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//! Stuff related to spin lattice. use ::ndarray::{prelude::*, NdIndex}; use ::rand::prelude::*; /// A struct encapsulating the spin lattice and all the operations performed /// on it. /// /// The lattice behaves like a torus - spins on opposite edges are considered /// each other's neighbors. pub struct Lattice { dims: [usize; 2], n_of_spins: i32, inner: Array2<i32>, neighbors: Array2<[[usize; 2]; 4]>, } impl Lattice { /// Create a new lattice of given dims with randomly generated spins. pub fn new(dims: [usize; 2]) -> Self { let inner = Array2::from_shape_fn(dims, |_| { *[-1, 1].choose(&mut SmallRng::from_entropy()).unwrap() }); Self::from_array(inner) } /// View inner array. pub fn inner( &self, ) -> ndarray::ArrayView<i32, ndarray::Dim<[ndarray::Ix; 2]>> { self.inner.view() } /// Create a new lattice from provided array. /// /// # Examples /// /// ``` /// # fn main() -> Result<(), Box<std::error::Error>> { /// # use ::ndarray::prelude::*; /// # use ::rand::prelude::*; /// # use ising_lib::prelude::*; /// let array = Array::from_shape_vec((2, 2), vec![1, -1, 1, -1])?; /// let lattice = Lattice::from_array(array); /// # Ok(()) /// # } /// ``` /// /// # Panics /// /// The function will panic if or if any of the spins has incorrect value /// (neither `-1` nor `1`). /// /// ```should_panic /// # fn main() -> Result<(), Box<std::error::Error>> { /// # use ::ndarray::prelude::*; /// # use ::rand::prelude::*; /// # use ising_lib::prelude::*; /// let array = Array::from_shape_vec((2, 2), vec![5, -1, 1, -1])?; /// // ↑ incorrect spin value /// let lattice = Lattice::from_array(array); /// # Ok(()) /// # } /// ``` pub fn from_array(array: Array2<i32>) -> Self { assert!( array.iter().all(|spin| *spin == 1 || *spin == -1), "Invalid spin value." ); let roll_index = |ix: usize, amt: i32, max: usize| { let max = max as i32; ((ix as i32 + amt + max) % max) as usize }; let (width, height) = array.dim(); let neighbors = Array2::from_shape_fn((width, height), |ix| { [ [roll_index(ix.0, 1, width), ix.1], // right [ix.0, roll_index(ix.1, 1, height)], // bottom [roll_index(ix.0, -1, width), ix.1], // left [ix.0, roll_index(ix.1, -1, height)], // top ] }); Lattice { dims: [width, height], inner: array, n_of_spins: width as i32 * height as i32, neighbors, } } /// Return lattice's dimensions. pub fn dims(&self) -> [usize; 2] { self.dims } /// Return the product of the `(ith, jth)` spin and the sum of all of its /// neighbors. fn spin_times_all_neighbors<I>(&self, ix: I) -> i32 where I: NdIndex<ndarray::Dim<[ndarray::Ix; 2]>> + Copy, { self.inner[ix] * self.neighbors[ix] .iter() .map(|n_ix| self.inner[*n_ix]) .sum::<i32>() } /// Return the product of the `(ith, jth)` spin and the sum of two of its /// neighbors (the right one and the bottom one). fn spin_times_two_neighbors<I>(&self, ix: I) -> i32 where I: NdIndex<ndarray::Dim<[ndarray::Ix; 2]>> + Copy, { self.inner[ix] * self.neighbors[ix][0..2] .iter() .map(|n_ix| self.inner[*n_ix]) .sum::<i32>() } /// Return the difference of energy that would be caused by /// flipping the `(ith, jth)` spin without actually doing it. /// Used to determine the probability of a flip. /// /// ```text /// Lattice before flip: Lattice after flip: /// ##| a|## ##| a|## /// -------- -------- /// b| s| c b|-s| c /// -------- -------- /// ##| d|## ##| d|## /// /// E_2 - E_1 = /// = ((-J) * (-s) * (a + b + c + d)) - ((-J) * s * (a + b + c + d)) = /// = -J * ((-s) - s) * (a + b + c + d) = /// = -J * -2 * s * (a + b + c + d) = /// = 2 * J * s * (a + b + c + d) /// ``` /// /// # Panics /// /// This function will panic if the index is out of bounds. /// /// ```should_panic /// # use ising_lib::prelude::*; /// let lattice = Lattice::new([10, 10]); /// let _ = lattice.measure_E_diff((42, 0)); /// ``` pub fn measure_E_diff<I>(&self, ix: I) -> f64 where I: NdIndex<ndarray::Dim<[ndarray::Ix; 2]>> + Copy, { 2.0 * f64::from(self.spin_times_all_neighbors(ix)) } /// Return the difference of energy that would be caused by /// flipping the `(ith, jth)` spin in the presence of an external magnetic /// field without actually doing it. Used to determine the probability /// of a flip. /// /// ```text /// Lattice: External magnetic field: /// ##| a|## ##|##|## /// -------- -------- /// b| s| c ##| h|## /// -------- -------- /// ##| d|## ##|##|## /// /// E_2 - E_1 = /// = ((-J) * (-s) * (a + b + c + d) - h * (-s)) - ((-J) * s * (a + b + c + d) - h * s) = /// = ((-s) - s) * (-J) * (a + b + c + d) + ((-s) - s) * (-h) = /// = -2 * s * ((-J) * (a + b + c + d) - h) = /// = 2 * s * (J * (a + b + c + d) + h) /// ``` /// /// # Panics /// /// This function will panic if the index is out of bounds. /// ```should_panic /// # use ising_lib::prelude::*; /// let lattice = Lattice::new([10, 10]); /// let _ = lattice.measure_E_diff((42, 0)); /// ``` pub fn measure_E_diff_with_h<I>(&self, ix: I, h: &Array2<f64>) -> f64 where I: NdIndex<ndarray::Dim<[ndarray::Ix; 2]>> + Copy, { 2.0 * (f64::from(self.spin_times_all_neighbors(ix)) + f64::from(self.inner[ix]) * h[ix]) } /// Return the energy of the lattice. /// /// ```text /// E = -J * ∑(s_i * s_j) /// ``` pub fn measure_E(&self) -> f64 { -f64::from( self.inner .indexed_iter() .map(|(ix, _)| self.spin_times_two_neighbors(ix)) .sum::<i32>(), ) } /// Return the energy of the lattice in the presence of an external magnetic /// field. /// /// ```text /// E = -J * ∑(s_i * s_j) - ∑(s_i * h_i) /// ``` pub fn measure_E_with_h(&self, h: &Array2<f64>) -> f64 { -f64::from( self.inner .indexed_iter() .map(|(ix, _)| self.spin_times_two_neighbors(ix)) .sum::<i32>(), ) - (self.inner.map(|s| f64::from(*s)) * h).sum() } /// Return the magnetization of the lattice. The magnetization is /// a value in range `[0.0, 1.0]` and it is the absolute value of the mean /// spin value. /// /// ```text /// I = 1/n * ∑s_i /// ``` pub fn measure_I(&self) -> f64 { f64::from(self.inner.sum().abs()) / f64::from(self.n_of_spins) } /// Flip the `(ith, jth)` spin. /// /// # Panics /// /// This function panics if the index is out of bounds. pub fn flip_spin<I>(&mut self, ix: I) where I: NdIndex<ndarray::Dim<[ndarray::Ix; 2]>> + Copy, { *self.inner.get_mut(ix).unwrap() *= -1; } /// Return a valid, randomly generated spin index. pub fn gen_random_index<R: RngCore>(&mut self, rng: &mut R) -> [usize; 2] { [ rng.gen_range(0, self.dims[0] as u64) as usize, rng.gen_range(0, self.dims[1] as u64) as usize, ] } } #[cfg(test)] mod test { use ::pretty_assertions::assert_eq; use super::*; fn float_error(x: f64, t: f64) -> f64 { (x - t).abs() / t } #[test] fn test_lattice_new() { let lattice = Lattice::new([17, 10]); assert_eq!(lattice.dims(), [17, 10]); } #[test] fn test_lattice_from_array() { let array = Array::from_shape_vec((2, 2), vec![1, -1, 1, -1]).unwrap(); let lattice = Lattice::from_array(array); assert_eq!(lattice.dims(), [2, 2]); } #[test] fn test_spin_times_neighbors() { let spins = [-1, -1, 1, 1, 1, 1, 1, 1, -1]; let array = Array::from_shape_vec((3, 3), spins.to_vec()).unwrap(); let lattice = Lattice::from_array(array); let product = lattice.spin_times_all_neighbors((1, 1)); assert_eq!(product, 2); } #[test] fn test_measure_E_difference() { let array = Array::from_shape_vec((3, 3), vec![-1, -1, 1, 1, 1, 1, -1, 1, 1]) .unwrap(); let lattice = Lattice::from_array(array); let E_diff = lattice.measure_E_diff((1, 1)); assert_eq!(E_diff, 4.0); } #[test] fn test_measure_E_difference_in_magnetic_field() { let array = Array::from_shape_vec((3, 3), vec![-1, -1, 1, 1, 1, 1, -1, 1, 1]) .unwrap(); let h = Array::from_shape_vec( (3, 3), vec![-1.0, -1.0, 1.0, 1.0, -7.0, 1.0, -1.0, 1.0, 1.0], ) .unwrap(); let lattice = Lattice::from_array(array); let E_diff = lattice.measure_E_diff_with_h((1, 1), &h); assert_eq!(E_diff, -10.0); } #[test] fn test_measure_E() { let array = Array::from_shape_vec((3, 3), vec![-1, -1, -1, 1, 1, -1, 1, 1, -1]) .unwrap(); let lattice = Lattice::from_array(array); let E = lattice.measure_E(); assert_eq!(E, -2.0); } #[test] fn test_measure_E_in_magnetic_field() { let array = Array::from_shape_vec((3, 3), vec![-1, -1, -1, 1, 1, -1, 1, 1, -1]) .unwrap(); let h = Array::from_shape_vec( (3, 3), vec![-1.0, -1.0, 1.0, 1.0, -7.0, 1.0, -1.0, 1.0, 1.0], ) .unwrap(); let lattice = Lattice::from_array(array); let E = lattice.measure_E_with_h(&h); assert_eq!(E, 5.0); } #[test] fn test_measure_I() { let array = Array::from_shape_vec((2, 2), vec![-1, -1, -1, 1]).unwrap(); let lattice = Lattice::from_array(array); let I = lattice.measure_I(); assert_eq!(I, 0.5); } #[test] fn test_flip_spin() { let array = Array::from_shape_vec( (3, 3), vec![-1, -1, -1, -1, 1, 1, -1, -1, 1], ) .unwrap(); let mut lattice = Lattice::from_array(array); let E_1 = lattice.measure_E(); lattice.flip_spin((1, 1)); let E_2 = lattice.measure_E(); assert!(float_error(E_2 - E_1, -4.0) < 0.01); } }