Biomedical Engineering Reference
In-Depth Information
The crucial point is that there is no momentum contribution to the partition function
in MC. The momentum contribution is essentially what we would expect for a system of
noninteracting particles.
9.2 The First 'Chemical' Monte Carlo Simulation
The first serious Monte Carlo calculation appears to be that of Metropolis
et al.
(1953)
(although the method was proposed independently by J.E. Meyer and by S. Ulam). In
fact, the Metropolis
et al.
paper marked the birth of computer simulation as a statistical
mechanical technique, and it preceded the molecular dynamics studies to be discussed in
Chapter 10. I cannot do better than repeat the words of the authors (i.e. quote their abstract):
A general method, suitable for fast computing machines, for investigating such properties as
equations of state for substances consisting of interacting individual molecules is described.
The method consists of a modified Monte Carlo integration over configuration space. Results
for the two-dimensional rigid-sphere system have been obtained on the LosAlamos MANIAC,
and are presented here. These results are compared to the free volume equation of state and to
a four-term virial coefficient expansion.
The rigid-sphere potential shown in Figure 9.3 is the simplest one imaginable; the spheres
cannot overlap, but their mutual potential energy is zero everywhere else.
0
Distance, R
Figure 9.3
Rigid sphere potential
The authors took a two-dimensional square array of 224 hard spheres (i.e. hard
discs
)
each with fixed radius σ . Here is the algorithm.
1. A starting configuration is chosen. Metropolis placed the 224 discs in a triangular lattice
of 14 particles per rowby 16 particles per column, alternate rows being displaced by 0.5σ .
2. Each disc is moved in succession in the
XY
plane according to the formula
X
→
X
+
αξ
1
Y
→
Y
+
αξ
2
