Sampling

An important component of MD is the ability to sample phase space well. Phase space is made up of points that represent the collected positions and momenta of the particles in the system.

N atoms = 6N dimensions comprised of 3N positions and 3N momenta.

Concretely these are stored in the history file [*.his]. Positions and velocities are written to this file at periodic intervals. The format of these file varies, but in principle we have the following information:

x₁, y₁, z₁, v_x1, v_y1, v_z1

x₂, y₂, z₂, v_x2, v_y2, v_z2

x₃, y₃, z₃, v_x3,…..

The collection of 6N points at time t represents a point in phase space, G. The instantaneous value of a property A is a function of G. An observable macroscopic property, A obtained from an MD simulation is a time average

A_obs = áAñ_time = áA(G(t))ñ_time

The value A depends on structure or velocities and hence on the point in phase space, G. The time average is discrete in a MD simulation. Thus, we have

The index t stands for a succession of time steps at the which positions and velocities have been saved. The major issue that needs to be resolved is how many time points are required to sufficiently determine an average property.

Statistical mechanics (SM) uses an average over a large number of systems, an ensemble to determine average properties. The Gibbs postulate is the statement that the ensemble average should equal the average thermodynamic property. The MD and SM point of view can be treated as equivalent according to the ergodic hypothesis. The ergodic hypothesis states that we can obtain the same average from a time dependent or an ensemble average.

The points in phase space are distributed according to a probability density r(G). If r_ens represents an equilibrium ensemble then it has no time dependence, ¶r_ens/¶t = 0. Thus, if a particular point in phase space G is removed, it is simultaneously replaced by an identical one. In the time-dependent picture as G(t) moves to G(t+1), another system arrives from G(t-1) to replace it. Care must be taken to ensure that the MD approach does not get stuck in a closed loop with a short recurrence time. This would be equivalent to averaging a small number of systems while excluding the majority.

Using this notation the ensemble average can be written

The probability density can be written in the form of a weight function w_ens(G)

The probability density is equal to the weight factor divided by the parition function Q that acts as a normalization factor. The partition function is a sum over states quantity that represents the average number of states accessible to the system at a given temperature. It is impractical to sample all of the states in the partition function by MD. Rather, we rely on sampling the most probable configurations and ignore those configurations that contribute a negligible amount to a thermodynamic average because their weight is so small.