6.2 Monte Carlo Method

In a canonical average such as

$$\langle a \rangle_{NVT} = \frac{1}{Q}\, \int a(\mbox{$\bf 1$... ...f 1$}..\mbox{$\bf N$})/kT\}\, d\mbox{$\bf 1$}..d\mbox{$\bf N$}$$ (6.3)

the denominator $Q \equiv \int \exp\{-E(\mbox{$\bf 1$}..\mbox{$\bf N$})/kT\}\, d\mbox{$\bf 1$}..d\mbox{$\bf N$}$ 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 $N$-particle configuration, $\mbox{$\bf\Gamma$}_{c} \equiv \left\{\mbox{$\bf 1$}..\mbox{$\bf N$} \right\}$, we generate a Markov chain of, say, $K$ configurations $\{ \mbox{$\bf\Gamma$}_{c}(k),\,k=1,\dots K\}$ such that the relative frequency of a configuration in the chain is proportional to the corresponding Boltzmann factor. An estimate for the mean value $\langle a \rangle$ is then

$$\langle a \rangle = \frac{1}{K} \sum_{k=1}^{K}a \left[ \mbox{$\bf\Gamma$}_{c}(k) \right]$$

Here is the procedure due to N. METROPOLIS:

Figure 6.2: 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 $E(k)$ and $E'$ are then restricted to $0$ or $\infty$, with Boltzmann factors $1$ or $0$, respectively. Here is the modified part of the MC procedure:

Figure 6.3: Monte Carlo for hard spheres

Yet another modification is needed for spin systems:

Figure 6.4: 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. 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 $T^{*}$; 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 $N=n.n$ spins $\sigma_{i}= \pm 1; \; i=1, \dots N$ be situated on the vertices of a two-dimensional square lattice. The interaction energy is defined by

$$E = \sum_{i} E_{i} = -\frac{1}{2} \sum_{i=1}^{N} \sum_{j=1}^{4} \sigma_{i} \sigma_{j}$$

where the sum over $j\/$ involves the 4 nearest neighbors of spin $i$. Periodic boundary conditions are assumed

• Write a Monte Carlo program to perform a biased random walk through configuration space.
• Determine the mean total moment $\langle M \rangle \equiv \langle \sum_{i} \sigma_{i} \rangle$ and its variance as a function of the quantity $1/kT$. Compare your results to literature data (e.g. BINDER, K.: Applications of the Monte Carlo Method in Statistical Physics. Springer, Berlin 1987).