Classical Monte Carlo Simulation

Abstract

In Monte Carlo simulation, our primary goal is to calculate an observable physical quantity X of a thermodynamic system in thermal equilibrium with a heat bath at temperature T. A macroscopic thermodynamic system consists of a large number of atoms or molecules, of the order of Avogadro number N _A ≈ 6.022 × 10²³ per mole. Moreover, in most general cases, the particles have complex interaction among themselves. The average property of a physical quantity of such a system is then determined not only by the large number of particles but also by the complex interaction among the particles. As per statistical mechanics, if the system is in thermal equilibrium with a heat bath at temperature T, the average of an observable quantity 〈X〉 can be calculated by evaluating the canonical partition function Z of the system and is given by $\langle X\rangle = \frac{1}{Z}\mathop \sum \limits_s {X_s}exp( - {E_s}/{k_B}T)$ where k _B is the Boltzmann constant. However, the difficulty in evaluating the exact partition function Z is two fold. First, there are large number of particles present in the system with many degrees of freedom. The calculation of the partition function Z usually leads to evaluation of an infinite series or an integral in higher dimension, a (6N) dimensional space. Second, there exists a complex interaction among the particles which gives rise to unexpected features in the macroscopic behaviour of the system. Monte Carlo (MC) simulation method can be employed in evaluating such averages. In a MC simulation, a reasonable number of states are generated randomly (instead of infinitely large number of all possible states) with their right Boltzmann weight and an average of a physical property is taken over those states only.