Microcanonical ensemble question

[Coffee_]

So from what I understood from some coure notes I've been reading, a microcanonical ensemble is a situation where we have an isolated system in thermal equilibrium with a constant given N,V,E - particles, volume,total energy.

I'm a bit confused. How I understood 'ensemble' is as a set of all the allowed microstates for a certain macrostate. This makes sense for distinguishable systems like a lattice, but makes no sense to me for an indistinguishable system like a gas.

Can someone explain to me what the difference is between distinguishable and indistinguishable systems is within this ensemble?

 

[vanhees71]

I don't understand what you mean by "dinstinguishable" and "indistinguishable" systems.

If you have, say, an ideal gas of indistinguishable particles, in the macrocanonical ensemble, the macrostate is defined by the exact number of gas particles (if conserved!) and total energy of the system (and of course external parameters like the volume, external fields, etc.). This macrostate can be realized by many states. Here you can take as a basis the N-particle Fock states with respect to the momentum-spin single-particle basis. These are also energy eigenstates (for non-interacting particles). Then the statistical operator is given as the one that maximizes the von Neumann entropy given these constraints.

The most simple way to get it is to work with a single-particle basis for a cubic finite volume with periodic boundary conditions. Then the momenta and energy are discretized and you can work with proper normalizable states. Now you take the energy-particle number eigenstates ##|E,N,\alpha \rangle##, where ##\alpha## is a discrete set of index variables parametrizing the degeneracy of these energy-particle-number eigenstates. The maximum entropy principle then gives that the equilibrium distribution is given by
$$\hat{\rho}_{E,N} = \frac{1}{Z} \sum_{\alpha} |E,N,\alpha \rangle \langle E,N,\alpha|.$$
The indistinguishability of the particles gives you constraints on the possible N-particle energy eigenstates, because they are given by totally antisymmetrized (symmetrized) product of ##N## single-particle states if your gas particles are fermions (bosons).

 

[Coffee_]

Thanks for the answer. Could I ask a quick other question on this topic since this question isn't really worth a repost?

If we use the gibbs-entropy formula on a canonical ensemble ## \Sigma p_i ln(p_i) ##, what are the microstates 'i' that need to be used here? Do the microstates have to contain the configuration of the reservoir, ''the heath bath'' as well?

For example, for the microcanonical ensemble the answer is no, the microstates that need to be counted in the entropy formula are only the states of the system that we are investigating. However in the canonical ensemble one does care about the reservoir or the heat bath as well though it seems, one for example does use the entropy of the heath bath in one of the derivation. The environment suddenly seems less irrelevant.

Also, do we in general take the positions of the particles as part of the microstates? That is if I keep all the particles at the same energy but slightly change the positions, in a microcanonical ensemble for example, does this count as a different microstate?

 

[vanhees71]

For thermal equilibrium the constraints in the maximum-entropy principle should be related with the conserved quantities of the system. In the grand-canonical ensemble you use the energy (in the reference frame, where the total momentum is 0, which defines the heatbath restframe in this case) and conserved charges (if there are any in the theory; in non-relativistic physics often the particle number is conserved and used).

The states, over which you sum are the simultaneous eigenstates of the conserved quantities you choose, e.g., energy-particle-number eigenstates. The entropy must be maximized under the constraint that the total probability of the system being in such a state is 1. Then you get equipartition of the states, i.e.,
$$p_i=1/\Omega$$
where ##\Omega## is the number of these eigenstates.

If a quantity has a continuous spectrum you must modify this a bit and take all states in an appropriate neighborhood of the given value of this quantity:

https://en.wikipedia.org/wiki/Microcanonical_ensemble#Quantum_mechanical

A good treatment is found in

F. Reif, Fundamentals of Statistical Mechanics