Statistical Mechanics Algorithms and Computations

by Krauth, Werner

• ISBN13: 9780198515364
• ISBN10: 0198515367
• Edition: 1st
• Format: Paperback
• Copyright: 2006-11-16
• Publisher: Oxford University Press

Summary

This book discusses the computational approach in modern statistical physics in a clear and accessible way and demonstrates its close relation to other approaches in theoretical physics. Individual chapters focus on subjects as diverse as the hard sphere liquid, classical spin models, single quantum particles and Bose-Einstein condensation. Contained within the chapters are in-depth discussions of algorithms, ranging from basic enumeration methods to modern Monte Carlo techniques. The emphasis is on orientation, with discussion of implementation details kept to a minimum. Illustrations, tables and concise printed algorithms convey key information, making the material very accessible. The book is completely self-contained and graphs and tables can readily be reproduced, requiring minimal computer code. Most sections begin at an elementary level and lead on to the rich and difficult problems of contemporary computational and statistical physics. The book will be of interest to a wide range of students, teachers and researchers in physics and the neighbouring sciences. An accompanying CD allows incorporation of the book's content (illustrations, tables, schematic programs) into the reader's own presentations.

Author Biography

Werner Krauth is Research Director at CNRS, Laboratoire de Physique de l'Ecole Normale Superieure, Paris, France.

Table of Contents

Monte Carlo methods	1	(80)
Popular games in Monaco	3	(24)
Direct sampling	3	(1)
Markov-chain sampling	4	(5)
Historical origins	9	(6)
Detailed balance	15	(6)
The Metropolis algorithm	21	(1)
A priori probabilities, triangle algorithm	22	(2)
Perfect sampling with Markov chains	24	(3)
Basic sampling	27	(17)
Real random numbers	27	(2)
Random integers, permutations, and combinations	29	(4)
Finite distributions	33	(2)
Continuous distributions and sample transformation	35	(2)
Gaussians	37	(2)
Random points in/on a sphere	39	(5)
Statistical data analysis	44	(18)
Sum of random variables, convolution	44	(4)
Mean value and variance	48	(4)
The central limit theorem	52	(3)
Data analysis for independent variables	55	(4)
Error estimates for Markov chains	59	(3)
Computing	62	(19)
Ergodicity	62	(1)
Importance sampling	63	(5)
Monte Carlo quality control	68	(2)
Stable distributions	70	(6)
Minimum number of samples	76	(1)
Exercises	77	(2)
References	79	(2)
Hard disks and spheres	81	(50)
Newtonian deterministic mechanics	83	(9)
Pair collisions and wall collisions	83	(3)
Chaotic dynamics	86	(1)
Observables	87	(3)
Periodic boundary conditions	90	(2)
Boltzmann's statistical mechanics	92	(16)
Direct disk sampling	95	(2)
Partition function for hard disks	97	(3)
Markov-chain hard-sphere algorithm	100	(3)
Velocities: the Maxwell distribution	103	(2)
Hydrodynamics: long-time tails	105	(3)
Pressure and the Boltzmann distribution	108	(11)
Bath-and-plate system	109	(2)
Piston-and-plate system	111	(2)
Ideal gas at constant pressure	113	(2)
Constant-pressure simulation of hard spheres	115	(4)
Large hard-sphere systems	119	(3)
Grid/cell schemes	119	(1)
Liquid--solid transitions	120	(2)
Cluster algorithms	122	(9)
Avalanches and independent sets	123	(2)
Hard-sphere cluster algorithm	125	(3)
Exercises	128	(2)
References	130	(1)
Density matrices and path integrals	131	(54)
Density matrices	133	(10)
The quantum harmonic oscillator	133	(2)
Free density matrix	135	(2)
Density matrices for a box	137	(2)
Density matrix in a rotating box	139	(4)
Matrix squaring	143	(6)
High-temperature limit, convolution	143	(2)
Harmonic oscillator (exact solution)	145	(3)
Infinitesimal matrix products	148	(1)
The Feynman path integral	149	(10)
Naive path sampling	150	(2)
Direct path sampling and the Levy construction	152	(3)
Periodic boundary conditions, paths in a box	155	(4)
Pair density matrices	159	(9)
Two quantum hard spheres	160	(2)
Perfect pair action	162	(5)
Many-particle density matrix	167	(1)
Geometry of paths	168	(17)
Paths in Fourier space	169	(5)
Path maxima, correlation functions	174	(3)
Classical random paths	177	(5)
Exercises	182	(2)
References	184	(1)
Bosons	185	(44)
Ideal bosons (energy levels)	187	(22)
Single-particle density of states	187	(3)
Trapped bosons (canonical ensemble)	190	(6)
Trapped bosons (grand canonical ensemble)	196	(4)
Large-N limit in the grand canonical ensemble	200	(5)
Differences between ensembles---fluctuations	205	(1)
Homogeneous Bose gas	206	(3)
The ideal Bose gas (density matrices)	209	(20)
Bosonic density matrix	209	(3)
Recursive counting of permutations	212	(1)
Canonical partition function of ideal bosons	213	(4)
Cycle-length distribution, condensate fraction	217	(2)
Direct-sampling algorithm for ideal bosons	219	(2)
Homogeneous Bose gas, winding numbers	221	(3)
Interacting bosons	224	(1)
Exercises	225	(2)
References	227	(2)
Order and disorder in spin systems	229	(38)
The Ising model---exact computations	231	(18)
Listing spin configurations	232	(2)
Thermodynamics, specific heat capacity, and magnetization	234	(2)
Listing loop configurations	236	(4)
Counting (not listing) loops in two dimensions	240	(7)
Density of states from thermodynamics	247	(2)
The Ising model---Monte Carlo algorithms	249	(10)
Local sampling methods	249	(3)
Heat bath and perfect sampling	252	(2)
Cluster algorithms	254	(5)
Generalized Ising models	259	(8)
The two-dimensional spin glass	259	(3)
Liquids as Ising-spin-glass models	262	(2)
Exercises	264	(2)
References	266	(1)
Entropic forces	267	(40)
Entropic continuum models and mixtures	269	(12)
Random clothes-pins	269	(4)
The Asakura--Oosawa depletion interaction	273	(4)
Binary mixtures	277	(4)
Entropic lattice model: dimers	281	(26)
Basic enumeration	281	(3)
Breadth-first and depth-first enumeration	284	(4)
Pfaffian dimer enumerations	288	(8)
Monte Carlo algorithms for the monomer--dimer problem	296	(3)
Monomer-dimer partition function	299	(4)
Exercises	303	(2)
References	305	(2)
Dynamic Monte Carlo methods	307	(30)
Random sequential deposition	309	(4)
Faster-than-the-clock algorithms	310	(3)
Dynamic spin algorithms	313	(8)
Spin-flips and dice throws	314	(3)
Accelerated algorithms for discrete systems	317	(2)
Futility	319	(2)
Disks on the unit sphere	321	(16)
Simulated annealing	324	(3)
Asymptotic densities and paper-cutting	327	(3)
Polydisperse disks and the glass transition	330	(1)
Jamming and planar graphs	331	(2)
Exercises	333	(2)
References	335	(2)
Acknowledgements	337	(2)
Index	339