We next consider the dynamics of a collection or ensemble of particles. In the previous sections, we have discussed the dynamics of one particle or systems that can be reformulated as involving only one particle. In general, the macroscopic world consists of many particles simultaneously interacting with one another. For example, the description of the air molecules in a closed room requires tracking at least 1027 particles. Clearly, to attempt to describe the motions of each individual particle would exhaust the computational power of any conceivable machine, not to mention that of the investigator. Therefore it is necessary to construct a means by which the collective behavior of an entire system can be assessed without relying on the complete description of the underlying histories of each individual particle in the ensemble. A statistical approach, called statistical mechanics, has been invented to treat the behavior of a large collection of particles. The basic principles and applications of equilibrium statistical mechanics are the subject of this chapter. In Chapter 6 we will consider treating systems that depart from equilibrium.

Density of States

A large collection of particles is often referred to as an ensemble. To describe the collective behavior of a large ensemble it is necessary to adopt a statistical approach. The best that we can expect to do is to predict, by using statistics, the average values of the macroscopic observables of interest, particularly the energy, the momentum, etc., and the probability that any given particle has a specific value of one of these observables, for example whether a particle has energy E + dE.