Ergodic Theory

a branch of dynamics. Ergodic theory arose in connection with the problem of replacing average values taken over a phase space by time averages in order to provide a mathematical substantiation of statistical mechanics.

The state of some physical system—for example, a particular volume of gas—is specified by the momenta and coordinates of the system’s component particles, that is, by 6N quantities, where N is the number of particles. It is convenient to imagine the possible states of the system as points in a space of 6N dimensions, which is called a phase space, and to imagine the evolution of the system in the course of time as a certain motion or trajectory in the phase space. As a rule, the various physical quantities associated with a given system—for example, temperature and pressure—are functions of the coordinates and momenta of the particles that constitute the system. In other words, the quantities are functions of a point in the phase space of the system. Such quantities are called phase-space functions.

When a theory is compared with an experiment, the computed values of various physical quantities must be compared with experimental data. Usually, only the average values of phase-space functions over all states corresponding to a given energy—that is, phase-space averages—are readily determined theoretically. On the other hand, a measurement of any physical quantity takes a finite time. Moreover, it takes a long time from the standpoint of the rates of molecular processes. Therefore, the result of any measurement is a time average—that is, an average along a trajectory—of the corresponding phase-space function. Thus, in order to compare experimental data with theoretical results, a substantiation of the replacement of time averages by phase-space averages must be provided.

A system in which the phase-space averages correspond to the time averages is called an ergodic system. The determination of the conditions under which a system is ergodic is the main problem of ergodic theory. The first attempts to establish conditions for the ergodicity of a physical system were made by L. Boltzmann. However, the first mathematically rigorous result was not obtained until 1931, when G. Birkhoff proved that a system is ergodic if and only if the system’s phase space cannot be decomposed into a union of two invariant sets (that is, sets consisting of complete trajectories), each of which has a positive volume. At the same time, Birkhoff proved the existence of time averages. Birkhoff’s investigations were continued and generalized in later works, such as the works of J. von Neumann, A. Ia. Khinchin, and N. M. Krylov and N. N. Bogoliubov. Ergodic theory has been developed essentially as a purely mathematical theory in the framework of the general theory of dynamic systems.

The results obtained in ergodic theory have not led to a complete solution of the problem of substantiating statistical mechanics. However, ergodic theory and the concept of ergodic systems play an important role in, for example, dynamics, the qualitative theory of differential equations, and the theory of stochastic processes.