# Erogodicity in a Monte Carlo simulation

Q1: What is the ergodicity and ergodicity breaking in a Monte Carlo simulation of a statistical physics problem?

Q2: How does one ensure that the ergodicity is maintained ?

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Ergodicity is a description of a system which has filled all degrees of freedom equally. For example, if you use MC method to simulate gas molecules, with constant initial velocity. The system will be ergodically distributed when the velocity follows the Maxwell-Boltzmann distribution. This is my understanding I'm sure there is a better definition involving entropy. Breaking this condition sounds like it implies a decrease in entropy. –  boyfarrell Apr 2 '13 at 13:22

In the context of a Monte Carlo (MC) simulation, ergodicity means that the algorithm that you use is designed in such a way that all points in the corresponding phase space (the one that contains the trajectory of your statistical ensemble) would be visited if the algorithm ran for an infinite amount of time. There is no way to prove that an algorithm is ergodic, as we just cannot let a simulation run infinitely. In the literature, you can find the concepts of balance and detailed balance. From a practitioner's point of view, if an algorithm fulfils detailed balance, it is safe to assume that the system behaves ergodic.

In general, you cannot show that a system is ergodic. In statistical physics, ergodicity is assumed for systems in thermal equilibrium, but this assumption cannot be proven (to my (limited) knowledge).

A good online resource for learning MC is this site with lecture notes. A good book that covers everything from the beginning is Tuckerman: Statistical Mechanics. A book that is more detailed on MC but requires solid knowledge in statistical physics is Frenkel & Smit: Understanding Molecular Simulation.

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