International Encyclopedia of Systems and Cybernetics

A system which will pass through every possible dynamical state compatible with its energy.

P. COVENEY explains: "… we can use "phases portraits" to show how an ergodic system behaves. But in this case, we portray the initial state of the system as a bundle of points in phase space, rather than a single point… In a non-ergodic system, the bundle retains its shape and moves in a periodic fashion over a limited portion of the space… In an ergodic system, the bundle maintains its shape but now roves around all parts of the space" (1990, p.52).

In a chaotic system, "…the bundle, whose volume must remain constant, spreads out into ever finer fibers, like a drop of ink spreading in water; eventually it invades every part of the space. This is a consequence of what is called LIOUVILLE's theorem… The total probability must be conserved (and add up to 1): the bundle behaves like an incompressible fluid drop. This an example of a "mixing ergodic flow" and manifests an approach to thermodynamic equilibrium when the time evolution ceases" (Ibid.).

An ergodic system is thus the one for which the global set of trajectories if well defined, while individual ones remain probabilistic.

About ergodicity of Markovian systems, G. PASK writes: "… the state (Note:i.e. the global state) of the Markovian system is invariant and it can be shown that the values in the distribution p* are independent of the initial state i. But any representative system can move away from any state to any other state. In the phase space the state points of the ensemble are in continual motion, because a representative system can always reach any of the states (none of the transitions are impossible), the average population, at any instant, being determined by p*. The equilibrium is called ergodic and the set of states which can be visited (in this case all the states) is called an ergodic set" (PASK., 1961 a p.122-3).

PASK shows furthermore that a system can be at the same time ergodic and cyclic.

As proved by KOLMOGOROV, a basically nonergodic system may however contain ergodic bundles or regions (S. DINER, 1992, p.355).