Understanding Ergodicity and Ensemble Averaging

1. How do you explain ergodicity?
2. What does ensemble average mean?
3. What is ergodic average?
4. What is the difference between time average and ensemble average?

How do you explain ergodicity?

In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense.

What does ensemble average mean?

In statistical mechanics, the ensemble average is defined as the mean of a quantity that is a function of the microstate of a system, according to the distribution of the system on its micro-states in this ensemble.

What is ergodic average?

The concepts of ergodicity and the ergodic hypothesis are central to applications of ergodic theory. The underlying idea is that for certain systems the time average of their properties is equal to the average over the entire space.

What is the difference between time average and ensemble average?

An ensemble average is a convenient theoretical concept since it is directly related to the probability density functions, which can be generally obtained by the theoretical analysis of a given physical system. On the other hand, a time average is more directly related to real experiments.