198 Molecular dynamics simulations
Consider a collection ofNclassical particles in a rectangular volumeL 1 ×L 2 ×L 3.
The particles interact with each other, and for simplicity we shall assume that the
interaction force can be written as a sum over pair forces,F(r), whose magnitude
depends only on the distance,r, between the particle pairs and which is directed
between them (see also the previous chapter). In that case the internal force (i.e. the
force due to interactions between the particles) acting on particle numberiis given as
Fi(R)=
∑
j=1,N;
j=i
F(|ri−rj|)ˆrij. (8.1)
Rdenotes the position coordinatesriof all particles in the notation introduced in
Section 7.2.1 (Pdenotes the momenta);rˆijis a unit vector directed alongrj−ri,
pointing from particleito particlej. In experimental situations there will be external
forces in addition to the internal ones – examples are gravitational forces and forces
due to the presence of boundaries. Neglecting these forces for the moment, we can
use (8.1) in the equations of motion:
d^2 ri(t)
dt^2
=
Fi(R)
mi
(8.2)
in whichmiis the mass of particlei. In this chapter we take the particles identical
unless stated otherwise. Molecular dynamics is the simulation technique in which
the equations (8.2) are solved numerically for a large collection of particles.
The solutions of the equations of motion describe the time evolution of a real
system although obviously the molecular dynamics approach is approximate for
the following reasons.
- First of all, instead of a quantum mechanical treatment we restrict ourselves to a
classical description for the sake of simplicity. In Chapter 9, we shall describe a
method in which ideas of the density functional description for quantum
many-particle systems (Chapter 5) are combined with the classical molecular
dynamics approach. The importance of the quantum effects depends strongly on
the particular type of system considered and on the physical parameters
(temperature, density ...).
- The forces between the particles are not known exactly: quantum mechanical
calculations from which they can be determined are subject to systematic errors
as a result of the neglect of correlation effects, as we have seen in previous
chapters. Usually these forces are given in a parametrised form, and the
parameters are determined either byab initiocalculations or by fitting the
results of simulations to experimental data. There exist systems for which the
forces are known to high precision, such as systems consisting of stars and
galaxies at large mutual distances and at nonrelativistic velocities where the
interaction is largely dominated by Newton’s gravitational 1/r^2 force.
