8.1 Introduction 199
Figure 8.1. Periodic boundary conditions for molecular dynamics. Each particle
interacts not only with every other particle in the system but also with all other
particles in the copies of the system. The arrows from the white particle point to
the nearest copies of the other particles in the system.
- Another approximation is inherent to most computer simulations aiming at a
description of the real world: the system sizes in such simulations are much
smaller than those of experimental systems. In the limit where the correlation
length is much smaller than the system size this does not matter too much, and
in the opposite regime, in which the correlation length exceeds the system size
we can use the finite-size scaling methods discussed in Chapter 5 in order to
extrapolate results for physical quantities in the finite system to those of the
infinite system (although second order transitions are seldom studied in
molecular dynamics because of the heavy demands on computing resources).
The finiteness of the system size is felt through the presence of the boundary.
The convention adopted in the vast majority of molecular simulations is to use
periodic boundary conditions (PBC) as it is assumed that for these boundary
conditions the behaviour of the system is most similar to that of a system of the
same size embedded in an infinite system. In fact, with periodic boundary
conditions the system of interest is surrounded by similar systems with exactly
the same configuration of particles at any time (see Figure 8.1). The interaction
between two particlesiandjis then given by the following expression:
FPBC(ri−rj)=
∑
n
F
∣
∣∣
∣∣
∣
ri−rj+
∑^3
μ= 1
Lμnμ
∣
∣∣
∣∣
∣
(8.3)
