This property allows one to apply the time correlator formalism to the trajectory in a molecular dynamics simulation

Time Correlator in Molecular Dynamics

It may be helpful to visualize the time correlation function as it is applied in a molecular dynamics simulation. In MD, the system is moved in discrete time intervals following Newton's equations of motion. At any time t we can calculate a property A(t). The time correlation function is the product of the property at t and at a time t later.

The angle brackets represent statistical averaging. In statistical mechanics we envision this averaging over many similar systems (the ensemble). We can use many separate time frames of molecular dynamics instead of many systems in the ensemble in the usual statistical mechanical approach in order to obtain useful time decays that can be analyzed. We can represent this be an integral

or by a summation

as it is done in practice.

For example, consider A to be the velocity v. At time zero v(0)×v(0) [or v(t)×v(t) at any time t in the simulation] is just the square of the magnitude of the velocity. Since it is the dot product we have

v(0)×v(0)= v_x(0)v_x(0) + v_y(0)v_y(0) + v_z(0)v_z(0)

At time t the particle has changed its velocity. At this time the value of the velocity auto-correlation function is

v(0)×v(t)= v_x(0)v_x(t) + v_y(0)v_y(t) + v_z(0)v_z(t)

Note that this is also equal to

v(0)×v(t)= |v(0)||v(t)|cosq.

Thus, as the angle between the velocity at time zero and velocity at time t approaches 90 degrees the value of v(0)×v(t) goes to zero.