Introduction to Monte Carlo Simulation Techniques

Abstract

Starting from the assumption that matter, for our purposes, consists of interacting particles obeying classical mechanics, and using the postulates of statistical mechanics, one can model any specific material as a system of particles provided one knows what the interactions between the particles are. However, whatever interactions are chosen the integrals that it is necessary to solve are formidable. For example, the average potential energy is

$$ <U>=\int{U({{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}}^{N}})}p({{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}^{N}})d{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}^{N}}, $$

(1)

where the probability density for a configuration of N distinguishable particles, ${{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}^{N}}\equiv({{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}_{1}},{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}_{2}},....,{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}_{N}})$, r₂,...., r_N) is

$$ {{Q}_{N}}=\int{exp\left[ -U({{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}}^{N}})/kT\right]d}{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}^{N}}. $$

(2)

where the configurational integral Q_N is

$$ {{Q}_{N}}=\int{exp\left[ -U({{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}}^{N}})/kT\right]}d{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}^{N}}. $$

(3)