International Encyclopedia of Systems and Cybernetics

2nd Edition, as published by Charles François 2004 Presented by the Bertalanffy Center for the Study of Systems Science Vienna for public access.


The International Encyclopedia of Systems and Cybernetics was first edited and published by the system scientist Charles François in 1997. The online version that is provided here was based on the 2nd edition in 2004. It was uploaded and gifted to the center by ASC president Michael Lissack in 2019; the BCSSS purchased the rights for the re-publication of this volume in 200?. In 2018, the original editor expressed his wish to pass on the stewardship over the maintenance and further development of the encyclopedia to the Bertalanffy Center. In the future, the BCSSS seeks to further develop the encyclopedia by open collaboration within the systems sciences. Until the center has found and been able to implement an adequate technical solution for this, the static website is made accessible for the benefit of public scholarship and education.


SYSTEM (Ergodic) 2)

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).


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Bertalanffy Center for the Study of Systems Science(2020).

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