# 3. Monte Carlo Method: Standard (NVT)

In a canonical average such as
 (3.1)

the denominator is usually unknown.

In the section on Biased Random Walks'' we learned that this is no hindrance for the calculation of averages:

Writing, for a certain -particle configuration, , we generate a Markov chain of, say, configurations such that the relative frequency of a configuration in the chain is proportional to the corresponding Boltzmann factor. An estimate for the mean value is then

Here is the procedure due to N. METROPOLIS:

In the case of hard disks or spheres the 3rd step in the above recipe must be modified. Values of and are then restricted to or , with Boltzmann factors or , respectively. Here is the modified part of the MC procedure:

Yet another modification is needed for spin systems:

PROJECT MC (FLUID): Write a subroutine MCSTEP which performs the basic Monte Carlo step as described in Fig. 3.1: selecting at random one of the LJ particles that were placed on a lattice by STARTCONFIG, displace it slightly and apply the PBC; then compute the new potential energy (using NIC!) and check whether the modified configuration is accepted or not, given a specific temperature ; if accepted, the next configuration is the modified one, otherwise the old configuration is retained for another step.

Write a main routine to combine the subroutines STARTCONF, ENERGY, and MCSTEP into a working MC program.

PROJECT MC (LATTICE): Let spins be situated on the vertices of a two-dimensional square lattice. The interaction energy is defined by

where the sum over involves the 4 nearest neighbors of spin . Periodic boundary conditions are assumed
• Write a Monte Carlo program to perform a biased random walk through configuration space.
• Determine the mean total moment and its variance as a function of the quantity . Compare your results to literature data (e.g. BINDER, K.: Applications of the Monte Carlo Method in Statistical Physics. Springer, Berlin 1987).

F. J. Vesely / University of Vienna