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Subsections
As we pointed out in the beginning of this chapter, the main purpose
of the computer simulations is to calculate thermodynamic properties.
They are calculated as averages of the corresponding functions that
depend on the particle coordinates. The averaging is done over the
configurations, produced during the MC run.
For many quantities, like magnetization or energy, the calculations
are obvious. Pressure is a little more tricky, and now we are going to
derive the expression for it.
As we remember from the thermodynamics:

where

For simplicity, we assume, that the simulation cell has cubic
shape
, and make
the coordinate change

Then the coordinate
of the particle i is
, and
|  |
(1) |
with
|  |
(2) |
Now after going back to the initial coordinates and substituting
, we see,
that the first term is just density, and the second is an ensemble
average of the sum of the force times coordinate.

The second term on the right hand side is called virial,
and this equation is often called the virial equation.
If the interaction potential is pairwise, it can be rearranged as
follows:

is the force, acting on the atom i from the atom j.
(Why?).
|  |
(3) |
Finally, we obtain
|  |
(4) |
Equation 4 should look familiar. Before, when we discussed
the theory of
liquids
, we derived the same
expression for pressure but in terms of the pairwise interaction
potential and g(r). Here we derived more general expression,
applicable even in the case when the interactions are not necessarily
pairwise. It also does not require calculations of g(r), that can
be very involved.
If we have a system of N particles at temperature T, and volume
V, the chemical potential
at large enough N can be written
as:

where QN is partition function for the system of N particles
in the volume N. The
-particle partition function is:
|  |
(5) |
Here we split the total system energy into the interaction UN+1
of the N+1-st particle with the other N particles, and the energy
of the N-particle system. The integral in the last
equation can be rewritten as the ensemble average of
over the ensemble of systems with N particles:

This is the basis for the so-called
- Widom test-particle method.
-
- Perform NVT MC simulation for the system of N particles
- Calculate the average of the
, by
inserting the ``ghost'' particle at random positions inside the box,
and calculating its interaction energy with the rest of the system,
as if it were real.
This technique works fine at moderate densities around the critical
point, but turns out to have problems at high densities around
liquid-solid coexistence, because the probability to insert the
``ghost'' at the place where the
is
very small.
The importance of the pair distribution function g(r) was discussed
in the chapter on the liquid-state
theory
. Its calculation in
the MC simulations is really straightforward. All we have to do is
just to count the number of particles at the distances between r and
r+dr from the chosen particle. In the homogeneous systems it is also
averaged over all particles. It is calculated as a histogram with the
bin width dr, and then is normalized by the area of the spherical
layer.
- Code example
- (to be added to the MC code). Notice, that it is
2D! Code.
In the previous lecture we mentioned, that it is very convenient to
cut the interaction potential after certain cut-off distance
rc. The thermodynamics properties has to be corrected for this
approximation. The expressions are listed below:
|  |
(6) |
Next: NPT Ensemble
Up: Calculation of Thermodynamic Properties.
Previous: Calculation of Thermodynamic Properties.
© 1997
Boris Veytsman
and Michael Kotelyanskii
Thu Nov 13 19:07:05 EST 1997