ergodic

Ergodic is a term used in mathematics, physics, and statistics to describe systems in which long-term time averages along a single trajectory reflect the corresponding averages over the whole space of states. In practice, ergodicity means that, given enough time, a system explores its available states in a way that makes time-based measurements representative of the ensemble.

In ergodic theory, which studies dynamical systems with a measure, a transformation T on a probability space is called ergodic if every T-invariant set has measure zero or one. Equivalently, for any integrable function f, the time average along orbits, defined by the limit of (1/N) sum_{n=0}^{N-1} f(T^n x) as N grows, equals the space average ∫ f dμ for almost every x. This is a fundamental result known as Birkhoff’s ergodic theorem.

In stochastic processes, ergodicity typically means that the process’s time averages converge to the corresponding ensemble averages. For a stationary process, ergodicity often means that the only events that are invariant under time shifts have probability 0 or 1. This implies that long-run observations of a single process sample the same distribution as the process’s probabilistic description.

In the context of Markov chains, ergodicity often requires the chain to be irreducible and aperiodic. Such chains possess a unique stationary distribution to which the chain converges from any starting state, and time averages converge to expectations with respect to that distribution.

In statistical mechanics, the ergodic hypothesis posits that, over long times, the dynamics of a many-particle system explore all accessible microstates of the same energy, allowing time averages to equal ensemble averages. It provides a bridge between microscopic dynamics and macroscopic thermodynamic quantities.

Examples include the Bernoulli shift, the irrational rotation on the circle, and the doubling map on the unit interval. Non-ergodic systems can have invariant subsets that prevent time averages from representing the whole space.