Computer Modeling in Physics

This text is based on the one-term lecture course (about twenty lectures) which has been given in 1988-1992 at Kiev University for forth-year students (that corresponds to last-year undergraduate or first-year graduate students) of the Radiophysical Department. The aim of the course was to explain in what a way a computer can be used for solution of physical problems. It is natural, therefore, that special attention in the course is devoted to the main methods of computer simulation, the Molecular Dynamics (MD) method and the Monte Carlo (MC) method, and also to the Stochastic Equations (SE) method which may be considered as an intermediate one between the MD and MC methods. However, besides the answer on the question in what a way, the principal aim of this course is an attempt to answer two additional important questions: when and why we should use computer modeling.

 

The draft of the course may be downloaded here:

Contents:

Preface
1    Introduction
1.1  Stages of modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  7
1.2  Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3  Choice of a computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  9
1.4  Some practical advices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  10
2    Stochastic Theory
2.1  The Henon-Heiles model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2  Driven pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3  Logistic map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4  The Feigenbaum theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5  Strange attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6  Kolmogorov-Sinai entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27
2.7  Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29
3    Molecular Dynamics Method
3.1  General statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35
3.2  Melting of two-dimensional crystals . . . . . . . . . . . . . . . . . . . . . . . . . 39
  3.2.1  Introductional remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
  3.2.2  Physical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 
  3.2.3  Numerical algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
  3.2.4  Results of simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3  Surface diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47
  3.3.1  Introductional remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47
  3.3.2  Diffusion of Na atom adsorbed on the surface of Na crystal . . . . . 47
3.4  Epitaxial crystal growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50
3.5  PP, PM and P3M methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52
3.6  Canonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4    Stochastic Equations
4.1  Langevin equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57
4.2  Numerical solution of the Langevin equation . . . . . . . . . . . . . . . . . . . .60
4.3  Generalized Langevin equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60
4.4  Fokker-Planck-Kramers equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.5  Kramers theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
  4.5.1  The problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
  4.5.2  Numerical solution of the FPK equation . . . . . . . . . . . . . . . . . . . . . 71
5    Monte Carlo Method
5.1  Generation of random numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
  5.1.1  Generation of the standard random numbers . . . . . . . . . . . . . . . . . . . 73
  5.1.2  Generation of random numbers with a given distribution . . . . . . . . . . .76
5.2  Some applications of the MC method . . . . . . . . . . . . . . . . . . . . . . . . . .79
  5.2.1  Percolation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79
  5.2.2  Penetration of neutrons through an absorbing plate . . . . . . . . . . . . . . .81
  5.2.3  Calculation of definite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .83
5.3  The MC method in statistical physics . . . . . . . . . . . . . . . . . . . . . . . . . . .85
  5.3.1  Idea of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .85
  5.3.2  Calculation scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .87
5.4  Modeling of phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89
  5.4.1  Continuum models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
  5.4.2  Lattice models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.5  Relaxation dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..95
5.6  Accelerating relaxation to equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . .100
5.7  Comparison of the MD and MK methods . . . . . . . . . . . . . . . . . . . . . . ..101
6    Integrable Systems
6.1  Fermi-Pasta-Ulam paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2  Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105
  6.2.1  Boussinesq's equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105
  6.2.2  Competition of dispersion and nonlinearity . . . . . . . . . . . . . . . . . . . . .106
  6.2.3  Waves of stationary shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109
  6.2.4  Atomic chain in an external periodic potential . . . . . . . . . . . . . . . . . . .111
  6.2.5  Solitonic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.3  Exactly integrable systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116
  6.3.1  Korteweg-de Vries equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
  6.3.2  Nonlinear SchrÄodinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .117
  6.3.3  Sine-Gordon equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
  6.3.4  Toda chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.4  Systems close to integrable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .122
  6.4.1  General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
  6.4.2  The Frenkel-Kontorova model and its generalizations . . . . . . . . . . . . .123
References

I will be grateful to everybody who could help to improve this course in any aspect - content, examples, English, figures, references, exersizes, details/errors, etc.

goto main Back to main page


Last updated on March 15, 2011 by O.Braun.  Copyright © by O.Braun