DEMOCRITUS

Statistical Mechanics

Modern science has revealed that matter is comprised of atoms, and the number of atoms present in even a tiny piece of matter is huge. A small beaker with only 18 c.c. of water will contain approximately 1.8 million, billion, billion atoms. It follows that any attempt to describe the physical properties of a material in terms of the motion of its individual atoms is doomed to failure. However, such astronomical numbers are perfect for a statistical treatment, by means of which the bulk properties (as opposed to individual atomic properties) may be determined with great accuracy. This is where the term statistical mechanics comes from: it is a fusion of mechanics with statistics. Fortunately, since it is the properties of bulk materials that are usually of interest, this treatment is exactly what is required.

Without statistical mechanics, little meaningful information could be obtained from molecular dynamics simulations. It provides a many powerful mathematical tools for drawing out useful properties from a mass of data. The most important tools are:

  1. Ensemble Averaging. This is important if we want to calculate the bulk properties of a system. Bulk properties are inevitably average properties, describing the average effect of all the atoms acting together. Ensemble averaging provides the prescription for calculating the averages properly.
  2. Correlation Functions. These are particularly valuable for drawing out relationships between the observable properties of a bulk system. Spatial correlations reveal how the system structure is organised, while time correlations reveal the dynamical processes that occur.
  3. Fluctuations. These describe how much a system with fixed average properties (i.e a system in equilibrium) can vary from the average at any instant. This variation is subtly linked to the physical processes that operate in the system.
  4. Distribution Functions. These describe how the properties of the bulk system are shared out among the component atoms. Equivalently they describe the probability of finding the bulk system in a given state.