Probability Distribution in Ensembles: Explained

[CassiopeiaA]

I am confused about the basic idea of probability distribution in ensembles.

Given macroscopic properties of the system, a system can have large number of micro states. But isn't the probability of finding a system in any of the micro state is equal? What is then the interpretation of this phase space probability distribution?

 

[MisterX]

All the points in phase space with the same energy have the same probability density in both the micro canonical and canonical ensembles. In the micro canonical ensemble, the probability density is zero everywhere except at some fixed energy or range of energies and the probability density is uniform in all the phase space with energies in the allowed range.

For the canonical ensemble, what we have is systems connected together, one which I will call "system 1" and one I will call the "heat bath". When we consider the phase space of the two combined systems, we assume that combined phase space distribution is given by the micro canonical ensemble with a fixed total energy. However when we consider only the distribution of system 1, we find that states with different [system 1] energies get a different weighting factor in their probability. In the thermodynamic limit, where the heat bath becomes infinitely large compared to system 1, the probabilities of different points in system 1's phase space are weighted by the Boltzmann factor.

So in conclusion, all the microstates are not equally likely. There might be a couple things I brushed over but I hope that helps.

 

[CassiopeiaA]

I think I require a little layman approach to my problem as there are problems in my fundamental understanding.

First of all what is the difference when we take micro states of a system in phase space and ensemble of the system in phase space?

Okay suppose I have a system with some N particles in it. And the system is defined by some total energy E. Now, this system can have large number of micro states and each state I can represent with a point on a 6N dimensional phase space. What I understood was, the particles are always in motion that's why it is possible to find particles in any of the micro state at a given instant of time. Now, why would the system prefer some micro state over other?
Is it possible that all the particle sin the system suddenly have velocity in a single direction at a given instant of time?

 

[MisterX]

First of all what is the difference when we take micro states of a system in phase space and ensemble of the system in phase space?

For a system with a fixed number of particles, a microstate is just a point in phase space. That is, a set of values for ##\mathbf{x}_1, \dots \mathbf{x}_N, \mathbf{p}_1, \dots, \mathbf{p}_N##. In equilibrium statistical mechanics each microstate has a probability density associated with it. In the micro canonical ensemble the probability density is given by one thing and in the canonical ensemble, it's given by something different.

What I understood was, the particles are always in motion that's why it is possible to find particles in any of the micro state at a given instant of time.

This is not exactly correct. The reason why we assume the probabilities of microstates at a given energy are uniform the micro canonical ensemble is twofold. The first reason is to simplify the problem. It has to do with the "ergotic hypothesis". It's a hypothesis that for a given system that might not actually be true, or might only be true on timescales so large that it's not a good assumption. But in many cases it's a useful assumption. The second reason is to remove any time dependence in the probability distribution. Equilibrium distributions don't change with time, and by assuming uniform probability for all states of a given energy, you assure the overall probability distribution in phase space does not change with time (as long as it's a system with energy conservation).

Now, why would the system prefer some micro state over other?

I already described one situation, where the system is in contact with another system which leads to some microstates being more probable than others. In general (outside of equilibrium statistical mechanics), it could simply be due to the initial conditions of the system.

Is it possible that all the particle sin the system suddenly have velocity in a single direction at a given instant of time?

Don't think about "suddenly", because time dynamics isn't considered at all in equilibrium statistical mechanics.

 

[CassiopeiaA]

In equilibrium statistical mechanics each microstate has a probability density associated with it. .

But according to principle of a priori, the probability of finding a system in any micro state is equal. Then what is the meaning of this probability distribution?