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6.2 Monte Carlo Method
In a canonical average such as
|
(6.3) |
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:
Figure 6.2:
Statistical-mechanical Monte Carlo for a model fluid
with continuous potential
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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:
Figure 6.3:
Monte Carlo for hard spheres
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Yet another modification is needed for spin systems:
Figure 6.4:
Monte Carlo simulation on an Ising lattice
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PROJECT MC (FLUID):
Write a subroutine MCSTEP which performs the basic Monte Carlo step
as described in Fig. 6.2: 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).
Next: 6.3 Molecular Dynamics Simulation
Up: 6. Simulation and Statistical
Previous: 6.1.2 Technical Matters
Franz J. Vesely Oct 2005
See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001