ERGODICITY
| Collection | International Encyclopedia of Systems and Cybernetics |
|---|---|
| Year | 2004 |
| Vol. (num.) | 2(1) |
| ID | ◀ 1165 ▶ |
| Object type | General information, Methodology or model |
Character of a system in which the average values of the variables evolve through time within the set of their statistically invariant values over all possible configurations.
Ergodicity reflects the existence of subjacent global and invariant conditions which determine the behavior of the variables only in a statistical way, but within limits.
Y. SINAI considers as the basic problem of ergodic theory: “the analysis of the causes generating the appearance of statistical laws in dynamic systems” (1992, p.69).
The behavior of an ergodic system evolves within a phases space that becomes divided into subspaces. As expressed by S. DINER: “Ergodic theory is basically a theory of the sets of trajectories” (1992, p.348).
As shown by BIRKHOFF (1931), the most significant characteristic of ergodicity is that the average behavior of the system over a sufficient lapse of time tends to be equal to the average configurations in space. This average on space is independent of the trajectory considered and the average in time is independent of the initial point. The behavior is thus not truly random and could be called para-random.
In consequence, the behavior of chaotic systems, with a number of initial conditions, tends to be ergodic.
See also
Ergodic relation, (according to ASHBY).