Basics of Monte Carlo Simulations

Monte Carlo simulations are, as indicated by the name, based on the idea of evolving a system by drawing random numbers. Unfortunately, statistically mean-ingftil configurations are typically confined to a small volume of phase space. To evolve a system within this volume we apply importance sampling, that is, we only sample states that actually contribute to statistical averages. 

To derive the relevant equations we consider our system to be in a particular state i. This state is in equilibrium with its environment if the probability flows in and out of this state are equal  

The system is in equilibrium if Eq. (1.11) applies to all states. Equation (1.11) is always fulfilled if the stricter condition  

For example, consider a local Monte Carlo scheme for a Lennard-Jones liquid in the NVT (constant particle number, volume, and temperature) ensemble We choose one particle at random, and move it a fixed distance away from the previous position in an arbitrary direction. Eor the reverse move (from j to ij 0j = at the two prefactors cancel out. Pi oc exp[— 3 (i)] according to the canonical Boltzmann distribution. Equation (1.13) indicates that the move is always accepted if the energy of the system is lowered by the displacement. If the energy increases, the move is accepted with probability exp (— 3AE), that is, we draw a random number between 0 and 1 and accept the move if the random number is smaller than exp (—PA ). 

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