# Statistical ensembles of various kinds

I was reading introductory statistical mechanics. My final aim is to understand the cumulant and cluster expansions.

The book I have is Atkins physical chemistry (I prefer it becuase it requires only a modest amount of statistics and probability). I got to the point where they derived the molecular partition function (q) for a fixed given total energy of the system (E). They did this by finding out the most probable configuration (n0,n1,n2,n3,n4,......) of the system such that n0 members are in the energy state -> e0, n1->e1, n2->e2, n3->e3, n4->e4... and so on.

N = n0 + n1 + n2 + n3 + .........
E = n0*e0 + n1*e1 + n2*e2 + n3*e3 .......

Sorry for the details but they're necessary for what I am going to ask.

Then they wanted to go to systems where the particles interact and wanted to extend the same idea. So they started defining ensembles. The canonical ensemble is a collection of members, where each member has the same N (no of particles), V(volume of member) and T(temperature of member). Also the total energy of the ensemble is fixed at E'. The analogy to derive the canonical partition function was given as this - consider the total number of participant members as N' such that the n'0 member is in energy state E0, n'1 ->E1, n'2 ->E2 n'3 ->E3, n'4 ->E4 and so on. And in analogy to the molecular partition function:

N' = n'0 + n'1 + n'2 + n'3 ....
E' = n'0*E0 + n'1*E1 + n'2*E2 + n'3*E3 ....

NOW - they say that these members in the ensemble are free to exchange energy between themselves -which creates a problem for me. If the N,V,T for each member is fixed then how can they exchange energy without change in T ?? are they already not in thermal equilibrium with each other? what am I missing?

Some other books said that canonical ensemble is similar to a closed system (which i understand well - no mass exchange but energy exchange allowed from surroundings) Is this analogy not incorrect because Atkins's book says total energy of ensemble is fixed at E'. Ohk, i can imagine that despite each member having fixed N,V,T for each member, they can still have different energies E1, E2, E3..... , because the interactions happening inside each members can be different. however, if they tried to exchange energy among each other, won't their temperatures change? it was shown in the derivation of molecular partition function that the distribution is only a function of temperature and total energy.

what is the essential extention to interacting particles here ?

1. is it that the ensmeble members (in analogy to particles in the non interacting system) can now interact, where as the particles were not allowed to interact while deriving molecular partition function, or

2. is it that the particles inside each member can interact within each member such that the member energies E'1, E'2, E'3 represent member energies after allowing intra member have interaction

## Answers and Replies

any help please..? before this gets locked! i want know what am i getting wrong.. how can you give energy to a canonical ensemble without changing its temperature?

Timo
Since you are asking for any help: The elements (states) in the canonical ensemble do not all have the same energy. That should explain how energy exchange is possible, where "energy exchange" in Atkins presumably means small fluctuations (didn't read all of your thread or the related Atkins pages). Beyond that, there are processes where even large quantities of energy can be transferred from and to a system without increasing temperature. The best-known process (and the only one I can think of at the moment) is the latent heat at first-order phase transitions (e.g. evaporation of water).