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Advanced Topics in Equilibrium Statistical
Mechanics
Glenn Fredrickson
5. Computer Simulation Methods
A. General Considerations
We have just seen that successful analytical theories for simple liquids like Ar
gon can be constructed by means of modern perturbation theories of liquids.
However, these calculations become more diﬃcult and tedious when one moves
to consider more complicated atomic and molecular ﬂuids. Moreover, comput
ers have gotten much faster since the 1970’s, so people today are more likely to
turn to
computer simulations of ﬂuids.
a1. Choice of Method
There are basically three methods used for liquids and complex ﬂuids:
Molecular dynamics
(MD) – One integrates Newton’s equations for a collec
tion of particles, keeping track of their positions and velocities at each time
step. Thermo properties are computed as time averages over dynamical
trajectories.
Brownian dynamics
(BD) – Similar to MD, except that stochastic force and
frictional drag terms are included. Usually used for colloids or polymers
when solvent can be treated as a continuum.
Monte Carlo
(MC) – A method for sampling the conFgurational integral by
means of a “pseudo” dynamical trajectory.
1
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Technique
Comments
atomic or molecular
MC (statics)
MC faster
liquid
MD (dynamics)
for statics
polymer solution
MC (statics)
coarsegrained
or melt
BD/MD (dynamics)
models often used
colloids
MC (statics)
solvent is
BD (dynamics)
continuum.
a2. Periodic BCs and Potential Truncation
Unfortunately, we can only aFord to run fairly small simulations with
N
<
∼
10
3
–
10
6
for atoms, much less for polymers! DiFerent
ensembles
can be used; in ±xed
N,V
we choose
to establish some
ρ
.Itisa
lsobesttochoose
N
consistent
with any known crystal structure, e.g., fcc argon:
N
=4
n
3
,
n
=1
,
2
,
3
,...
To minimize “±nite size” eFects in the simulation cell (mostly surface eForts),
it is typical to impose periodic BCs
:
When it comes to computing
forces
, one also uses the
“minimum image
convention”
:
1. translate a cell of the same size (L) and shape to be centered around the
molecule of interest.
2. Sum the forces with the
N
−
1 molecules in the translated cell. These are
the closest
periodic images of these molecules.
²requently, one also
truncates
the potential at some cutoF distance
r
c
(e.g.,
2
.
5
σ
). It is crucial that
r
c
≤
L/
2
for consistency with the minimum image convention as shown in the attached
±gure:
2
B. The MD Method
The basic idea here is to solve Newton’s equations of motion:
m
¨
r
i
=
F
i
i
=1
,
2
,...,N
forward in time, starting from some set of initial positions
{
r
i
}
and velocities
{
˙
r
i
}
. Many Fnite di±erence algorithms are available; the simplest and most
popular is the Verlet algorithm
. Start with:
r
i
(
t
±
δt
)=
r
i
(
t
)
±
˙
r
i
(
t
)+
2
2!
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This document was uploaded on 05/18/2010.
 Spring '09

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