Ergodicity

In mathematics, especially in ergodic theory, ergodicity is a way of saying that a dynamical system behaves as one indivisible statistical system, rather than being composed of statistically distinguishable subsystems. More precisely, a measure-preserving dynamical system is ergodic if every invariant measurable set has either measure zero or full measure. Equivalently, the system cannot be decomposed, up to sets of measure zero, into two smaller invariant parts of positive measure. Ergodic theorems relate this condition to the equality of time averages and space averages. Under suitable hypotheses, the time average of an observable along almost every orbit is equal to its space average. Ergodicity itself, however, is not the same as randomness, mixing, chaos, or the assertion that every individual orbit visits every part of the space. Ergodicity may also be described in terms of ergodic measures: an invariant probability measure is ergodic if it cannot be decomposed into a nontrivial convex combination of other invariant probability measures. Ergodic systems occur in many areas of physics, geometry, probability theory, and dynamical systems. The origins of the subject lie in statistical physics, where Ludwig Boltzmann formulated the ergodic hypothesis in connection with the foundations of statistical mechanics.