H problem in statistical mechanics

[kof9595995]

If we are dealing with a canonical ensemble, the extensive variables are S and V; the thermodynamic limit corresponds to holding the conjugate variables T and p constant and letting S and V tend to infinity. The particle number N, is held fixed by definition. If we use a grand canonical ensemble, N is also extensive, and the thing which is held constant is the chemical potential \mu. Your statement about holding the specific volume (i.e. density) constant is equivalent to holding \mu constant in the non-interacting limit, and approximate the same in the weakly interacting limit.

I'm not really familiar with these, because all discussions about thermodynamic limit I've read are based on grand canonical ensemble, do you have any online resource about it so that I could read in more details?

 

[genneth]

I'm not really familiar with these, because all discussions about thermodynamic limit I've read are based on grand canonical ensemble, do you have any online resource about it so that I could read in more details?

I'm afraid nothing really comes to mind at the moment. This is the sort of detail which tends to develop in people independently with time (relative to when they first learn statistical mechanics), because the usual textbook path through the topic will avoid it. A suggestion, which I learned from our local Soviet expert, is Landau and Lifschitz. The Soviet era books tended to have been written completely independently and are therefore most likely to provide orthogonal information to the usual textbooks. Didn't someone in this thread already mention that this pathology of hydrogen atom statistical mechanics is discussed in somewhere Russian?

 

[DrDu]

Yes, I recommended Landau Lifshetz too. I think the divergence of S and Z is unavoidable for infinite volume limit.
The thermodynamic limit is taken as V to infinity at fixed particle density.
Total energy and entropy diverge in that limit but energy density and entropy density approach a constant value.
However, the case of a single atom in an infinite volume is still special as it corresponds to vanishing energy and particle density.

 

[kof9595995]

Ok, thank you guys, I'll have a look at it.