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,
the relative frequency of a configuration in the chain is proportional
to the corresponding Boltzmann factor.
An estimate for the mean value
Here is the procedure due to N. METROPOLIS:
Statistical-mechanical Monte Carlo for a model fluid
with continuous potential
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:
Monte Carlo for hard spheres
Yet another modification is needed for spin systems:
Monte Carlo simulation on an Ising lattice
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):
be situated on the
vertices of a two-dimensional square lattice. The interaction energy is
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
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).