3. Monte Carlo Method: Standard (NVT)

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

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:

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

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