Biomedical Engineering Reference
In-Depth Information
where α is some (arbitrary) maximum allowed distance and ξ
i
are random numbers
between
1. This means that after we move a particle, it is equally likely to be
anywhere within a square of side 2σ centred about its original position.
3. At the end of the moves, the discs are examined for overlap. In the case that the move
did indeed put one disc on top of another disc, the particles are returned to their original
positions. This is a primitive form of the technique known as
importance sampling
that
is now a key feature of modern MC simulations.
4. We then calculate any useful data for averaging. In this model, Φ
−
1 and
+
0 for all allowed
conformations, but we might be interested for example in the disc-disc separations in
order to investigate the radial distribution functions to be discussed in later chapters.
5. After a large number of allowed configurations have been examined (and the calculation
has lost all knowledge of its starting configuration), we stop and do the final averaging.
The choice of α is actually important here. If α is large then discs invariably overlap
and if α is small then the conformations do not change much and in either case we have
to perform a very large number of iterations to get a meaningful average. It is normally
chosen so that 50% of the trial moves are accepted.
6. Finally the acid test: compare with experiment. It is rather difficult to think of a real-life
system that could be realistically represented by the rigid-sphere model but the authors
did their best; they used the virial theorem of Clausius to derive an equation of state for
the rigid discs
=
Nk
B
T
1
πσ
2
n
pA
=
+
2
where <
n
> is the average density of other particles at the surface of a particle and
A
the area. They then calculated <
n
> from the MC simulation and compared with the
equation of state.
9.3
Importance Sampling
In the case of a more complicated problem (even a modest Lennard-Jones potential), a
more sophisticated treatment of importance sampling is needed for the following reason.
The average potential energy can indeed be obtained from the configuration integral
Φ (
q
) exp
d
q
Φ (
q
)
k
B
T
−
=
exp
d
q
Φ
(9.3)
Φ (
q
)
k
B
T
−
and each integral can indeed be approximated by a finite sum of
M
terms in MC
i
=
1
Φ
i
(
q
) exp
Φ
i
(
q
)
k
B
T
M
−
Φ
≈
i
=
1
exp
(9.4)
Φ
i
(
q
)
k
B
T
M
−
