# Regarding Monte-Carlo in Statistical Mechanics

Suppose we have a system in contact with a bath at temerature,$\beta^{-1}$.It is initially in a particular configuration.Now,we take a trial configuration.If the energy is lower,it is sure to move to that state.If it is higher,it has a probability of $e^{-\beta\delta E}$ for moving to that state(where $\delta E$ is the difference between the final and initial energies).We have arrived at this rule of dynamics by abiding to detailed balance.And we say,that this dynamics respects the probability law,$P(C)\backsim e^{-\beta E(C)}$ (we are talking about C-th configuration).

Now,what I'm asking is, can you say from the dynamics that the probability law is respected?Well,we derived the dynamics using the law,but can you give me an example where we consider an ensemble of systems all initially in some configuration,and then the configurations changing with time according to the above dynamics,and we can calculate the fraction of systems in the ensemble in state 'C' and find it out to be that given by the probability law.