International Encyclopedia of Systems and Cybernetics

The character of the behavior of a system whose average transit between successive states in time statistically equates the probabilities of the different possible states.

L. BRILLOUIN enounced the ergodic hypothesis, already introduced by BOLTZMANN, in the following way:" The average of some relation, measured through a period of time, must be equal to the average of the same relation, taken on the surface of constant energy".

According to BRILLOUIN, the idea was "to follow some determined trajectory and to show that it progressively covers the surface of constant energy, in the way that a wire covers a coil or a thread the surface of a ball".

However there are some difficulties, thus stated by BRILLOUIN: "Some peculiar trajectories are periodic, close on themselves and do not cover the surface in a uniform way. Others reach bifurcation points where different probabilities must be introduced for different branches" (1959, p.189).

The mentally unconfortable mix of determinism and randomness appears clearly in the former comment. The same problem is reflected in other comments by W.R. ASHBY about "equilibrium in a MARKOV chain" (1956, nr.9/6) where it can be seen that, within a population, the distribution of differently characterized sub-groups tends at any moment to be equal to the mean behavior through time of these groups. Such systems are neither perfectly deterministic, nor perfectly random: individual randomness is limited by the characteristics of the individuals, which define the limits of their autonomous and collective randomness in relation to the specific environmental conditions and to the global mix of the different types of autonomous individuals.

These intricated situations seems to be related to:

1. the general theory of deterministic chaos: the ergodic relation holds only so far as the system maintains itself within the limits of its attractor.

2. the concept of organizational closure as it might be applied to populations (in a very general sense).

3. and possibly the concept of self-criticality in composite systems.

These systems seem to possess a kind of collective, but blurred memory of their initial conditions, which become mixed up with time, but are never totally erased.

The POINCARÉ section seems to have been the first specific ergodic relation, characteristic of nearly chaotic systems, being also useful for the study of periodic trajectories.