## Ergodicity March 26, 2006

What does it mean for a system to be *ergodic* ? On the surface, we could say that a system is ergodic if the *time average* of one path taken by the system , equals the *ensemble average*, the cut across all paths at some instant in time. For illustrative purposes, think of each path as a squiggly curve, starting from the origin. When looking at multiple paths, we have multiple squiggly lines coming out of the origin. The *y-*value or *ordinate* in this case is the system state, and the *x*-value or *abscissa* is time.

In order for this to happen, the state transitions of the system have to be such that each state of the system is likely to be visited infinitely often. This cannot happen if there is some state, which when entered, one cannot depart from.

How does ergodicity relate to fixed-points ?

## One Response to “Ergodicity”

abbMarch 27th, 2006 at 12:57 pm | Permalink“For Markov processes with a finite number of possible states (for example, networks with a finite number of neurons N), ergodicity means that there is some time after which, whatever the initial state was, one has a nonzero possibility of being in any state. In other words, the system `spreads out’ over all of its phase space if we wait long enough. An ergodic system can be shown to have a unique stationary distribution to which it will converge from any initial state.”