nonergodic

Nonergodic describes a property of a system, process, or model in which the long-time behavior of a single trajectory does not generally reflect the statistical properties of the whole system. In particular, time averages taken along one evolution do not necessarily equal ensemble averages computed over the space of all states. This contrasts with ergodic systems, where time and ensemble averages coincide for almost every starting point.

In the language of measure-preserving dynamics, a system with transformation T on a probability space (X, F, μ) is ergodic if every T-invariant set has measure 0 or 1; equivalently, the time average of any integrable function f converges to its space average μ(f) for μ-almost every initial state. Nonergodicity means this is not the case: there exist nontrivial invariant sets or the time averages depend on where the system began, so a single trajectory may never sample all regions of state space.

Causes and manifestations include the presence of multiple ergodic components separated by barriers or constraints, long-lived correlations, aging in glassy or disordered materials, or quenched randomness that prevents full exploration of states. In stochastic processes, nonergodicity can arise when the stationary distribution decomposes into several ergodic components. In Markov chains, a chain with more than one closed communicating class is nonergodic.

Examples range from a particle in a double-well potential at low temperature to spin glasses, or certain random walks in random environments. The concept is important because many standard predictions rely on ergodicity; for nonergodic systems, time averages from a single run may not reflect equilibrium properties, necessitating non-equilibrium analysis or an explicit accounting of ergodic components.