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Ensemble Averages
The ensemble is a central concept in statistical mechanics. Imagine that a given molecular system is replicated many times over, so that we have an enormous number of copies, each possessing the same physical characteristics of temperature, density, number of atoms and so on. Since we are interested in the bulk properties of the system, it is not necessary for these replicas to have exactly the same atomic positions and velocities. In other words the replicas are allowed to differ microscopically, while retaining the same general properties. Such a collection of replicated systems is called an ensemble.
Because of the way the ensemble is constructed, if a snapshot of all the replicas is taken at the same instant, we will find that they differ in the instantaneous values of their bulk properties. This phenomenon is called fluctuation. Thus the true value of any particular bulk property must be calculated as an average over all the replicas. This is what is meant by an ensemble average, and the instantaneous values are said to fluctuate about the mean value.
Molecular dyamics proceeds by a numerical integration of the equations of motion. Each time step generates a new arrangement of the atoms (called a configuration) and new instantaneous values for bulk properties such as temperature, pressure, configuration energy etc. To determine the true or thermodynamic values of these variables requires an ensemble average. In molecular dynamics this is achieved be performing the average over successive configurations generated by the simulation. In doing this we are making an implict assumption that an ensemble average (which relates to many replicas of the system) is the same as an average over time of one replica (the system we are simulating). This assumption is known as the Ergodic Hypothesis. Fortunately it seems to be generally true, provided a long enough time is taken in the average. However it has not yet been rigorously proved mathematically.
Examples of properties that can be calculated as ensemble averages include:
In fact, almost everything of interest that can be obtained through a molecular dynamics simulation, requires the calculation of some kind of ensemble average!
