The following is from Ole Peters (Ph.D. in theoretical physics from Imperial College London in 2004) website;
http://tuvalu.santafe.edu/~ole/risk.html

"A fresh perspective on expectation values emerged in the development of statistical mechanics in the 19th century^1.

It was noted that there are two distinct averages or expectations, one being the construct of Fermat -- an arithmetic mean of all

possible states of the universe; the other being a time average -- the state that results from following a single realization of a system

along its trajectory through time. Both procedures average out the fluctuations from stochastic systems, the former by considering

infinitely many systems (or universes) in parallel, the latter by considering an infinite amount of time, or infinitely many time slices.

^1 Lebowitz, J. and O. Penrose. Modern ergodic theory.  Physics Today 26, 2, 23--29 (1973)

TEDxGoodenoughCollege - Ole Peters - Time and Chance, http://www.youtube.com/watch?v=LGqOH3sYmQA

The following is from the Impirical College London website;
http://www3.imperial.ac.uk/newsandeventspggrp/imperialcollege/newssummary/news_17-12-2010-16-3-49?newsid=95474

"The ensemble average is the most commonly used approach. It is based on imagining multiple scenarios that all begin from the same starting conditions, and then averaging their outcomes. The alternative, time averaging, imagines all possible scenarios playing out over time. As we live on a timeline, previous decisions cannot be undone as time passes. Any new decision constrains our choices when making subsequent ones. Time averaging provides the more accurate prediction for the real world outcome of an investment decision. Today's study shows that, in the investment world, the differences in the results from these two approaches are critical: time averaging inherently incorporates a measure of risk, but ensemble averaging does not. This means that ensemble averaging consistently undervalues risk by underestimating the effects of time on investments and overestimating the degree of choice that investors have. It also encourages excessive leveraging of investments, which itself accentuates fluctuations in the market, increases market volatility, and imparts a negative drift in the market that helps drive investors into negative equity."

Non-ergodicity (ie Geometric Brownian Motions)= describes a situation when the ensemble average is not

the same as the time average. (Any ergodic system is stationary).

For example a market where share prices go up by 10% and then fall by 10% is an example of a non-ergodic system.

The growth rate of the ensemble average (arithmetic average) is 0% per round

The growth rate of the time-average (geometric average) is approximately -0.5% per round.

To illustrate the basic concept futher consider that you are faced with the following fair coin toss (you start with 100 USD):

50% chance you get head=win 50%

50% chance you get tails= lose 40%

In maple code this will translate to: