Equilibration

  Every time the state of the system changes, the system will be ``out of equilibrium'' for a while. We are referring here to thermodynamic equilibrium. By this, it is meant that the indicators of the system state--of which many are described below--are not stationary (that is, fluctuating around a fixed value), but relaxing towards a new value (that is, fluctuating around a value which is slowly drifting with time).

The state change may be induced by us or spontaneous. It is induced by us when we change a parameter of the simulation--such as the temperature, or the density--thereby perturbing the system, and then wait for a new equilibrium to be reached. It is spontaneous when, for instance, the system undergoes a phase transition, thereby moving from one equilibrium state to another.

In all cases, we usually want equilibrium to be reached before starting performing measurements on the system. A physical quantity A generally approaches its equilibrium value exponentially with time:  

$$A(t) = A_\circ + C \exp(-t/\tau)$$ (9)

where A(t) indicates here a physical quantities averaged over a short time to get rid of instantaneous fluctuations, but not of its long-term drift. The relevant variable is here the relaxation time $\tau$.We may have cases where $\tau$ is of the order of hundreds of time steps, allowing us to see A(t) converge to $A_\circ$ and make a direct measurement of the equilibrium value. In the opposite case, $\tau$ could be much larger than our overall simulation time scale; for instance, of the order of one second. In this case, we do not see any relaxation occurring during the run, and molecular dynamics hardly seems a valid technique to use in such a situation. In intermediate situations, we may see the drift but we cannot wait long enough to observe convergency of A(t) to $A_\circ$.In these cases, one can often obtain an estimate for $A_\circ$by applying (3.1) on the available data, even if the final point is still relatively far from it.