Ergodic hypothesis

The assumption that a system’s long-term trajectory explores its accessible phase space in a representative way, allowing long-time averages to stand in for averages over possible states.

Ergodic theory: The mathematical study of measure-preserving transformations and the long-term behavior of their trajectories. It formalizes when a system’s orbit samples its state space in statistically representative ways.
Dynamical system: A system whose state changes over time according to a specified rule. Ergodicity is a property of the evolution rule acting on states.
Statistical mechanics: The framework that explains macroscopic properties through the statistical behavior of microscopic states. It often substitutes ensemble predictions for measurements collected over time.
Mixing (mathematics): A dynamical property in which correlations between events decay as their time separation grows. Mixing is stronger than ergodicity and implies it under standard measure-preserving conditions.
Many-body localization: A phase in disordered interacting quantum systems that fails to transport energy and retains local memory. It provides a prominent setting where conventional ergodic behavior can break down.
Invariant measure: A measure unchanged by the evolution of a dynamical system. Ergodic averages are defined relative to a measure preserved by the system’s dynamics.
Trajectory: The sequence or curve of states produced by a system’s evolution from an initial state. The hypothesis asks how one such path samples accessible states over time.
Microcanonical ensemble: The statistical ensemble assigning equal weight to states with fixed energy and conserved quantities. It supplies a natural phase-space average for isolated equilibrium systems.
Nonergodic system: A dynamical system whose invariant regions prevent typical trajectories from sampling the full space under consideration. Such systems show why a single trajectory need not reproduce the full ensemble average.
Eigenstate thermalization hypothesis: The proposal that individual energy eigenstates encode thermal expectation values for local observables. It explains quantum thermalization through eigenstate structure rather than classical trajectory exploration.

Linked from 10 pages