1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
use pyo3::prelude::*;
use numpy::
{
IntoPyArray,
PyArrayDyn,
PyReadonlyArrayDyn
};
pub fn register_module(py: Python<'_>, parent_module: &PyModule) -> PyResult<()>
{
let isometric = PyModule::new(py, "isometric")?;
super::legendre::py::register_module(py, isometric)?;
parent_module.add_submodule(isometric)?;
isometric.add_class::<FJC>()?;
Ok(())
}
/// The freely-jointed chain (FJC) model thermodynamics in the isometric ensemble.
#[pyclass]
#[derive(Copy, Clone)]
pub struct FJC
{
/// The mass of each hinge in the chain in units of kg/mol.
#[pyo3(get)]
pub hinge_mass: f64,
/// The length of each link in the chain in units of nm.
#[pyo3(get)]
pub link_length: f64,
/// The number of links in the chain.
#[pyo3(get)]
pub number_of_links: u8,
/// The thermodynamic functions of the model in the isometric ensemble approximated using a Legendre transformation.
#[pyo3(get)]
pub legendre: super::legendre::py::FJC
}
#[pymethods]
impl FJC
{
#[new]
pub fn init(number_of_links: u8, link_length: f64, hinge_mass: f64) -> Self
{
FJC
{
hinge_mass,
link_length,
number_of_links,
legendre: super::legendre::py::FJC::init(number_of_links, link_length, hinge_mass)
}
}
/// The expected force as a function of the applied end-to-end length and temperature,
///
/// .. math::
/// f(\xi, T) = \frac{\partial \psi}{\partial\xi} = \frac{kT}{\xi} + \frac{kT}{\ell_b}\left(\frac{1}{2} - \frac{1}{N_b}\right)\frac{\sum_{s=0}^{s_\mathrm{max}}(-1)^s\binom{N_b}{s}\left(m - \frac{s}{N_b}\right)^{N_b - 3}}{\sum_{s=0}^{s_\mathrm{max}}(-1)^s\binom{N_b}{s}\left(m - \frac{s}{N_b}\right)^{N_b - 2}},
///
/// where :math:`m\equiv(1 - \xi/N_b\ell_b)/2` and :math:`s_\mathrm{max}/N_b\leq m\leq (s_\mathrm{max}+1)/N_b`.
///
/// Args:
/// end_to_end_length (numpy.ndarray): The end-to-end length :math:`\xi`.
/// temperature (float): The temperature :math:`T`.
///
/// Returns:
/// numpy.ndarray: The force :math:`f`.
///
pub fn force<'py>(&self, py: Python<'py>, end_to_end_length: PyReadonlyArrayDyn<f64>, temperature: f64) -> &'py PyArrayDyn<f64>
{
end_to_end_length.as_array().mapv(|end_to_end_length: f64| super::force(&self.number_of_links, &self.link_length, &end_to_end_length, &temperature)).into_pyarray(py)
}
/// The expected nondimensional force as a function of the applied nondimensional end-to-end length per link,
///
/// .. math::
/// \eta(\gamma) = \frac{\partial\vartheta}{\partial\gamma} = \frac{1}{N_b\gamma} + \left(\frac{1}{2} - \frac{1}{N_b}\right)\frac{\sum_{s=0}^{s_\mathrm{max}}(-1)^s\binom{N_b}{s}\left(m - \frac{s}{N_b}\right)^{N_b - 3}}{\sum_{s=0}^{s_\mathrm{max}}(-1)^s\binom{N_b}{s}\left(m - \frac{s}{N_b}\right)^{N_b - 2}},
///
/// where :math:`m\equiv(1 - \gamma)/2` and :math:`s_\mathrm{max}/N_b\leq m\leq (s_\mathrm{max}+1)/N_b`.
///
/// Args:
/// nondimensional_end_to_end_length_per_link (numpy.ndarray): The nondimensional end-to-end length per link :math:`\gamma\equiv \xi/N_b\ell_b`.
///
/// Returns:
/// numpy.ndarray: The nondimensional force :math:`\eta\equiv\beta f\ell_b`.
///
pub fn nondimensional_force<'py>(&self, py: Python<'py>, nondimensional_end_to_end_length_per_link: PyReadonlyArrayDyn<f64>) -> &'py PyArrayDyn<f64>
{
nondimensional_end_to_end_length_per_link.as_array().mapv(|nondimensional_end_to_end_length_per_link: f64| super::nondimensional_force(&self.number_of_links, &nondimensional_end_to_end_length_per_link)).into_pyarray(py)
}
/// The Helmholtz free energy as a function of the applied end-to-end length and temperature,
///
/// .. math::
/// \psi(\xi, T) = -kT\ln Q(\xi, T).
///
/// Args:
/// end_to_end_length (numpy.ndarray): The end-to-end length :math:`\xi`.
/// temperature (float): The temperature :math:`T`.
///
/// Returns:
/// numpy.ndarray: The Helmholtz free energy :math:`\psi`.
///
pub fn helmholtz_free_energy<'py>(&self, py: Python<'py>, end_to_end_length: PyReadonlyArrayDyn<f64>, temperature: f64) -> &'py PyArrayDyn<f64>
{
end_to_end_length.as_array().mapv(|end_to_end_length: f64| super::helmholtz_free_energy(&self.number_of_links, &self.link_length, &self.hinge_mass, &end_to_end_length, &temperature)).into_pyarray(py)
}
/// The Helmholtz free energy per link as a function of the applied end-to-end length and temperature.
///
/// Args:
/// end_to_end_length (numpy.ndarray): The end-to-end length :math:`\xi`.
/// temperature (float): The temperature :math:`T`.
///
/// Returns:
/// numpy.ndarray: The Helmholtz free energy per link :math:`\psi/N_b`.
///
pub fn helmholtz_free_energy_per_link<'py>(&self, py: Python<'py>, end_to_end_length: PyReadonlyArrayDyn<f64>, temperature: f64) -> &'py PyArrayDyn<f64>
{
end_to_end_length.as_array().mapv(|end_to_end_length: f64| super::helmholtz_free_energy_per_link(&self.number_of_links, &self.link_length, &self.hinge_mass, &end_to_end_length, &temperature)).into_pyarray(py)
}
/// The relative Helmholtz free energy as a function of the applied end-to-end length and temperature,
///
/// .. math::
/// \Delta\psi(\xi, T) = kT\ln\left[\frac{P_\mathrm{eq}(0)}{P_\mathrm{eq}(\xi)}\right].
///
/// Args:
/// end_to_end_length (numpy.ndarray): The end-to-end length :math:`\xi`.
/// temperature (float): The temperature :math:`T`.
///
/// Returns:
/// numpy.ndarray: The relative Helmholtz free energy :math:`\Delta\psi\equiv\psi(\xi,T)-\psi(0,T)`.
///
pub fn relative_helmholtz_free_energy<'py>(&self, py: Python<'py>, end_to_end_length: PyReadonlyArrayDyn<f64>, temperature: f64) -> &'py PyArrayDyn<f64>
{
end_to_end_length.as_array().mapv(|end_to_end_length: f64| super::relative_helmholtz_free_energy(&self.number_of_links, &self.link_length, &end_to_end_length, &temperature)).into_pyarray(py)
}
/// The relative Helmholtz free energy per link as a function of the applied end-to-end length and temperature.
///
/// Args:
/// end_to_end_length (numpy.ndarray): The end-to-end length :math:`\xi`.
/// temperature (float): The temperature :math:`T`.
///
/// Returns:
/// numpy.ndarray: The relative Helmholtz free energy per link :math:`\Delta\psi/N_b`.
///
pub fn relative_helmholtz_free_energy_per_link<'py>(&self, py: Python<'py>, end_to_end_length: PyReadonlyArrayDyn<f64>, temperature: f64) -> &'py PyArrayDyn<f64>
{
end_to_end_length.as_array().mapv(|end_to_end_length: f64| super::relative_helmholtz_free_energy_per_link(&self.number_of_links, &self.link_length, &end_to_end_length, &temperature)).into_pyarray(py)
}
/// The nondimensional Helmholtz free energy as a function of the applied nondimensional end-to-end length per link and temperature.
///
/// Args:
/// nondimensional_end_to_end_length_per_link (numpy.ndarray): The nondimensional end-to-end length per link :math:`\gamma\equiv \xi/N_b\ell_b`.
/// temperature (float): The temperature :math:`T`.
///
/// Returns:
/// numpy.ndarray: The nondimensional Helmholtz free energy :math:`N_b\vartheta=\beta\psi`.
///
pub fn nondimensional_helmholtz_free_energy<'py>(&self, py: Python<'py>, nondimensional_end_to_end_length_per_link: PyReadonlyArrayDyn<f64>, temperature: f64) -> &'py PyArrayDyn<f64>
{
nondimensional_end_to_end_length_per_link.as_array().mapv(|nondimensional_end_to_end_length_per_link: f64| super::nondimensional_helmholtz_free_energy(&self.number_of_links, &self.link_length, &self.hinge_mass, &nondimensional_end_to_end_length_per_link, &temperature)).into_pyarray(py)
}
/// The nondimensional Helmholtz free energy per link as a function of the applied nondimensional end-to-end length per link and temperature.
///
/// Args:
/// nondimensional_end_to_end_length_per_link (numpy.ndarray): The nondimensional end-to-end length per link :math:`\gamma\equiv \xi/N_b\ell_b`.
/// temperature (float): The temperature :math:`T`.
///
/// Returns:
/// numpy.ndarray: The nondimensional Helmholtz free energy per link :math:`\vartheta\equiv\beta\psi/N_b`.
///
pub fn nondimensional_helmholtz_free_energy_per_link<'py>(&self, py: Python<'py>, nondimensional_end_to_end_length_per_link: PyReadonlyArrayDyn<f64>, temperature: f64) -> &'py PyArrayDyn<f64>
{
nondimensional_end_to_end_length_per_link.as_array().mapv(|nondimensional_end_to_end_length_per_link: f64| super::nondimensional_helmholtz_free_energy_per_link(&self.number_of_links, &self.link_length, &self.hinge_mass, &nondimensional_end_to_end_length_per_link, &temperature)).into_pyarray(py)
}
/// The nondimensional relative Helmholtz free energy as a function of the applied nondimensional end-to-end length per link,
///
/// .. math::
/// \beta\Delta\psi(\gamma) = \ln\left[\frac{\mathscr{P}_\mathrm{eq}(0)}{\mathscr{P}_\mathrm{eq}(\gamma)}\right].
///
/// Args:
/// nondimensional_end_to_end_length_per_link (numpy.ndarray): The nondimensional end-to-end length per link :math:`\gamma\equiv \xi/N_b\ell_b`.
///
/// Returns:
/// numpy.ndarray: The nondimensional relative Helmholtz free energy :math:`N_b\Delta\vartheta=\beta\Delta\psi`.
///
pub fn nondimensional_relative_helmholtz_free_energy<'py>(&self, py: Python<'py>, nondimensional_end_to_end_length_per_link: PyReadonlyArrayDyn<f64>) -> &'py PyArrayDyn<f64>
{
nondimensional_end_to_end_length_per_link.as_array().mapv(|nondimensional_end_to_end_length_per_link: f64| super::nondimensional_relative_helmholtz_free_energy(&self.number_of_links, &nondimensional_end_to_end_length_per_link)).into_pyarray(py)
}
/// The nondimensional relative Helmholtz free energy per link as a function of the applied nondimensional end-to-end length per link,
///
/// .. math::
/// \Delta\vartheta(\gamma) = \ln\left[\frac{\mathscr{P}_\mathrm{eq}(0)}{\mathscr{P}_\mathrm{eq}(\gamma)}\right]^{1/N_b}.
///
/// Args:
/// nondimensional_end_to_end_length_per_link (numpy.ndarray): The nondimensional end-to-end length per link :math:`\gamma\equiv \xi/N_b\ell_b`.
///
/// Returns:
/// numpy.ndarray: The nondimensional relative Helmholtz free energy per link :math:`\Delta\vartheta\equiv\beta\Delta\psi/N_b`.
///
pub fn nondimensional_relative_helmholtz_free_energy_per_link<'py>(&self, py: Python<'py>, nondimensional_end_to_end_length_per_link: PyReadonlyArrayDyn<f64>) -> &'py PyArrayDyn<f64>
{
nondimensional_end_to_end_length_per_link.as_array().mapv(|nondimensional_end_to_end_length_per_link: f64| super::nondimensional_relative_helmholtz_free_energy_per_link(&self.number_of_links, &nondimensional_end_to_end_length_per_link)).into_pyarray(py)
}
/// The equilibrium probability density of end-to-end vectors as a function of the end-to-end length,
///
/// .. math::
/// P_\mathrm{eq}(\xi) = \frac{e^{-\beta\psi(\xi, T)}}{4\pi\int e^{-\beta\psi(\xi', T)} \,{\xi'}{}^2 d\xi'} = \frac{1}{8\pi\ell_b^2\xi}\frac{N_b^{N_b - 2}}{(N_b - 2)!}\sum_{s=0}^{s_\mathrm{max}}(-1)^s\binom{N_b}{s}\left(m - \frac{s}{N_b}\right)^{N_b - 2},
///
/// where :math:`m\equiv(1 - \xi/N_b\ell_b)/2` and :math:`s_\mathrm{max}/N_b\leq m\leq (s_\mathrm{max}+1)/N_b`.
///
/// Args:
/// end_to_end_length (numpy.ndarray): The end-to-end length :math:`\xi`.
///
/// Returns:
/// numpy.ndarray: The equilibrium probability density :math:`P_\mathrm{eq}`.
///
pub fn equilibrium_distribution<'py>(&self, py: Python<'py>, end_to_end_length: PyReadonlyArrayDyn<f64>) -> &'py PyArrayDyn<f64>
{
end_to_end_length.as_array().mapv(|end_to_end_length: f64| super::equilibrium_distribution(&self.number_of_links, &self.link_length, &end_to_end_length)).into_pyarray(py)
}
/// The nondimensional equilibrium probability density of nondimensional end-to-end vectors per link as a function of the nondimensional end-to-end length per link,
///
/// .. math::
/// \mathscr{P}_\mathrm{eq}(\gamma) = \frac{e^{-\Delta\vartheta(\gamma)}}{4\pi\int e^{-\Delta\vartheta(\gamma')} \,{\gamma'}{}^2 d\gamma'} = \frac{1}{8\pi\gamma}\frac{N_b^{N_b}}{(N_b - 2)!}\sum_{s=0}^{s_\mathrm{max}}(-1)^s\binom{N_b}{s}\left(m - \frac{s}{N_b}\right)^{N_b - 2},
///
/// where :math:`m\equiv(1 - \gamma)/2` and :math:`s_\mathrm{max}/N_b\leq m\leq (s_\mathrm{max}+1)/N_b`.
///
/// Args:
/// nondimensional_end_to_end_length_per_link (numpy.ndarray): The nondimensional end-to-end length per link :math:`\gamma\equiv \xi/N_b\ell_b`.
///
/// Returns:
/// numpy.ndarray: The nondimensional equilibrium probability density :math:`\mathscr{P}_\mathrm{eq}\equiv (N_b\ell_b)^3 P_\mathrm{eq}`.
///
pub fn nondimensional_equilibrium_distribution<'py>(&self, py: Python<'py>, nondimensional_end_to_end_length_per_link: PyReadonlyArrayDyn<f64>) -> &'py PyArrayDyn<f64>
{
nondimensional_end_to_end_length_per_link.as_array().mapv(|nondimensional_end_to_end_length_per_link: f64| super::nondimensional_equilibrium_distribution(&self.number_of_links, &nondimensional_end_to_end_length_per_link)).into_pyarray(py)
}
/// The equilibrium probability density of end-to-end lengths as a function of the end-to-end length, given by :footcite:t:`treloar1949physics` as
///
/// .. math::
/// g_\mathrm{eq}(\xi) = 4\pi\xi^2 P_\mathrm{eq}(\xi) = \frac{\xi}{2\ell_b^2}\frac{N_b^{N_b-2}}{(N_b - 2)!}\sum_{s=0}^{s_\mathrm{max}}(-1)^s\binom{N_b}{s}\left(m - \frac{s}{N_b}\right)^{N_b - 2},
///
/// where :math:`m\equiv(1 - \xi/N_b\ell_b)/2` and :math:`s_\mathrm{max}/N_b\leq m\leq (s_\mathrm{max}+1)/N_b`.
///
/// Args:
/// end_to_end_length (numpy.ndarray): The end-to-end length :math:`\xi`.
///
/// Returns:
/// numpy.ndarray: The equilibrium probability density :math:`g_\mathrm{eq}`.
///
pub fn equilibrium_radial_distribution<'py>(&self, py: Python<'py>, end_to_end_length: PyReadonlyArrayDyn<f64>) -> &'py PyArrayDyn<f64>
{
end_to_end_length.as_array().mapv(|end_to_end_length: f64| super::equilibrium_radial_distribution(&self.number_of_links, &self.link_length, &end_to_end_length)).into_pyarray(py)
}
/// The nondimensional equilibrium probability density of nondimensional end-to-end lengths per link as a function of the nondimensional end-to-end length per link,
///
/// .. math::
/// \mathscr{g}_\mathrm{eq}(\gamma) = 4\pi\gamma^2 \mathscr{P}_\mathrm{eq}(\gamma) = \frac{\gamma}{2}\frac{N_b^{N_b}}{(N_b - 2)!}\sum_{s=0}^{s_\mathrm{max}}(-1)^s\binom{N_b}{s}\left(m - \frac{s}{N_b}\right)^{N_b - 2},
///
/// where :math:`m\equiv(1 - \gamma)/2` and :math:`s_\mathrm{max}/N_b\leq m\leq (s_\mathrm{max}+1)/N_b`.
///
/// Args:
/// nondimensional_end_to_end_length_per_link (numpy.ndarray): The nondimensional end-to-end length per link :math:`\gamma\equiv \xi/N_b\ell_b`.
///
/// Returns:
/// numpy.ndarray: The nondimensional equilibrium probability density :math:`\mathscr{g}_\mathrm{eq}\equiv N_b\ell_b g_\mathrm{eq}`.
///
pub fn nondimensional_equilibrium_radial_distribution<'py>(&self, py: Python<'py>, nondimensional_end_to_end_length_per_link: PyReadonlyArrayDyn<f64>) -> &'py PyArrayDyn<f64>
{
nondimensional_end_to_end_length_per_link.as_array().mapv(|nondimensional_end_to_end_length_per_link: f64| super::nondimensional_equilibrium_radial_distribution(&self.number_of_links, &nondimensional_end_to_end_length_per_link)).into_pyarray(py)
}
}