Statistical Mechanics and Computations

Code from the course: Statistical Mechanics and Computations. It includes exercises from the textbook Statistical Mechanics: Algorithms and Computations by Werner Krauth.

From the Course

#	Folder	File	Description
1			Monte Carlo Algorithms: Direct Sampling & Markov Chain sampling
	Lecture	direct_pi.py	Algorithm 1.1 - compute pi using direct sampling
		direct_pi_multirun.py	Compute pi using multiple runs of direct sampling
		markov_pi.py	Calculate pi using Markov Chain Monte Carlo
		markov_pi_multirun.py	Calculate pi using multiple runs of Markov Chain Monte Carlo
		pebble_basic.py	Demonstrate neighbour table for MCMC-Algorithm 1.6
		pebble_basic_inhomogeneous.py	Demonstrate neighbour table for MCMC-Algorithm 1.6 with inhomogeneous probabilities
	Tutorial	pebble_basic.py	Move pebble using neighbour Table 1.3
		pebble_basic_movie.py	Move pebble using neighbour Table 1.3
		pebble_basic_multirun.py	Move pebble using neighbour Table 1.3
		pebble_dual_eigen.py	Two games: Illustrate conversion of reducible matrix to irreducible+aperiodic to make motion ergodic
		pebble_dual_movie.py	'''Two games: Illustrate conversion of reducible matrix to irreducible+aperiodic to make motion ergodic'''
		pebble_multirun_all_histogram.py	Show probability for reaching other cells after 0, 1, 2, ... moves
		pebble_multirun_histogram.py	Show probability for reaching other cells after 0, 1, 2, ... moves
		pebble_recurrent_eigen.py	Dual pebble: make aperiodic
		pebble_recurrent_movie.py	Dual pebble: make aperiodic
		pebble_transfer.py	Model Monte Carlo simulation as a transfer matrix, illustrating speed of convergence
		pebble_transfer_eigen.py	Eigenvalues and eigenvectors of transfer matrix, illustrating speed of convergence
		pebble_transfer_sub.py	Model Monte Carlo simulation as a transfer matrix; subtract equilibrium value to show speed of convergence
	Homework	exerciseB.py	Shows that the error of markov_pi follows the law: const / sqrt(N_trials)for large N_trials. The constant is larger (sometimes much larger) than 1.642 and it depends on the stepsize delta.
		exerciseC.py	Bunching method: compute the error in markov_pi.py from a single run and without knowing the mathematical value of pi.
		exerciseC1.py	Bunching method: compute the error in markov_pi.py from a single run and without knowing the mathematical value of pi
		exerciseC3.py	Bunching method: compute the error in markov_pi.py from a single run and without knowing the mathematical value of pi
2			Hard Disks: from classical mechanics to statistical mechancs
	Lecture	direct_disks_box.py	Direct sampling of disks in box, tabula rasa
		direct_disks_box_a1.py	Direct sampling of disks in box, tabula rasa: investigate success rate
		direct_disks_box_movie.py	Direct sampling of disks in box, tabula rasa, plotted as movie
		direct_disks_box_multirun.py	Direct sampling of disks in box, tabula rasa, multiple runs
		direct_disks_box_multirun_b1.py	Generate histogram of x positions by direct sampling of disks in box, tabula rasa
		event_disks_box.py	Algorithm 2.1: event-driven molecular dynamics for hard disks in a box
		event_disks_box_a3.py	Algorithm 2.1: event-driven molecular dynamics for hard disks in a box. Repeat
		event_disks_box_b3.py	Algorithm 2.1: event-driven molecular dynamics for hard disks in a box. Repeat and generate histogram
		event_disks_box_movie.py	Algorithm 2.1: event-driven molecular dynamics for hard disks in a box. Repeat. Plotted as movie
		markov_disks_box.py	Algorithm 2.2: generating a hard disk configuration from another one using a Markov chain
		markov_disks_box_a2.py	Algorithm 2.2: generating a hard disk configuration from another one using a Markov chain
		markov_disks_box_movie.py	Algorithm 2.2: generating a hard disk configuration from another one using a Markov chain
		markov_disks_box_multirun_b2.py	Algorithm 2.2: generating a hard disk configuration from another one using a Markov chain
	Tutorial	direct_discrete.py	Create configuration of rods using tabula rasa to give equiprobable distribution
		direct_disks_any.py	Compute acceptance probability for hard disks as a function of density
		direct_disks_box.py	Algorithm 2.1: event-driven molecular dynamics for hard disks in a box
		direct_disks_box_slow.py	Algorithm 2.1: event-driven molecular dynamics for hard disks in a box. Build full set of disks, including invalid, then cull.
		direct_disks_multirun.py	Algorithm 2.1: event-driven molecular dynamics for hard disks in a box with periodic boundary conditions: multiple runs to collects stats
		direct_disks_multirun_movie.py	Algorithm 2.1: event-driven molecular dynamics for hard disks in a box with periodic boundary conditions: multiple runs to collects stats
		random_sequential_discrete.py	Create configuration of rods without tabula rasa: does not give equiprobable distribution
		random_sequential_discrete_movie.py	Create configuration of rods without tabula rasa: does not give equiprobable distribution
3			Entropic Interactions and Phase Transitions
	Lecture	direct_pins.py	Algorithm 6.1: direct sampling
		direct_pins_improved.py	Algorithm 6.1: direct sampling
		direct_pins_movie.py	Algorithm 6.1: direct sampling
		direct_pins_noreject.py	Algorithm 6.2: rejection free sampling
		direct_pins_noreject_movie.py	Algorithm 6.2: rejection free sampling
	Tutorial	direct_pins.py	Algorithm 6.1: direct sampling
		direct_pins_density.py
		direct_pins_movie.py
		direct_pins_noreject.py
		direct_pins_noreject_movie.py
		direct_pins_noreject_periodic.py
		direct_pins_noreject_periodic_pair.py
	Homework	my_markov_disks.py
		preparation1.py
		preparation2.py
4			Sampling and Integration
	Lecture	direct_sphere_3d.py
		direct_sphere_3d_movie.py
		direct_surface.py
		direct_surface_3d.py
		direct_surface_3d_movie.py
		gauss_2d.py
		gauss_2d_movie.py
		gauss_3d.py
		gauss_3d_movie.py
		gauss_test.py
		gauss_test_movie.py
		naive_gauss.py
		naive_gauss_movie.py
	Tutorial	basic_use_random.py
		gamma_transform.py
		gauss_transform.py
		markov_gauss.py
		markov_gauss_movie.py
		markov_inv_sqrt.py
		markov_inv_sqrt_movie.py
		naive_ran.py
		reject_direct_gauss_cut.py
		reject_inv_sqrt_cut.py
		tower_discrete.py
		walker_test.py
	Homework	data-bunch.py
		markov_hypersphere.py
		markov_hypersphere_C2.py
		markov_sphere_3D.py
		markov_sphere_4D.py
5			Quantum Statistical Mechancs 1/3: Density Matrices and Path Integrals
	Lecture	harmonic_wavefunction.py
		harmonic_wavefunction_check.py
		harmonic_wavefunction_check_movie.py
		harmonic_wavefunction_movie.py
		matrix_square_harmonic.py
		matrix_square_harmonic_movie.py
		naive_harmonic_path.py
		naive_harmonic_path_movie.py
	Tutorial	free_periodic_complex_exp.py
		free_periodic_complex_exp_movie.py
		free_periodic_sine_cosine.py
		free_periodic_sine_cosine_movie.py
		harmonic_trotter_movie.py
		quantum_time_evolution.py
	Homework	a2.py
		markov_gauss_a1.py
		markov_gauss_movie_a1.py
		matrix_square_anharmonic_c1.py
		matrix_square_anharmonic_c3.py
		matrix_square_harmonic_b1.py
		path_Integral_b2.py
		path_Integral_c2.py
6			Quantum Statistical Mechancs 2/3: Lévy Quantum Paths
	Lecture	continuous_random_walk.py	Construct path using a simple random walk (endpoint is not held fixed)
		levy_free_path.py	Simple simplementation of Levy Free Path Algorithm
		levy_harmonic_path.py
		levy_harmonic_path_3D.py	Lévy flight in 3D
		levy_harmonic_path_movie.py
		naive_harmonic_path.py	Construct path using direct sanpling (inefficient)
		naive_path_slice.py
		naive_path_slice_movie.py
		trivial_free_path.py	Generate a random walk, then pull back as described in Lecture 6, 20:12. Compare result with Lévy free path
	Tutorial	naive_boson_trap.py	Enumerate energies in a simple harmonic trap
		naive_single_particle.py	Enumerate states for Bosons, i.e. indistinguisgable particles
	Homework	a1.py
		a2.py
		a3.py
		b1.py
		b2.py
		b3.py
		c1.py
		c2.py
7			Quantum Statistical Mechancs 3/3: Bose-Einstein Condensation
	Lecture	markov_harmonic_bosons.py
		markov_harmonic_bosons_movie.py
		permutation_sample.py
	Tutorial	canonic_harmonic_recursion.py
		canonic_harmonic_recursion_movie.py
		direct_harmonic_bosons.py
		markov_harmonic_bosons.py
		permutation_sample.py
		permutation_sample_cycle.py
	Homework	a1.py
		a2.py
		a2a.py
		a3.py
		b1.py
		b2.py
		c1.py
8			Ising Model: Enumerations and Monte-Carlo Algorithms
	Lecture	cluster_ising.py
		cluster_ising_movie.py
		energy_ising.py
		enumerate_ising.py
		enumerate_ising_mod.py
		enumerate_ising_movie.py
		gray.py
		markov_ising.py
		markov_ising_movie.py
		thermo_ising.py
		thermo_ising_movie.py
	Tutorial	heatbath_ising.py
		heatbath_ising_random_map.py
		heatbath_ising_random_map_movie.py
	Homework	A1.py
		A2.py
		B1.py
		B2.py
		C1.py
		C2.py
9			Dynamic Monte Carlo, Simulated Annealing
	Lecture	dynamic_ising.py
		dynamic_ising_patch.py
		fast_spin_dynamics.py
		fast_throw.py
		naive_spin_dynamics.py
		naive_spin_dynamics_movie.py
		naive_throw.py
	Tutorial	direct_sphere_disks.py
		direct_sphere_disks_any.py
		direct_sphere_disks_any_movie.py
		direct_sphere_disks_movie.py
		example_pylab_visualization.py
		markov_sphere_disks.py
		resize_disks.py
		simulated_annealing.py
		simulated_annealing_movie.py
	Homework	A1.py
		A2.py
		B1.py
		C1.py
		C2.py
10			The Alpha and Omega of Monte Carlo
	Lecture	direct_gamma.py	Integral of x**gamma, illustrating need for importance sampling
		direct_gamma_average.py	Integral of x**gamma, illustrating need for importance sampling
		direct_gamma_average_movie.py	Histogram of Integral of x**gamma, illustrating need for importance sampling
		direct_gamma_average_rescaled.py
		direct_gamma_average_rescaled_movie.py	Integral of x**gamma: rescale, plot histograms, and compare with Lévy distribution
		direct_gamma_running.py
		direct_gamma_running_movie.py
		direct_needle.py	Buffon's experiment (with cheat)
		direct_needle_patch.py	Buffon's experiment

From the Book

#		File	Problem/Algorithm/Description
-		template.py	Template for python programs
1		smac.py	Useful functions: BoxMuller, CircleThrowing, and SphereGenerator
1.1		direct-plot.py	Implement Algorithm 1.1. Plot error and investigate relationship with N.
-		markov-pi.py	Implement Algorithm 1.2. Plot error and rejection rate.
-		direct.py	1.3 Store state in file
-		smacfiletoken.py
-		markov-discrete-pebble.py	1.4 Use table
-		large-markov.py
-		transfer.m	1.8 Eigenvalues of transfer matrix
1.2		permutation.py	1.9 Sample permutations using Alg. 1.11 and verify that it generate all 120 permutations of 5 elements equally often
-		permutation-histogram.py
-		naivegauss.py	1.12 Gauss
-		direct-surface.py	Monte Carlo simulation of Exercise 2.11 of Chaosbook: in higher dimensions, any two vectors are nearly orthogonal
1.3		binomialconvolution.py	1.18 Binomial Convolution
1.4.2		direct-gamma.py	1.22 Importance sampling:Implement Algorithm 1.29, subtract mean value for each sample, and generate histograms of the average of N samples and the rescaled averages.
-		direct-gamma-zeta.py
-		markov-zeta.py	Algorithm 1.31, use of Markov Chain to detect non-integrable singularity.
1.4.4		levy-convolution.py	Algorithm 1.32
2.1.4		spheres2.py	Exercise 2-4: Sinai's system of two large sphere in a box. Show histogram of positions.

2.2.1||pair-time.py|Algorithm 2.2: Pair collision time for two particles -||md.py|Algorithm 2.3 Pair collision -||md-viz.py|Visualize data generated by md.py -||md-plot.py|Visualize output from md.cpp. Plot distribution of distances from wall, and compare energy histogram with Bolzmann distribution. -|md||Algorithm 2.3 Pair collision -|md|Makefile|Algorithm 2.3 Pair collision -|md|configuration.cpp|Manage a box full of particles -|md|configuration.hpp|Manage a box full of particles -|md|md.cpp|Algorithm 2.3 Pair collision -|md|md.hpp|Algorithm 2.3 Pair collision -|md|particle.cpp|Represents one single particle -|md|particle.hpp|Represents one single particle -||direct-disks.py|Exercise 2.6: directly sample the positions of 4 disks in a square box without periodic boundary conditions, for different covering densities -||hist-plot.py|Exercise 2.6: ... -||geometry.py|Used to implement periodic and aperiodic boundary conditions in Exercises 2.6-2.8 2.2.2||directDisksAny.py| 2.2.3|| 3.1||harmonic_wavefunction.py|Exercise 3.1: verify orthonormality of the solutions to Schroedinger's equation for Simple Harmonic Oscillator -||harmonic_density.py|Exercise 3.2: determine density matrix 3.2||matrix-square.py|Exercise 3.4: implement Alg 3.3, matrix-square -||matrix-square-check.py|Exercise 3.4: check results of matrix squaring against exact solution. -||poeschl-teller.py|Exercise 3.5: plot Poeschl-Teller potential and investigate density matrix and partition function 3.3|||The Feynman path integral 3.4|||Pair density matrices 3.5|||Geometry of Paths 4||| Bosons 5.1||gray.py|Algorithm 5.2: Gray code for spins {1,...N}. -||enumerate-ising.py|Algorithm 5.3: single flip enumeration for the Ising model. -||ising.py|Algorithm 5.3: Single spin-slip enumeration for Ising model -||ising-stats.py|Figure 6.6 - plot data from ising.py -|ising|catch.hpp|Support for Test harness for enumerate-ising.cpp -|ising|enumerate-ising.cpp|Algorithm 5.3: single flip enumeration for the Ising model. -|ising|enumerate-ising.hpp|Algorithm 5.3: single flip enumeration for the Ising model. -|ising|gray.hpp|Greycode for enumerate-ising.cpp, plus iterator for neighbours -|ising|Makefile|Makefile for enumerate-ising.cpp -|ising|nbr.hpp|Calculate neighbours for enumerate-ising.cpp -|ising|test-gray.cpp|Tests for gray.hpp -|ising|test-nbr.cpp|Tests for nbr.cpp -|ising|tests.cpp|Test harness for enumerate-ising.cpp 5.2|||The Ising model - Monte-Carlo algorithm 5.3|||Generalized Ising models 6|||Entropic Forces 7|||Dynamic Monte Carlo Methods

Exam

File	Description
Q1.py
Q4-1.py
Q4-2.py
Q7.py
Q10.py
Q11.py

Miscellaneous

File	Description
walker.py	Used to create README.MD from file tree