Biomedical Engineering Reference
In-Depth Information
If we calculate the mutual potential energy Φ
i
(
q
) for a random array of particles, then the
array will have a Boltzmann weight of
exp
Φ
i
(
q
)
k
B
T
−
and if some particles are close together Φ
i
will be large and the Boltzmann factor small.Most
of phase space corresponds to nonphysical configurations with very high energies and only
for a very small proportion of phase space does the Boltzmann factor have an appreciable
value. This means that most of the computing time will be spent on configurations that give
negligible contributions to properties of interest.
Metropolis
et al.
(1953) therefore proposed a modified MC scheme where, instead of
choosing configurations randomly then weighting themwith exp(
−
Φ/
k
B
T
), configurations
were chosen with a probability distribution of exp(
−
Φ/
k
B
T
) and then weighted equally.
This adds a branch to the algorithm as follows.
1. Before making a move, we calculate the energy change of the system caused by the
move. If the energy change is negative, the move is allowed.
2. If the energy change is positive, we allow the move but with a probability of
exp(
−
Φ/
k
B
T
); to do this, we generate a random number ξ
3
between 0 and 1, and if
ξ
3
< exp
Φ
k
B
T
−
we allow the move to take place.
3. Otherwise we leave the particle in its old position and move on to the next particle.
4. Whether the move is allowed or not, we consider that we have generated a new
configuration for the purposes of taking averages.
This summarizes the
Metropolis Monte Carlo
(MMC) technique. The crucial feature of
MMC is that it biases the generation of configurations towards those that make significant
contributions to the configuration integral. It generates configurations with a probability
exp
Φ (
q
)
k
B
T
−
and then counts each of them equally. By contrast the simple MC integration method
generates states with equal probability and then assigns them this weight.
9.4 Periodic Box
Figure 9.4 shows a box of argon atoms, and examination reveals two problems.
1. Atoms near the edges of the box will experience quite different resultant forces from
those atoms near the centre of the box.
2. Second, a MC move may well place one of the atoms outside the box and this will alter
the density of the system under study.
The periodic box concept, illustrated in Figure 9.5, gives a solution to these problems.
We appeal to the ensemble concept of statistical thermodynamics, and surround our system
