Computational Physics

200 Molecular dynamics simulations

whereLμare vectors along the edges of the rectangular system volume and the

first sum on the right hand side is over all vectorsnwith integer coefficientsnμ.

The forceFis directed along the line connecting particleiand the image

particlerj−

∑ 3

μ= 1 Lμnμaccording to the convention ofEq. (8.1). Of course,

calculating terms of this infinite sum until convergence is achieved is a

time-consuming procedure, and in the next section we shall consider techniques

for approximating this sum efficiently.

• The time average must obviously be evaluated over a finite time. For liquid
  argon, which is the most widely studied system in molecular dynamics because
  simple Lennard–Jones pair forces yield results which are in very good
  agreement with experiment, the typical time step used in the numerical
  integration of the equations of motion is about 10−^14 seconds, which means that
  for the∼ 105 integration steps which can usually be carried out in a reasonable
  amount of computer time, the total simulation is restricted to about
  10 −^9 seconds. The correlation time of the system should therefore be much
  smaller than this. There is also a limitation in time because of the finite size of
  the system. This might in principle become noticeable when the particles have
  travelled on average more than half the linear system size, but in practice such
  effects occur at much longer time scales, of the order of therecurrence time, the
  time after which the system returns to the initial configuration (in continuum
  mechanics, this is called thePoincaré time).


• The numerical integration algorithm is not infinitely accurate. This forces us to
  make some optimum choice between speed and accuracy: the larger the
  integration time step, the more inaccurate the results of the simulation. In fact, the
  system will follow a trajectory in phase space which deviates from the trajectory
  it would follow in reality. The effect on the physical quantities as measured in the
  simulation is of course related to this deviation in the course of time.
  We may summarise by saying that MD is – in principle – a direct simulation
  of a many-particle system but we have seen that, just as with any computational
  technique in physics, MD simulations must be carried out with considerable care.
  It is furthermore advisable to carry out reference tests for systems for which exact
  results exist or for which there is an extensive literature for comparison.

8.2 Molecular dynamics at constant energy

In the previous section we sketched the molecular dynamics method briefly for the

simplest case in which the equations of motion for a collection of particles are solved

for forces depending on the relative positions of the particles only. In that case energy

and momentum are conserved.^1 Trivially, the particle number and system volume are

(^1) The angular momentum is not conserved because of the periodic boundary conditions breaking the
spherical symmetry of the interactions.