8 Molecular dynamics simulations
8.1 Introduction
In the previous chapter we saw that the experimental values of physical quantities
of a many-particle system can be found as an ensemble average. Experimental
systems are so large that it is impossible to determine this ensemble average by
summing over all the accessible states in a computer. There exist essentially two
methods for determining these physical quantities as statistical averages over a
restricted set of states: the molecular dynamics and Monte Carlo methods. Imagine
that we have a random sample of, say, 10^7 configurations of the system which are
all compatible with the values of the system parameters. For such a large number
we expect averages of physical quantities over the sample to be rather close to the
ensemble average. It is unfortunately impossible to generate such a random sample;
however, we can generate a sample consisting of a large number of configurations
which are determined successively from each other and are hence correlated. This
is done in the molecular dynamics and Monte Carlo methods. The latter will be
described in Chapter 10.
Molecular dynamics is a widely used method for studying classical many-particle
systems. It consists essentially of integrating the equations of motion of the system
numerically. It can therefore be viewed as a simulation of the system as it develops
over a period of time. The system moves in phase space along its physical trajectory
as determined by the equations of motion, whereas in the Monte Carlo method it
follows a (directed) random walk. The great advantage of the MD method is that it
not only provides a way to evaluate expectation values of static physical quantities;
dynamical phenomena, such as transport of heat or charge, or relaxation of systems
far from equilibrium can also be studied.
In this section we discuss the general principles of the molecular dynamics
method. In the following sections more details will be given and special tech-
niques will be discussed. There exists a vast research literature on this subject and
there are some review papers and books [ 1 – 5 ].
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