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 ?