """Function to test the nondimensionalization
        of the relative Helmholtz free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_small * \
                (0.5 + (0.5 - np.random.rand()))
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            nondimensional_relative_helmholtz_free_energy = \
                model.nondimensional_relative_helmholtz_free_energy(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness
                )
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            relative_helmholtz_free_energy = \
                model.relative_helmholtz_free_energy(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            residual_abs = \
                relative_helmholtz_free_energy \
                / parameters.boltzmann_constant/temperature \
                - nondimensional_relative_helmholtz_free_energy
            residual_rel = \
                residual_abs / \
                nondimensional_relative_helmholtz_free_energy
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_relative_helmholtz_free_energy_per_link(self):
        """Function to test the nondimensionalization
        of the relative Helmholtz free energy per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_small * \
                (0.5 + (0.5 - np.random.rand()))
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            nondimensional_relative_helmholtz_free_energy_per_link = \
                model.nondimensional_relative_helmholtz_free_energy_per_link(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness
                )
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            relative_helmholtz_free_energy_per_link = \
                model.relative_helmholtz_free_energy_per_link(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            residual_abs = \
                relative_helmholtz_free_energy_per_link \
                / parameters.boltzmann_constant/temperature \
                - nondimensional_relative_helmholtz_free_energy_per_link
            residual_rel = \
                residual_abs / \
                nondimensional_relative_helmholtz_free_energy_per_link
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_gibbs_free_energy(self):
        """Function to test the nondimensionalization
        of the Gibbs free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            nondimensional_gibbs_free_energy = \
                model.nondimensional_gibbs_free_energy(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness,
                    temperature
                )
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            gibbs_free_energy = \
                model.gibbs_free_energy(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            residual_abs = \
                gibbs_free_energy / \
                parameters.boltzmann_constant/temperature \
                - nondimensional_gibbs_free_energy
            residual_rel = \
                residual_abs / \
                nondimensional_gibbs_free_energy
            self.assertLessEqual(
                np.abs(residual_abs),
                parameters.abs_tol
            )
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_gibbs_free_energy_per_link(self):
        """Function to test the nondimensionalization
        of the Gibbs free energy per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            nondimensional_gibbs_free_energy_per_link = \
                model.nondimensional_gibbs_free_energy_per_link(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness,
                    temperature
                )
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            gibbs_free_energy_per_link = \
                model.gibbs_free_energy_per_link(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            residual_abs = \
                gibbs_free_energy_per_link / \
                parameters.boltzmann_constant/temperature \
                - nondimensional_gibbs_free_energy_per_link
            residual_rel = \
                residual_abs / \
                nondimensional_gibbs_free_energy_per_link
            self.assertLessEqual(
                np.abs(residual_abs),
                parameters.abs_tol
            )
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_relative_gibbs_free_energy(self):
        """Function to test the nondimensionalization
        of the relative Gibbs free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            nondimensional_relative_gibbs_free_energy = \
                model.nondimensional_relative_gibbs_free_energy(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness
                )
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            relative_gibbs_free_energy = \
                model.relative_gibbs_free_energy(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            residual_abs = \
                relative_gibbs_free_energy / \
                parameters.boltzmann_constant/temperature \
                - nondimensional_relative_gibbs_free_energy
            residual_rel = \
                residual_abs / \
                nondimensional_relative_gibbs_free_energy
            self.assertLessEqual(
                np.abs(residual_abs),
                parameters.abs_tol
            )
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_relative_gibbs_free_energy_per_link(self):
        """Function to test the nondimensionalization
        of the relative Gibbs free energy per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            nondimensional_relative_gibbs_free_energy_per_link = \
                model.nondimensional_relative_gibbs_free_energy_per_link(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness
                )
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            relative_gibbs_free_energy_per_link = \
                model.relative_gibbs_free_energy_per_link(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            residual_abs = \
                relative_gibbs_free_energy_per_link / \
                parameters.boltzmann_constant/temperature \
                - nondimensional_relative_gibbs_free_energy_per_link
            residual_rel = \
                residual_abs / \
                nondimensional_relative_gibbs_free_energy_per_link
            self.assertLessEqual(
                np.abs(residual_abs),
                parameters.abs_tol
            )
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )


class PerLink(unittest.TestCase):
    """Class for per-linkness tests.

    """
    def test_end_to_end_length(self):
        """Function to test the per-linkness
        of the end-to-end length.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            end_to_end_length = \
                model.end_to_end_length(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            end_to_end_length_per_link = \
                model.end_to_end_length_per_link(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            residual_abs = \
                end_to_end_length \
                / number_of_links \
                - end_to_end_length_per_link
            residual_rel = \
                residual_abs / \
                end_to_end_length_per_link
            self.assertLessEqual(
                np.abs(residual_abs),
                parameters.abs_tol
            )
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_nondimensional_end_to_end_length(self):
        """Function to test the per-linkness
        of the nondimensional end-to-end length.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            nondimensional_end_to_end_length = \
                model.nondimensional_end_to_end_length(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness
                )
            nondimensional_end_to_end_length_per_link = \
                model.nondimensional_end_to_end_length_per_link(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness
                )
            residual_abs = \
                nondimensional_end_to_end_length \
                / number_of_links \
                - nondimensional_end_to_end_length_per_link
            residual_rel = \
                residual_abs / \
                nondimensional_end_to_end_length_per_link
            self.assertLessEqual(
                np.abs(residual_abs),
                parameters.abs_tol
            )
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_helmholtz_free_energy(self):
        """Function to test the per-linkness
        of the Helmholtz free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_small * \
                (0.5 + (0.5 - np.random.rand()))
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            helmholtz_free_energy = \
                model.helmholtz_free_energy(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            helmholtz_free_energy_per_link = \
                model.helmholtz_free_energy_per_link(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            residual_abs = \
                helmholtz_free_energy \
                / number_of_links \
                - helmholtz_free_energy_per_link
            residual_rel = \
                residual_abs / \
                helmholtz_free_energy_per_link
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_relative_helmholtz_free_energy(self):
        """Function to test the per-linkness
        of the relative Helmholtz free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_small * \
                (0.5 + (0.5 - np.random.rand()))
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            relative_helmholtz_free_energy = \
                model.relative_helmholtz_free_energy(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            relative_helmholtz_free_energy_per_link = \
                model.relative_helmholtz_free_energy_per_link(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            residual_abs = \
                relative_helmholtz_free_energy \
                / number_of_links \
                - relative_helmholtz_free_energy_per_link
            residual_rel = \
                residual_abs / \
                relative_helmholtz_free_energy_per_link
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_nondimensional_helmholtz_free_energy(self):
        """Function to test the per-linkness
        of the nondimensional Helmholtz free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_small * \
                (0.5 + (0.5 - np.random.rand()))
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            nondimensional_helmholtz_free_energy = \
                model.nondimensional_helmholtz_free_energy(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness,
                    temperature
                )
            nondimensional_helmholtz_free_energy_per_link = \
                model.nondimensional_helmholtz_free_energy_per_link(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness,
                    temperature
                )
            residual_abs = \
                nondimensional_helmholtz_free_energy \
                / number_of_links \
                - nondimensional_helmholtz_free_energy_per_link
            residual_rel = \
                residual_abs / \
                nondimensional_helmholtz_free_energy_per_link
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_nondimensional_relative_helmholtz_free_energy(self):
        """Function to test the per-linkness
        of the nondimensional Helmholtz free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_small * \
                (0.5 + (0.5 - np.random.rand()))
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            nondimensional_relative_helmholtz_free_energy = \
                model.nondimensional_relative_helmholtz_free_energy(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness
                )
            nondimensional_relative_helmholtz_free_energy_per_link = \
                model.nondimensional_relative_helmholtz_free_energy_per_link(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness
                )
            residual_abs = \
                nondimensional_relative_helmholtz_free_energy \
                / number_of_links \
                - nondimensional_relative_helmholtz_free_energy_per_link
            residual_rel = \
                residual_abs / \
                nondimensional_relative_helmholtz_free_energy_per_link
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_gibbs_free_energy(self):
        """Function to test the per-linkness
        of the Gibbs free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            gibbs_free_energy = \
                model.gibbs_free_energy(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            gibbs_free_energy_per_link = \
                model.gibbs_free_energy_per_link(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            residual_abs = \
                gibbs_free_energy \
                / number_of_links \
                - gibbs_free_energy_per_link
            residual_rel = \
                residual_abs / \
                gibbs_free_energy_per_link
            self.assertLessEqual(
                np.abs(residual_abs),
                parameters.abs_tol
            )
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_relative_gibbs_free_energy(self):
        """Function to test the per-linkness
        of the relative Gibbs free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            relative_gibbs_free_energy = \
                model.relative_gibbs_free_energy(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            relative_gibbs_free_energy_per_link = \
                model.relative_gibbs_free_energy_per_link(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            residual_abs = \
                relative_gibbs_free_energy \
                / number_of_links \
                - relative_gibbs_free_energy_per_link
            residual_rel = \
                residual_abs / \
                relative_gibbs_free_energy_per_link
            self.assertLessEqual(
                np.abs(residual_abs),
                parameters.abs_tol
            )
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_nondimensional_gibbs_free_energy(self):
        """Function to test the per-linkness
        of the nondimensional Gibbs free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            nondimensional_gibbs_free_energy = \
                model.nondimensional_gibbs_free_energy(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness,
                    temperature
                )
            nondimensional_gibbs_free_energy_per_link = \
                model.nondimensional_gibbs_free_energy_per_link(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness,
                    temperature
                )
            residual_abs = \
                nondimensional_gibbs_free_energy \
                / number_of_links \
                - nondimensional_gibbs_free_energy_per_link
            residual_rel = \
                residual_abs / \
                nondimensional_gibbs_free_energy_per_link
            self.assertLessEqual(
                np.abs(residual_abs),
                parameters.abs_tol
            )
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_nondimensional_relative_gibbs_free_energy(self):
        """Function to test the per-linkness
        of the nondimensional relative Gibbs free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            nondimensional_relative_gibbs_free_energy = \
                model.nondimensional_relative_gibbs_free_energy(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness
                )
            nondimensional_relative_gibbs_free_energy_per_link = \
                model.nondimensional_relative_gibbs_free_energy_per_link(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness
                )
            residual_abs = \
                nondimensional_relative_gibbs_free_energy \
                / number_of_links \
                - nondimensional_relative_gibbs_free_energy_per_link
            residual_rel = \
                residual_abs / \
                nondimensional_relative_gibbs_free_energy_per_link
            self.assertLessEqual(
                np.abs(residual_abs),
                parameters.abs_tol
            )
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )


class Relative(unittest.TestCase):
    """Class for relativeness tests.

    """
    def test_helmholtz_free_energy(self):
        """Function to test the relativeness
        of the Helmholtz free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_small * \
                (0.5 + (0.5 - np.random.rand()))
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            helmholtz_free_energy = \
                model.helmholtz_free_energy(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            helmholtz_free_energy_0 = \
                model.helmholtz_free_energy(
                    np.array(
                        parameters.zero *
                        number_of_links*link_length
                    ),
                    potential_stiffness,
                    temperature
                )
            relative_helmholtz_free_energy = \
                model.relative_helmholtz_free_energy(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            residual_abs = \
                helmholtz_free_energy \
                - helmholtz_free_energy_0 \
                - relative_helmholtz_free_energy
            residual_rel = \
                residual_abs / \
                helmholtz_free_energy_0
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_helmholtz_free_energy_per_link(self):
        """Function to test the relativeness
        of the Helmholtz free energy per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_small * \
                (0.5 + (0.5 - np.random.rand()))
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            helmholtz_free_energy_per_link = \
                model.helmholtz_free_energy_per_link(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            helmholtz_free_energy_per_link_0 = \
                model.helmholtz_free_energy_per_link(
                    np.array(
                        parameters.zero *
                        number_of_links*link_length
                    ),
                    potential_stiffness,
                    temperature
                )
            relative_helmholtz_free_energy_per_link = \
                model.relative_helmholtz_free_energy_per_link(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            residual_abs = \
                helmholtz_free_energy_per_link \
                - helmholtz_free_energy_per_link_0 \
                - relative_helmholtz_free_energy_per_link
            residual_rel = \
                residual_abs / \
                helmholtz_free_energy_per_link_0
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_nondimensional_helmholtz_free_energy(self):
        """Function to test the relativeness
        of the nondimensional Helmholtz free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_small * \
                (0.5 + (0.5 - np.random.rand()))
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            nondimensional_helmholtz_free_energy = \
                model.nondimensional_helmholtz_free_energy(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness,
                    temperature
                )
            nondimensional_helmholtz_free_energy_0 = \
                model.nondimensional_helmholtz_free_energy(
                    np.array(
                        parameters.zero
                    ),
                    nondimensional_potential_stiffness,
                    temperature
                )
            nondimensional_relative_helmholtz_free_energy = \
                model.nondimensional_relative_helmholtz_free_energy(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness
                )
            residual_abs = \
                nondimensional_helmholtz_free_energy \
                - nondimensional_helmholtz_free_energy_0 \
                - nondimensional_relative_helmholtz_free_energy
            residual_rel = \
                residual_abs / \
                nondimensional_helmholtz_free_energy_0
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_nondimensional_helmholtz_free_energy_per_link(self):
        """Function to test the relativeness
        of the nondimensional Helmholtz free energy per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_small * \
                (0.5 + (0.5 - np.random.rand()))
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            nondimensional_helmholtz_free_energy_per_link = \
                model.nondimensional_helmholtz_free_energy_per_link(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness,
                    temperature
                )
            nondimensional_helmholtz_free_energy_per_link_0 = \
                model.nondimensional_helmholtz_free_energy_per_link(
                    np.array(
                        parameters.zero
                    ),
                    nondimensional_potential_stiffness,
                    temperature
                )
            nondimensional_relative_helmholtz_free_energy_per_link = \
                model.nondimensional_relative_helmholtz_free_energy_per_link(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness
                )
            residual_abs = \
                nondimensional_helmholtz_free_energy_per_link \
                - nondimensional_helmholtz_free_energy_per_link_0 \
                - nondimensional_relative_helmholtz_free_energy_per_link
            residual_rel = \
                residual_abs / \
                nondimensional_helmholtz_free_energy_per_link_0
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_gibbs_free_energy(self):
        """Function to test the relativeness
        of the Gibbs free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            gibbs_free_energy = \
                model.gibbs_free_energy(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            gibbs_free_energy_0 = \
                model.gibbs_free_energy(
                    np.array(
                        parameters.zero*number_of_links*link_length
                    ),
                    potential_stiffness,
                    temperature
                )
            relative_gibbs_free_energy = \
                model.relative_gibbs_free_energy(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            residual_abs = \
                gibbs_free_energy \
                - gibbs_free_energy_0 \
                - relative_gibbs_free_energy
            residual_rel = \
                residual_abs / \
                gibbs_free_energy_0
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_gibbs_free_energy_per_link(self):
        """Function to test the relativeness
        of the Gibbs free energy per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            gibbs_free_energy_per_link = \
                model.gibbs_free_energy_per_link(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            gibbs_free_energy_per_link_0 = \
                model.gibbs_free_energy_per_link(
                    np.array(
                        parameters.zero*number_of_links*link_length
                    ),
                    potential_stiffness,
                    temperature
                )
            relative_gibbs_free_energy_per_link = \
                model.relative_gibbs_free_energy_per_link(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            residual_abs = \
                gibbs_free_energy_per_link \
                - gibbs_free_energy_per_link_0 \
                - relative_gibbs_free_energy_per_link
            residual_rel = \
                residual_abs / \
                gibbs_free_energy_per_link_0
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_nondimensional_gibbs_free_energy(self):
        """Function to test the relativeness
        of the nondimensional Gibbs free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            nondimensional_gibbs_free_energy = \
                model.nondimensional_gibbs_free_energy(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness,
                    temperature
                )
            nondimensional_gibbs_free_energy_0 = \
                model.nondimensional_gibbs_free_energy(
                    np.array(
                        parameters.zero*number_of_links*link_length
                    ),
                    nondimensional_potential_stiffness,
                    temperature
                )
            nondimensional_relative_gibbs_free_energy = \
                model.nondimensional_relative_gibbs_free_energy(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness
                )
            residual_abs = \
                nondimensional_gibbs_free_energy \
                - nondimensional_gibbs_free_energy_0 \
                - nondimensional_relative_gibbs_free_energy
            residual_rel = \
                residual_abs / \
                nondimensional_gibbs_free_energy_0
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )

    def test_nondimensional_gibbs_free_energy_per_link(self):
        """Function to test the relativeness
        of the nondimensional Gibbs free energy per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            nondimensional_gibbs_free_energy_per_link = \
                model.nondimensional_gibbs_free_energy_per_link(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness,
                    temperature
                )
            nondimensional_gibbs_free_energy_per_link_0 = \
                model.nondimensional_gibbs_free_energy_per_link(
                    np.array(
                        parameters.zero*number_of_links*link_length
                    ),
                    nondimensional_potential_stiffness,
                    temperature
                )
            nondimensional_relative_gibbs_free_energy_per_link = \
                model.nondimensional_relative_gibbs_free_energy_per_link(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness
                )
            residual_abs = \
                nondimensional_gibbs_free_energy_per_link \
                - nondimensional_gibbs_free_energy_per_link_0 \
                - nondimensional_relative_gibbs_free_energy_per_link
            residual_rel = \
                residual_abs / \
                nondimensional_gibbs_free_energy_per_link_0
            self.assertLessEqual(
                np.abs(residual_rel),
                parameters.rel_tol
            )


class Zero(unittest.TestCase):
    """Class for zero tests.

    """
    def test_relative_helmholtz_free_energy(self):
        """Function to test the zeros
        of the relative Helmholtz free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            relative_helmholtz_free_energy_0 = \
                model.relative_helmholtz_free_energy(
                    np.array(
                        parameters.zero
                    ),
                    potential_stiffness,
                    temperature
                )
            self.assertLessEqual(
                np.abs(relative_helmholtz_free_energy_0),
                parameters.boltzmann_constant*temperature*number_of_links *
                parameters.zero
            )

    def test_relative_helmholtz_free_energy_per_link(self):
        """Function to test the zeros
        of the relative Helmholtz free energy per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            relative_helmholtz_free_energy_per_link_0 = \
                model.relative_helmholtz_free_energy_per_link(
                    np.array(
                        parameters.zero
                    ),
                    potential_stiffness,
                    temperature
                )
            self.assertLessEqual(
                np.abs(relative_helmholtz_free_energy_per_link_0),
                parameters.boltzmann_constant*temperature *
                parameters.zero
            )

    def test_nondimensional_relative_helmholtz_free_energy(self):
        """Function to test the zeros
        of the nondimensional relative Helmholtz free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            nondimensional_relative_helmholtz_free_energy_0 = \
                model.nondimensional_relative_helmholtz_free_energy(
                    np.array(
                        parameters.zero
                    ),
                    nondimensional_potential_stiffness
                )
            self.assertLessEqual(
                np.abs(
                    nondimensional_relative_helmholtz_free_energy_0
                ),
                number_of_links *
                parameters.zero
            )

    def test_nondimensional_relative_helmholtz_free_energy_per_link(self):
        """Function to test the zeros
        of the nondimensional relative Helmholtz free energy per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            nondimensional_relative_helmholtz_free_energy_per_link_0 = \
                model.nondimensional_relative_helmholtz_free_energy_per_link(
                    np.array(
                        parameters.zero
                    ),
                    nondimensional_potential_stiffness
                )
            self.assertLessEqual(
                np.abs(
                    nondimensional_relative_helmholtz_free_energy_per_link_0
                ),
                parameters.zero
            )

    def test_relative_gibbs_free_energy(self):
        """Function to test the zeros
        of the relative Gibbs free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            relative_gibbs_free_energy_0 = \
                model.relative_gibbs_free_energy(
                    np.array(
                        parameters.zero
                    ),
                    potential_stiffness,
                    temperature
                )
            self.assertLessEqual(
                np.abs(relative_gibbs_free_energy_0),
                parameters.boltzmann_constant*temperature*number_of_links *
                parameters.zero
            )

    def test_relative_gibbs_free_energy_per_link(self):
        """Function to test the zeros
        of the relative Gibbs free energy per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            relative_gibbs_free_energy_per_link_0 = \
                model.relative_gibbs_free_energy_per_link(
                    np.array(
                        parameters.zero
                    ),
                    potential_stiffness,
                    temperature
                )
            self.assertLessEqual(
                np.abs(relative_gibbs_free_energy_per_link_0),
                parameters.boltzmann_constant*temperature *
                parameters.zero
            )

    def test_nondimensional_relative_gibbs_free_energy(self):
        """Function to test the zeros
        of the nondimensional relative Gibbs free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            nondimensional_relative_gibbs_free_energy_0 = \
                model.nondimensional_relative_gibbs_free_energy(
                    np.array(
                        parameters.zero
                    ),
                    nondimensional_potential_stiffness
                )
            self.assertLessEqual(
                np.abs(
                    nondimensional_relative_gibbs_free_energy_0
                ),
                number_of_links *
                parameters.zero
            )

    def test_nondimensional_relative_gibbs_free_energy_per_link(self):
        """Function to test the zeros
        of the nondimensional relative Gibbs free energy per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            nondimensional_relative_gibbs_free_energy_per_link_0 = \
                model.nondimensional_relative_gibbs_free_energy_per_link(
                    np.array(
                        parameters.zero
                    ),
                    nondimensional_potential_stiffness
                )
            self.assertLessEqual(
                np.abs(
                    nondimensional_relative_gibbs_free_energy_per_link_0
                ),
                parameters.zero
            )


class Connection(unittest.TestCase):
    """Class for connection tests.

    """
    def test_force(self):
        """Function to test the connection
        of the force.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                0.5*parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                0.5*parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            force = \
                model.force(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            h_step = parameters.rel_tol*number_of_links*link_length
            force_from_derivative = (
                model.relative_helmholtz_free_energy(
                    np.array(
                        potential_distance + 0.5*h_step
                    ),
                    potential_stiffness,
                    temperature
                    ) -
                model.relative_helmholtz_free_energy(
                    np.array(
                        potential_distance - 0.5*h_step
                    ),
                    potential_stiffness,
                    temperature
                )
            )/h_step
            residual_abs = \
                force \
                - force_from_derivative
            residual_rel = \
                residual_abs / \
                force
            self.assertLessEqual(
                np.abs(residual_rel), h_step
            )

    def test_nondimensional_force(self):
        """Function to test the connection
        of the nondimensional force.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                0.5*parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                0.5*parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            nondimensional_force = \
                model.nondimensional_force(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness
                )
            h_step = parameters.rel_tol
            nondimensional_force_from_derivative = (
                model.nondimensional_relative_helmholtz_free_energy_per_link(
                    np.array(
                        nondimensional_potential_distance + 0.5*h_step
                    ),
                    nondimensional_potential_stiffness
                    ) -
                model.nondimensional_relative_helmholtz_free_energy_per_link(
                    np.array(
                        nondimensional_potential_distance - 0.5*h_step
                    ),
                    nondimensional_potential_stiffness
                )
            )/h_step
            residual_abs = \
                nondimensional_force \
                - nondimensional_force_from_derivative
            residual_rel = \
                residual_abs / \
                nondimensional_force
            self.assertLessEqual(
                np.abs(residual_rel), h_step
            )

    def test_end_to_end_length(self):
        """Function to test the connection
        of the end-to-end length.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                0.5*parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                0.5*parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            end_to_end_length = \
                model.end_to_end_length(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            h_step = parameters.rel_tol*number_of_links*link_length
            end_to_end_length_from_derivative = \
                -1.0/potential_stiffness*(
                    model.relative_gibbs_free_energy(
                        np.array(
                            potential_distance + 0.5*h_step
                        ),
                        potential_stiffness,
                        temperature
                    ) -
                    model.relative_gibbs_free_energy(
                        np.array(
                            potential_distance - 0.5*h_step
                        ),
                        potential_stiffness,
                        temperature
                    )
                )/h_step
            residual_abs = \
                end_to_end_length \
                - end_to_end_length_from_derivative
            residual_rel = \
                residual_abs / \
                end_to_end_length
            self.assertLessEqual(
                np.abs(residual_rel), h_step
            )

    def test_end_to_end_length_per_link(self):
        """Function to test the connection
        of the end-to-end length per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                0.5*parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                0.5*parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            potential_distance = \
                nondimensional_potential_distance * \
                number_of_links*link_length
            potential_stiffness = \
                nondimensional_potential_stiffness * \
                parameters.boltzmann_constant*temperature / \
                link_length**2
            end_to_end_length_per_link = \
                model.end_to_end_length_per_link(
                    np.array(potential_distance),
                    potential_stiffness,
                    temperature
                )
            h_step = parameters.rel_tol*number_of_links*link_length
            end_to_end_length_per_link_from_derivative = \
                -1.0/potential_stiffness*(
                    model.relative_gibbs_free_energy_per_link(
                        np.array(
                            potential_distance + 0.5*h_step
                        ),
                        potential_stiffness,
                        temperature
                    ) -
                    model.relative_gibbs_free_energy_per_link(
                        np.array(
                            potential_distance - 0.5*h_step
                        ),
                        potential_stiffness,
                        temperature
                    )
                )/h_step
            residual_abs = \
                end_to_end_length_per_link \
                - end_to_end_length_per_link_from_derivative
            residual_rel = \
                residual_abs / \
                end_to_end_length_per_link
            self.assertLessEqual(
                np.abs(residual_rel), h_step
            )

    def test_nondimensional_end_to_end_length(self):
        """Function to test the connection
        of the nondimensional end-to-end length.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                0.5*parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                0.5*parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            nondimensional_end_to_end_length = \
                model.nondimensional_end_to_end_length(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness
                )
            h_step = parameters.rel_tol
            nondimensional_end_to_end_length_from_derivative = \
                -1.0/nondimensional_potential_stiffness/number_of_links*(
                    model.nondimensional_relative_gibbs_free_energy(
                        np.array(
                            nondimensional_potential_distance + 0.5*h_step
                        ),
                        nondimensional_potential_stiffness
                    ) -
                    model.nondimensional_relative_gibbs_free_energy(
                        np.array(
                            nondimensional_potential_distance - 0.5*h_step
                        ),
                        nondimensional_potential_stiffness
                    )
                )/h_step
            residual_abs = \
                nondimensional_end_to_end_length \
                - nondimensional_end_to_end_length_from_derivative
            residual_rel = \
                residual_abs / \
                nondimensional_end_to_end_length
            self.assertLessEqual(
                np.abs(residual_rel), h_step
            )

    def test_nondimensional_end_to_end_length_per_link(self):
        """Function to test the connection
        of the nondimensional end-to-end length per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                np.random.randint(
                    parameters.number_of_links_minimum,
                    high=parameters.number_of_links_maximum
                )
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance = \
                parameters. \
                nondimensional_potential_distance_reference + \
                0.5*parameters. \
                nondimensional_potential_distance_scale * \
                (0.5 - np.random.rand())
            nondimensional_potential_stiffness = \
                parameters. \
                nondimensional_potential_stiffness_reference + \
                0.5*parameters. \
                nondimensional_potential_stiffness_scale * \
                (0.5 - np.random.rand())
            nondimensional_end_to_end_length_per_link = \
                model.nondimensional_end_to_end_length_per_link(
                    np.array(nondimensional_potential_distance),
                    nondimensional_potential_stiffness
                )
            h_step = parameters.rel_tol
            nondimensional_end_to_end_length_per_link_from_derivative = \
                -1.0/nondimensional_potential_stiffness/number_of_links*(
                    model.nondimensional_relative_gibbs_free_energy_per_link(
                        np.array(
                            nondimensional_potential_distance + 0.5*h_step
                        ),
                        nondimensional_potential_stiffness
                    ) -
                    model.nondimensional_relative_gibbs_free_energy_per_link(
                        np.array(
                            nondimensional_potential_distance - 0.5*h_step
                        ),
                        nondimensional_potential_stiffness
                    )
                )/h_step
            residual_abs = \
                nondimensional_end_to_end_length_per_link \
                - nondimensional_end_to_end_length_per_link_from_derivative
            residual_rel = \
                residual_abs / \
                nondimensional_end_to_end_length_per_link
            self.assertLessEqual(
                np.abs(residual_rel), h_step
            )


class StrongPotential(unittest.TestCase):
    """Class for the modified canonical ensemble asymptotics
    under strong potential tests.

    """
    def test_force(self):
        """Function to test the behavior
        of the force.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())

            def residual_rel(nondimensional_potential_stiffness):
                potential_stiffness = nondimensional_potential_stiffness / \
                    link_length**2 * \
                    parameters.boltzmann_constant*temperature

                def integrand_numerator(end_to_end_length):
                    return (
                        model.asymptotic.strong_potential.
                        force(
                            end_to_end_length,
                            potential_stiffness,
                            temperature
                        ) -
                        model.
                        force(
                            end_to_end_length,
                            potential_stiffness,
                            temperature
                        )
                    )**2

                def integrand_denominator(end_to_end_length):
                    return (
                        model.
                        force(
                            end_to_end_length,
                            potential_stiffness,
                            temperature
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.zero*number_of_links*link_length,
                    parameters.nondimensional_potential_distance_small *
                    number_of_links*link_length,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.zero*number_of_links*link_length,
                    parameters.nondimensional_potential_distance_small *
                    number_of_links*link_length,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_large
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_large *
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(log_log_slope/2.0 + 1.0),
                parameters.log_log_tol
            )

    def test_nondimensional_force(self):
        """Function to test the behavior
        of the nondimensional force.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )

            def residual_rel(nondimensional_potential_stiffness):

                def integrand_numerator(
                    nondimensional_end_to_end_length_per_link
                ):
                    return (
                        model.asymptotic.strong_potential.
                        nondimensional_force(
                            nondimensional_end_to_end_length_per_link,
                            nondimensional_potential_stiffness
                        ) -
                        model.
                        nondimensional_force(
                            nondimensional_end_to_end_length_per_link,
                            nondimensional_potential_stiffness
                        )
                    )**2

                def integrand_denominator(
                    nondimensional_end_to_end_length_per_link
                ):
                    return (
                        model.
                        nondimensional_force(
                            nondimensional_end_to_end_length_per_link,
                            nondimensional_potential_stiffness
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.zero,
                    parameters.nondimensional_potential_distance_small,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.zero,
                    parameters.nondimensional_potential_distance_small,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_large
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_large *
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(log_log_slope/2.0 + 1.0),
                parameters.log_log_tol
            )

    def test_helmholtz_free_energy(self):
        """Function to test the behavior
        of the Helmholtz free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())

            def residual_rel(nondimensional_potential_stiffness):
                potential_stiffness = nondimensional_potential_stiffness / \
                    link_length**2 * \
                    parameters.boltzmann_constant*temperature

                def integrand_numerator(end_to_end_length):
                    return (
                        model.asymptotic.strong_potential.
                        helmholtz_free_energy(
                            end_to_end_length,
                            potential_stiffness,
                            temperature
                        ) -
                        model.asymptotic.strong_potential.
                        helmholtz_free_energy(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small *
                                number_of_links*link_length
                            ),
                            potential_stiffness,
                            temperature
                        ) -
                        model.
                        helmholtz_free_energy(
                            end_to_end_length,
                            potential_stiffness,
                            temperature
                        ) +
                        model.
                        helmholtz_free_energy(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small *
                                number_of_links*link_length
                            ),
                            potential_stiffness,
                            temperature
                        )
                    )**2

                def integrand_denominator(end_to_end_length):
                    return (
                        model.
                        helmholtz_free_energy(
                            end_to_end_length,
                            potential_stiffness,
                            temperature
                        ) -
                        model.
                        helmholtz_free_energy(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small *
                                number_of_links*link_length
                            ),
                            potential_stiffness,
                            temperature
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.zero*number_of_links*link_length,
                    parameters.nondimensional_potential_distance_small *
                    number_of_links*link_length,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.zero*number_of_links*link_length,
                    parameters.nondimensional_potential_distance_small *
                    number_of_links*link_length,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_large
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_large *
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(log_log_slope/2.0 + 1.0),
                parameters.log_log_tol
            )

    def test_helmholtz_free_energy_per_link(self):
        """Function to test the behavior
        of the Helmholtz free energy per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())

            def residual_rel(nondimensional_potential_stiffness):
                potential_stiffness = nondimensional_potential_stiffness / \
                    link_length**2 * \
                    parameters.boltzmann_constant*temperature

                def integrand_numerator(end_to_end_length):
                    return (
                        model.asymptotic.strong_potential.
                        helmholtz_free_energy_per_link(
                            end_to_end_length,
                            potential_stiffness,
                            temperature
                        ) -
                        model.asymptotic.strong_potential.
                        helmholtz_free_energy_per_link(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small *
                                number_of_links*link_length
                            ),
                            potential_stiffness,
                            temperature
                        ) -
                        model.
                        helmholtz_free_energy_per_link(
                            end_to_end_length,
                            potential_stiffness,
                            temperature
                        ) +
                        model.
                        helmholtz_free_energy_per_link(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small *
                                number_of_links*link_length
                            ),
                            potential_stiffness,
                            temperature
                        )
                    )**2

                def integrand_denominator(end_to_end_length):
                    return (
                        model.
                        helmholtz_free_energy_per_link(
                            end_to_end_length,
                            potential_stiffness,
                            temperature
                        ) -
                        model.
                        helmholtz_free_energy_per_link(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small *
                                number_of_links*link_length
                            ),
                            potential_stiffness,
                            temperature
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.zero*number_of_links*link_length,
                    parameters.nondimensional_potential_distance_small *
                    number_of_links*link_length,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.zero*number_of_links*link_length,
                    parameters.nondimensional_potential_distance_small *
                    number_of_links*link_length,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_large
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_large *
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(log_log_slope/2.0 + 1.0),
                parameters.log_log_tol
            )

    def test_relative_helmholtz_free_energy(self):
        """Function to test the behavior
        of the relative Helmholtz free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())

            def residual_rel(nondimensional_potential_stiffness):
                potential_stiffness = nondimensional_potential_stiffness / \
                    link_length**2 * \
                    parameters.boltzmann_constant*temperature

                def integrand_numerator(end_to_end_length):
                    return (
                        model.asymptotic.strong_potential.
                        relative_helmholtz_free_energy(
                            end_to_end_length,
                            potential_stiffness,
                            temperature
                        ) -
                        model.asymptotic.strong_potential.
                        relative_helmholtz_free_energy(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small *
                                number_of_links*link_length
                            ),
                            potential_stiffness,
                            temperature
                        ) -
                        model.
                        relative_helmholtz_free_energy(
                            end_to_end_length,
                            potential_stiffness,
                            temperature
                        ) +
                        model.
                        relative_helmholtz_free_energy(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small *
                                number_of_links*link_length
                            ),
                            potential_stiffness,
                            temperature
                        )
                    )**2

                def integrand_denominator(end_to_end_length):
                    return (
                        model.
                        relative_helmholtz_free_energy(
                            end_to_end_length,
                            potential_stiffness,
                            temperature
                        ) -
                        model.
                        relative_helmholtz_free_energy(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small *
                                number_of_links*link_length
                            ),
                            potential_stiffness,
                            temperature
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.zero*number_of_links*link_length,
                    parameters.nondimensional_potential_distance_small *
                    number_of_links*link_length,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.zero*number_of_links*link_length,
                    parameters.nondimensional_potential_distance_small *
                    number_of_links*link_length,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_large
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_large *
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(log_log_slope/2.0 + 1.0),
                parameters.log_log_tol
            )

    def test_relative_helmholtz_free_energy_per_link(self):
        """Function to test the behavior
        of the relative Helmholtz free energy per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())

            def residual_rel(nondimensional_potential_stiffness):
                potential_stiffness = nondimensional_potential_stiffness / \
                    link_length**2 * \
                    parameters.boltzmann_constant*temperature

                def integrand_numerator(end_to_end_length):
                    return (
                        model.asymptotic.strong_potential.
                        relative_helmholtz_free_energy_per_link(
                            end_to_end_length,
                            potential_stiffness,
                            temperature
                        ) -
                        model.asymptotic.strong_potential.
                        relative_helmholtz_free_energy_per_link(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small *
                                number_of_links*link_length
                            ),
                            potential_stiffness,
                            temperature
                        ) -
                        model.
                        relative_helmholtz_free_energy_per_link(
                            end_to_end_length,
                            potential_stiffness,
                            temperature
                        ) +
                        model.
                        relative_helmholtz_free_energy_per_link(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small *
                                number_of_links*link_length
                            ),
                            potential_stiffness,
                            temperature
                        )
                    )**2

                def integrand_denominator(end_to_end_length):
                    return (
                        model.
                        relative_helmholtz_free_energy_per_link(
                            end_to_end_length,
                            potential_stiffness,
                            temperature
                        ) -
                        model.
                        relative_helmholtz_free_energy_per_link(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small *
                                number_of_links*link_length
                            ),
                            potential_stiffness,
                            temperature
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.zero*number_of_links*link_length,
                    parameters.nondimensional_potential_distance_small *
                    number_of_links*link_length,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.zero*number_of_links*link_length,
                    parameters.nondimensional_potential_distance_small *
                    number_of_links*link_length,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_large
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_large *
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(log_log_slope/2.0 + 1.0),
                parameters.log_log_tol
            )

    def test_nondimensional_helmholtz_free_energy(self):
        """Function to test the behavior
        of the nondimensional Helmholtz free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())

            def residual_rel(nondimensional_potential_stiffness):

                def integrand_numerator(
                    nondimensional_end_to_end_length_per_link
                ):
                    return (
                        model.asymptotic.strong_potential.
                        nondimensional_helmholtz_free_energy(
                            nondimensional_end_to_end_length_per_link,
                            nondimensional_potential_stiffness,
                            temperature
                        ) -
                        model.asymptotic.strong_potential.
                        nondimensional_helmholtz_free_energy(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small
                            ),
                            nondimensional_potential_stiffness,
                            temperature
                        ) -
                        model.
                        nondimensional_helmholtz_free_energy(
                            nondimensional_end_to_end_length_per_link,
                            nondimensional_potential_stiffness,
                            temperature
                        ) +
                        model.
                        nondimensional_helmholtz_free_energy(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small
                            ),
                            nondimensional_potential_stiffness,
                            temperature
                        )
                    )**2

                def integrand_denominator(
                    nondimensional_end_to_end_length_per_link
                ):
                    return (
                        model.
                        nondimensional_helmholtz_free_energy(
                            nondimensional_end_to_end_length_per_link,
                            nondimensional_potential_stiffness,
                            temperature
                        ) -
                        model.
                        nondimensional_helmholtz_free_energy(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small
                            ),
                            nondimensional_potential_stiffness,
                            temperature
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.zero,
                    parameters.nondimensional_potential_distance_small,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.zero,
                    parameters.nondimensional_potential_distance_small,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_large
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_large *
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(log_log_slope/2.0 + 1.0),
                parameters.log_log_tol
            )

    def test_nondimensional_helmholtz_free_energy_per_link(self):
        """Function to test the behavior
        of the nondimensional Helmholtz free energy per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())

            def residual_rel(nondimensional_potential_stiffness):

                def integrand_numerator(
                    nondimensional_end_to_end_length_per_link
                ):
                    return (
                        model.asymptotic.strong_potential.
                        nondimensional_helmholtz_free_energy_per_link(
                            nondimensional_end_to_end_length_per_link,
                            nondimensional_potential_stiffness,
                            temperature
                        ) -
                        model.asymptotic.strong_potential.
                        nondimensional_helmholtz_free_energy_per_link(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small
                            ),
                            nondimensional_potential_stiffness,
                            temperature
                        ) -
                        model.
                        nondimensional_helmholtz_free_energy_per_link(
                            nondimensional_end_to_end_length_per_link,
                            nondimensional_potential_stiffness,
                            temperature
                        ) +
                        model.
                        nondimensional_helmholtz_free_energy_per_link(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small
                            ),
                            nondimensional_potential_stiffness,
                            temperature
                        )
                    )**2

                def integrand_denominator(
                    nondimensional_end_to_end_length_per_link
                ):
                    return (
                        model.
                        nondimensional_helmholtz_free_energy_per_link(
                            nondimensional_end_to_end_length_per_link,
                            nondimensional_potential_stiffness,
                            temperature
                        ) -
                        model.
                        nondimensional_helmholtz_free_energy_per_link(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small
                            ),
                            nondimensional_potential_stiffness,
                            temperature
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.zero,
                    parameters.nondimensional_potential_distance_small,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.zero,
                    parameters.nondimensional_potential_distance_small,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_large
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_large *
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(log_log_slope/2.0 + 1.0),
                parameters.log_log_tol
            )

    def test_nondimensional_relative_helmholtz_free_energy(self):
        """Function to test the behavior
        of the nondimensional relative Helmholtz free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )

            def residual_rel(nondimensional_potential_stiffness):

                def integrand_numerator(
                    nondimensional_end_to_end_length_per_link
                ):
                    return (
                        model.asymptotic.strong_potential.
                        nondimensional_relative_helmholtz_free_energy(
                            nondimensional_end_to_end_length_per_link,
                            nondimensional_potential_stiffness
                        ) -
                        model.asymptotic.strong_potential.
                        nondimensional_relative_helmholtz_free_energy(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small
                            ),
                            nondimensional_potential_stiffness
                        ) -
                        model.
                        nondimensional_relative_helmholtz_free_energy(
                            nondimensional_end_to_end_length_per_link,
                            nondimensional_potential_stiffness
                        ) +
                        model.
                        nondimensional_relative_helmholtz_free_energy(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small
                            ),
                            nondimensional_potential_stiffness
                        )
                    )**2

                def integrand_denominator(
                    nondimensional_end_to_end_length_per_link
                ):
                    return (
                        model.
                        nondimensional_relative_helmholtz_free_energy(
                            nondimensional_end_to_end_length_per_link,
                            nondimensional_potential_stiffness
                        ) -
                        model.
                        nondimensional_relative_helmholtz_free_energy(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small
                            ),
                            nondimensional_potential_stiffness
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.zero,
                    parameters.nondimensional_potential_distance_small,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.zero,
                    parameters.nondimensional_potential_distance_small,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_large
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_large *
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(log_log_slope/2.0 + 1.0),
                parameters.log_log_tol
            )

    def test_nondimensional_relative_helmholtz_free_energy_per_link(self):
        """Function to test the behavior
        of the nondimensional relative Helmholtz free energy per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )

            def residual_rel(nondimensional_potential_stiffness):

                def integrand_numerator(
                    nondimensional_end_to_end_length_per_link
                ):
                    return (
                        model.asymptotic.strong_potential.
                        nondimensional_relative_helmholtz_free_energy_per_link(
                            nondimensional_end_to_end_length_per_link,
                            nondimensional_potential_stiffness
                        ) -
                        model.asymptotic.strong_potential.
                        nondimensional_relative_helmholtz_free_energy_per_link(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small
                            ),
                            nondimensional_potential_stiffness
                        ) -
                        model.
                        nondimensional_relative_helmholtz_free_energy_per_link(
                            nondimensional_end_to_end_length_per_link,
                            nondimensional_potential_stiffness
                        ) +
                        model.
                        nondimensional_relative_helmholtz_free_energy_per_link(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small
                            ),
                            nondimensional_potential_stiffness
                        )
                    )**2

                def integrand_denominator(
                    nondimensional_end_to_end_length_per_link
                ):
                    return (
                        model.
                        nondimensional_relative_helmholtz_free_energy_per_link(
                            nondimensional_end_to_end_length_per_link,
                            nondimensional_potential_stiffness
                        ) -
                        model.
                        nondimensional_relative_helmholtz_free_energy_per_link(
                            np.array(
                                parameters.
                                nondimensional_potential_distance_small
                            ),
                            nondimensional_potential_stiffness
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.zero,
                    parameters.nondimensional_potential_distance_small,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.zero,
                    parameters.nondimensional_potential_distance_small,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_large
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_large *
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(log_log_slope/2.0 + 1.0),
                parameters.log_log_tol
            )


class WeakPotential(unittest.TestCase):
    """Class for the modified canonical ensemble asymptotics
    under weak potential tests.

    """
    def test_end_to_end_length(self):
        """Function to test the behavior
        of the end-to-end length.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())

            def residual_rel(nondimensional_potential_stiffness):
                potential_stiffness = nondimensional_potential_stiffness / \
                    link_length**2 * \
                    parameters.boltzmann_constant*temperature

                def integrand_numerator(nondimensional_potential_distance):
                    potential_distance = nondimensional_potential_distance * \
                        number_of_links*link_length
                    return (
                        model.
                        end_to_end_length(
                            potential_distance,
                            potential_stiffness,
                            temperature
                        ) -
                        model.asymptotic.weak_potential.
                        end_to_end_length(
                            potential_distance,
                            potential_stiffness,
                            temperature
                        )
                    )**2

                def integrand_denominator(nondimensional_potential_distance):
                    potential_distance = nondimensional_potential_distance * \
                        number_of_links*link_length
                    return (
                        model.
                        end_to_end_length(
                            potential_distance,
                            potential_stiffness,
                            temperature
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_small
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(residual_rel_1),
                parameters.nondimensional_potential_stiffness_small
            )
            self.assertLessEqual(
                np.abs(residual_rel_2),
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            self.assertLessEqual(
                np.abs(log_log_slope + 1.0),
                parameters.log_log_tol
            )

    def test_end_to_end_length_per_link(self):
        """Function to test the behavior
        of the end-to-end length per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())

            def residual_rel(nondimensional_potential_stiffness):
                potential_stiffness = nondimensional_potential_stiffness / \
                    link_length**2 * \
                    parameters.boltzmann_constant*temperature

                def integrand_numerator(nondimensional_potential_distance):
                    potential_distance = nondimensional_potential_distance * \
                        number_of_links*link_length
                    return (
                        model.
                        end_to_end_length_per_link(
                            potential_distance,
                            potential_stiffness,
                            temperature
                        ) -
                        model.asymptotic.weak_potential.
                        end_to_end_length_per_link(
                            potential_distance,
                            potential_stiffness,
                            temperature
                        )
                    )**2

                def integrand_denominator(nondimensional_potential_distance):
                    potential_distance = nondimensional_potential_distance * \
                        number_of_links*link_length
                    return (
                        model.
                        end_to_end_length_per_link(
                            potential_distance,
                            potential_stiffness,
                            temperature
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_small
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(residual_rel_1),
                parameters.nondimensional_potential_stiffness_small
            )
            self.assertLessEqual(
                np.abs(residual_rel_2),
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            self.assertLessEqual(
                np.abs(log_log_slope + 1.0),
                parameters.log_log_tol
            )

    def test_nondimensional_end_to_end_length(self):
        """Function to test the behavior
        of the nondimensional end-to-end length.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )

            def residual_rel(nondimensional_potential_stiffness):

                def integrand_numerator(nondimensional_potential_distance):
                    return (
                        model.
                        nondimensional_end_to_end_length(
                            nondimensional_potential_distance,
                            nondimensional_potential_stiffness
                        ) -
                        model.asymptotic.weak_potential.
                        nondimensional_end_to_end_length(
                            nondimensional_potential_distance,
                            nondimensional_potential_stiffness
                        )
                    )**2

                def integrand_denominator(nondimensional_potential_distance):
                    return (
                        model.
                        nondimensional_end_to_end_length(
                            nondimensional_potential_distance,
                            nondimensional_potential_stiffness,
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_small
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(residual_rel_1),
                parameters.nondimensional_potential_stiffness_small
            )
            self.assertLessEqual(
                np.abs(residual_rel_2),
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            self.assertLessEqual(
                np.abs(log_log_slope + 1.0),
                parameters.log_log_tol
            )

    def test_nondimensional_end_to_end_length_per_link(self):
        """Function to test the behavior
        of the nondimensional end-to-end length per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )

            def residual_rel(nondimensional_potential_stiffness):

                def integrand_numerator(nondimensional_potential_distance):
                    return (
                        model.
                        nondimensional_end_to_end_length_per_link(
                            nondimensional_potential_distance,
                            nondimensional_potential_stiffness
                        ) -
                        model.asymptotic.weak_potential.
                        nondimensional_end_to_end_length_per_link(
                            nondimensional_potential_distance,
                            nondimensional_potential_stiffness
                        )
                    )**2

                def integrand_denominator(nondimensional_potential_distance):
                    return (
                        model.
                        nondimensional_end_to_end_length_per_link(
                            nondimensional_potential_distance,
                            nondimensional_potential_stiffness,
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_small
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(residual_rel_1),
                parameters.nondimensional_potential_stiffness_small
            )
            self.assertLessEqual(
                np.abs(residual_rel_2),
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            self.assertLessEqual(
                np.abs(log_log_slope + 1.0),
                parameters.log_log_tol
            )

    def test_gibbs_free_energy(self):
        """Function to test the behavior
        of the Gibbs free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            potential_distance_ref = \
                np.array(
                    parameters.nondimensional_potential_distance_large_1
                )*number_of_links*link_length

            def residual_rel(nondimensional_potential_stiffness):
                potential_stiffness = nondimensional_potential_stiffness / \
                    link_length**2 * \
                    parameters.boltzmann_constant*temperature

                def integrand_numerator(nondimensional_potential_distance):
                    potential_distance = nondimensional_potential_distance * \
                        number_of_links*link_length
                    return (
                        model.
                        gibbs_free_energy(
                            potential_distance,
                            potential_stiffness,
                            temperature
                        ) -
                        model.
                        gibbs_free_energy(
                            np.array(potential_distance_ref),
                            potential_stiffness,
                            temperature
                        ) -
                        model.asymptotic.weak_potential.
                        gibbs_free_energy(
                            potential_distance,
                            potential_stiffness,
                            temperature
                        ) +
                        model.asymptotic.weak_potential.
                        gibbs_free_energy(
                            np.array(potential_distance_ref),
                            potential_stiffness,
                            temperature
                        )
                    )**2

                def integrand_denominator(nondimensional_potential_distance):
                    potential_distance = nondimensional_potential_distance * \
                        number_of_links*link_length
                    return (
                        model.
                        gibbs_free_energy(
                            potential_distance,
                            potential_stiffness,
                            temperature
                        ) -
                        model.
                        gibbs_free_energy(
                            np.array(potential_distance_ref),
                            potential_stiffness,
                            temperature
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_small
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(residual_rel_1),
                parameters.nondimensional_potential_stiffness_small
            )
            self.assertLessEqual(
                np.abs(residual_rel_2),
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            self.assertLessEqual(
                np.abs(log_log_slope + 1.0),
                parameters.log_log_tol
            )

    def test_gibbs_free_energy_per_link(self):
        """Function to test the behavior
        of the Gibbs free energy per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            potential_distance_ref = \
                np.array(
                    parameters.nondimensional_potential_distance_large_1
                )*number_of_links*link_length

            def residual_rel(nondimensional_potential_stiffness):
                potential_stiffness = nondimensional_potential_stiffness / \
                    link_length**2 * \
                    parameters.boltzmann_constant*temperature

                def integrand_numerator(nondimensional_potential_distance):
                    potential_distance = nondimensional_potential_distance * \
                        number_of_links*link_length
                    return (
                        model.
                        gibbs_free_energy_per_link(
                            potential_distance,
                            potential_stiffness,
                            temperature
                        ) -
                        model.
                        gibbs_free_energy_per_link(
                            np.array(potential_distance_ref),
                            potential_stiffness,
                            temperature
                        ) -
                        model.asymptotic.weak_potential.
                        gibbs_free_energy_per_link(
                            potential_distance,
                            potential_stiffness,
                            temperature
                        ) +
                        model.asymptotic.weak_potential.
                        gibbs_free_energy_per_link(
                            np.array(potential_distance_ref),
                            potential_stiffness,
                            temperature
                        )
                    )**2

                def integrand_denominator(nondimensional_potential_distance):
                    potential_distance = nondimensional_potential_distance * \
                        number_of_links*link_length
                    return (
                        model.
                        gibbs_free_energy_per_link(
                            potential_distance,
                            potential_stiffness,
                            temperature
                        ) -
                        model.
                        gibbs_free_energy_per_link(
                            np.array(potential_distance_ref),
                            potential_stiffness,
                            temperature
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_small
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(residual_rel_1),
                parameters.nondimensional_potential_stiffness_small
            )
            self.assertLessEqual(
                np.abs(residual_rel_2),
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            self.assertLessEqual(
                np.abs(log_log_slope + 1.0),
                parameters.log_log_tol
            )

    def test_relative_gibbs_free_energy(self):
        """Function to test the behavior
        of the relative Gibbs free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            potential_distance_ref = \
                np.array(
                    parameters.nondimensional_potential_distance_large_1
                )*number_of_links*link_length

            def residual_rel(nondimensional_potential_stiffness):
                potential_stiffness = nondimensional_potential_stiffness / \
                    link_length**2 * \
                    parameters.boltzmann_constant*temperature

                def integrand_numerator(nondimensional_potential_distance):
                    potential_distance = nondimensional_potential_distance * \
                        number_of_links*link_length
                    return (
                        model.
                        relative_gibbs_free_energy(
                            potential_distance,
                            potential_stiffness,
                            temperature
                        ) -
                        model.
                        relative_gibbs_free_energy(
                            np.array(potential_distance_ref),
                            potential_stiffness,
                            temperature
                        ) -
                        model.asymptotic.weak_potential.
                        relative_gibbs_free_energy(
                            potential_distance,
                            potential_stiffness,
                            temperature
                        ) +
                        model.asymptotic.weak_potential.
                        relative_gibbs_free_energy(
                            np.array(potential_distance_ref),
                            potential_stiffness,
                            temperature
                        )
                    )**2

                def integrand_denominator(nondimensional_potential_distance):
                    potential_distance = nondimensional_potential_distance * \
                        number_of_links*link_length
                    return (
                        model.
                        relative_gibbs_free_energy(
                            potential_distance,
                            potential_stiffness,
                            temperature
                        ) -
                        model.
                        relative_gibbs_free_energy(
                            np.array(potential_distance_ref),
                            potential_stiffness,
                            temperature
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_small
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(residual_rel_1),
                parameters.nondimensional_potential_stiffness_small
            )
            self.assertLessEqual(
                np.abs(residual_rel_2),
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            self.assertLessEqual(
                np.abs(log_log_slope + 1.0),
                parameters.log_log_tol
            )

    def test_relative_gibbs_free_energy_per_link(self):
        """Function to test the behavior
        of the relative Gibbs free energy per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            potential_distance_ref = \
                np.array(
                    parameters.nondimensional_potential_distance_large_1
                )*number_of_links*link_length

            def residual_rel(nondimensional_potential_stiffness):
                potential_stiffness = nondimensional_potential_stiffness / \
                    link_length**2 * \
                    parameters.boltzmann_constant*temperature

                def integrand_numerator(nondimensional_potential_distance):
                    potential_distance = nondimensional_potential_distance * \
                        number_of_links*link_length
                    return (
                        model.
                        relative_gibbs_free_energy_per_link(
                            potential_distance,
                            potential_stiffness,
                            temperature
                        ) -
                        model.
                        relative_gibbs_free_energy_per_link(
                            np.array(potential_distance_ref),
                            potential_stiffness,
                            temperature
                        ) -
                        model.asymptotic.weak_potential.
                        relative_gibbs_free_energy_per_link(
                            potential_distance,
                            potential_stiffness,
                            temperature
                        ) +
                        model.asymptotic.weak_potential.
                        relative_gibbs_free_energy_per_link(
                            np.array(potential_distance_ref),
                            potential_stiffness,
                            temperature
                        )
                    )**2

                def integrand_denominator(nondimensional_potential_distance):
                    potential_distance = nondimensional_potential_distance * \
                        number_of_links*link_length
                    return (
                        model.
                        relative_gibbs_free_energy_per_link(
                            potential_distance,
                            potential_stiffness,
                            temperature
                        ) -
                        model.
                        relative_gibbs_free_energy_per_link(
                            np.array(potential_distance_ref),
                            potential_stiffness,
                            temperature
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_small
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(residual_rel_1),
                parameters.nondimensional_potential_stiffness_small
            )
            self.assertLessEqual(
                np.abs(residual_rel_2),
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            self.assertLessEqual(
                np.abs(log_log_slope + 1.0),
                parameters.log_log_tol
            )

    def test_nondimensional_gibbs_free_energy(self):
        """Function to test the behavior
        of the nondimensional Gibbs free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            nondimensional_potential_distance_ref = \
                np.array(
                    parameters.nondimensional_potential_distance_large_1
                )

            def residual_rel(nondimensional_potential_stiffness):

                def integrand_numerator(nondimensional_potential_distance):
                    return (
                        model.
                        nondimensional_gibbs_free_energy(
                            nondimensional_potential_distance,
                            nondimensional_potential_stiffness,
                            temperature
                        ) -
                        model.
                        nondimensional_gibbs_free_energy(
                            np.array(nondimensional_potential_distance_ref),
                            nondimensional_potential_stiffness,
                            temperature
                        ) -
                        model.asymptotic.weak_potential.
                        nondimensional_gibbs_free_energy(
                            nondimensional_potential_distance,
                            nondimensional_potential_stiffness,
                            temperature
                        ) +
                        model.asymptotic.weak_potential.
                        nondimensional_gibbs_free_energy(
                            np.array(nondimensional_potential_distance_ref),
                            nondimensional_potential_stiffness,
                            temperature
                        )
                    )**2

                def integrand_denominator(nondimensional_potential_distance):
                    return (
                        model.
                        nondimensional_gibbs_free_energy(
                            nondimensional_potential_distance,
                            nondimensional_potential_stiffness,
                            temperature
                        ) -
                        model.
                        nondimensional_gibbs_free_energy(
                            np.array(nondimensional_potential_distance_ref),
                            nondimensional_potential_stiffness,
                            temperature
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_small
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(residual_rel_1),
                parameters.nondimensional_potential_stiffness_small
            )
            self.assertLessEqual(
                np.abs(residual_rel_2),
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            self.assertLessEqual(
                np.abs(log_log_slope + 1.0),
                parameters.log_log_tol
            )

    def test_nondimensional_gibbs_free_energy_per_link(self):
        """Function to test the behavior
        of the nondimensional Gibbs free energy per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            temperature = \
                parameters.temperature_reference + \
                parameters.temperature_scale*(0.5 - np.random.rand())
            nondimensional_potential_distance_ref = \
                np.array(
                    parameters.nondimensional_potential_distance_large_1
                )

            def residual_rel(nondimensional_potential_stiffness):

                def integrand_numerator(nondimensional_potential_distance):
                    return (
                        model.
                        nondimensional_gibbs_free_energy_per_link(
                            nondimensional_potential_distance,
                            nondimensional_potential_stiffness,
                            temperature
                        ) -
                        model.
                        nondimensional_gibbs_free_energy_per_link(
                            np.array(nondimensional_potential_distance_ref),
                            nondimensional_potential_stiffness,
                            temperature
                        ) -
                        model.asymptotic.weak_potential.
                        nondimensional_gibbs_free_energy_per_link(
                            nondimensional_potential_distance,
                            nondimensional_potential_stiffness,
                            temperature
                        ) +
                        model.asymptotic.weak_potential.
                        nondimensional_gibbs_free_energy_per_link(
                            np.array(nondimensional_potential_distance_ref),
                            nondimensional_potential_stiffness,
                            temperature
                        )
                    )**2

                def integrand_denominator(nondimensional_potential_distance):
                    return (
                        model.
                        nondimensional_gibbs_free_energy_per_link(
                            nondimensional_potential_distance,
                            nondimensional_potential_stiffness,
                            temperature
                        ) -
                        model.
                        nondimensional_gibbs_free_energy_per_link(
                            np.array(nondimensional_potential_distance_ref),
                            nondimensional_potential_stiffness,
                            temperature
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_small
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(residual_rel_1),
                parameters.nondimensional_potential_stiffness_small
            )
            self.assertLessEqual(
                np.abs(residual_rel_2),
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            self.assertLessEqual(
                np.abs(log_log_slope + 1.0),
                parameters.log_log_tol
            )

    def test_nondimensional_relative_gibbs_free_energy(self):
        """Function to test the behavior
        of the nondimensional relative Gibbs free energy.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance_ref = \
                np.array(
                    parameters.nondimensional_potential_distance_large_1
                )

            def residual_rel(nondimensional_potential_stiffness):

                def integrand_numerator(nondimensional_potential_distance):
                    return (
                        model.
                        nondimensional_relative_gibbs_free_energy(
                            nondimensional_potential_distance,
                            nondimensional_potential_stiffness
                        ) -
                        model.
                        nondimensional_relative_gibbs_free_energy(
                            np.array(nondimensional_potential_distance_ref),
                            nondimensional_potential_stiffness
                        ) -
                        model.asymptotic.weak_potential.
                        nondimensional_relative_gibbs_free_energy(
                            nondimensional_potential_distance,
                            nondimensional_potential_stiffness
                        ) +
                        model.asymptotic.weak_potential.
                        nondimensional_relative_gibbs_free_energy(
                            np.array(nondimensional_potential_distance_ref),
                            nondimensional_potential_stiffness
                        )
                    )**2

                def integrand_denominator(nondimensional_potential_distance):
                    return (
                        model.
                        nondimensional_relative_gibbs_free_energy(
                            nondimensional_potential_distance,
                            nondimensional_potential_stiffness
                        ) -
                        model.
                        nondimensional_relative_gibbs_free_energy(
                            np.array(nondimensional_potential_distance_ref),
                            nondimensional_potential_stiffness
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_small
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(residual_rel_1),
                parameters.nondimensional_potential_stiffness_small
            )
            self.assertLessEqual(
                np.abs(residual_rel_2),
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            self.assertLessEqual(
                np.abs(log_log_slope + 1.0),
                parameters.log_log_tol
            )

    def test_nondimensional_relative_gibbs_free_energy_per_link(self):
        """Function to test the behavior
        of the nondimensional relative Gibbs free energy per link.

        """
        for _ in range(parameters.number_of_loops):
            number_of_links = \
                parameters.number_of_links_maximum - \
                parameters.number_of_links_minimum
            link_length = \
                parameters.link_length_reference + \
                parameters.link_length_scale*(0.5 - np.random.rand())
            hinge_mass = \
                parameters.hinge_mass_reference + \
                parameters.hinge_mass_scale*(0.5 - np.random.rand())
            model = FJC(
                number_of_links,
                link_length,
                hinge_mass
            )
            nondimensional_potential_distance_ref = \
                np.array(
                    parameters.nondimensional_potential_distance_large_1
                )

            def residual_rel(nondimensional_potential_stiffness):

                def integrand_numerator(nondimensional_potential_distance):
                    return (
                        model.
                        nondimensional_relative_gibbs_free_energy_per_link(
                            nondimensional_potential_distance,
                            nondimensional_potential_stiffness
                        ) -
                        model.
                        nondimensional_relative_gibbs_free_energy_per_link(
                            np.array(nondimensional_potential_distance_ref),
                            nondimensional_potential_stiffness
                        ) -
                        model.asymptotic.weak_potential.
                        nondimensional_relative_gibbs_free_energy_per_link(
                            nondimensional_potential_distance,
                            nondimensional_potential_stiffness
                        ) +
                        model.asymptotic.weak_potential.
                        nondimensional_relative_gibbs_free_energy_per_link(
                            np.array(nondimensional_potential_distance_ref),
                            nondimensional_potential_stiffness
                        )
                    )**2

                def integrand_denominator(nondimensional_potential_distance):
                    return (
                        model.
                        nondimensional_relative_gibbs_free_energy_per_link(
                            nondimensional_potential_distance,
                            nondimensional_potential_stiffness
                        ) -
                        model.
                        nondimensional_relative_gibbs_free_energy_per_link(
                            np.array(nondimensional_potential_distance_ref),
                            nondimensional_potential_stiffness
                        )
                    )**2

                numerator = integrate(
                    integrand_numerator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                denominator = integrate(
                    integrand_denominator,
                    parameters.nondimensional_potential_distance_large_1,
                    parameters.nondimensional_potential_distance_large_2,
                    parameters.points)
                return np.sqrt(numerator/denominator)

            residual_rel_1 = residual_rel(
                parameters.nondimensional_potential_stiffness_small
            )
            residual_rel_2 = residual_rel(
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            log_log_slope = np.log(residual_rel_2/residual_rel_1) / \
                np.log(parameters.log_log_scale)
            self.assertLessEqual(
                np.abs(residual_rel_1),
                parameters.nondimensional_potential_stiffness_small
            )
            self.assertLessEqual(
                np.abs(residual_rel_2),
                parameters.nondimensional_potential_stiffness_small /
                parameters.log_log_scale
            )
            self.assertLessEqual(
                np.abs(log_log_slope + 1.0),
                parameters.log_log_tol
            )