ergodisch

Ergodisch, or ergodic, is a term used in mathematics, physics and statistics to describe a property of a dynamical system or stochastic process. In an ergodic system, long-term temporal behavior along a single trajectory reflects the statistical properties of the whole system when viewed as an ensemble. This idea underpins the practical use of time series data to infer ensemble statistics.

In mathematical terms, consider a measure-preserving transformation T on a probability space (X, B, μ). The system is ergodic if every T-invariant measurable set has μ-measure 0 or 1. Equivalently, any T-invariant integrable function is almost surely constant. A central consequence is Birkhoff’s ergodic theorem: for any integrable function f, the time average along almost every orbit converges to the space (ensemble) average ∫ f dμ. This formalizes the notion that, for ergodic systems, the average over time along a single trajectory equals the average over the entire ensemble.

In physics, the ergodic hypothesis posits that, at equilibrium, time averages of observables equal their ensemble averages. Ergodicity is distinct from, yet related to, mixing and other forms of statistical independence; a system may be ergodic without being strongly chaotic, and some systems are non-ergodic, displaying dependence on initial conditions or aging.

Applications span statistical mechanics, dynamical systems, and probability theory, where ergodicity justifies replacing ensemble averages with time averages in experiments and simulations.