Connecting the microcanonical with the (grand) canonical ensemble

[vanhees71]

Also answering vanHees!

Maybe one could start with chapter 3 in Reif and the loop back to the first 2 chapters in case that's necessary.
How do we do this? Just start a thread and hope for interest?

Which Reif? The Berkeley physics course one or the more advanced standalone textbook? I'm fine with both.

 

[vanhees71]

I certainly use Stirling's formula a lot, so all my results can only be valid for large N. I agree completely.

One big problem occurs in section 6 when I try to specify what the chemical potential and the density of states actually is for an ideal gas. That's where I switch to an integral with an infinite upper limit although the total energy is finite. I guess that's related to what you point out.
I really only use an infinite upper limit so that I can evaluate the integrals for total energy and particle number.
Until section 6 I only consider a finite set of discrete energy levels, which ought to mean that I can keep the highest energy level below the total energy simply by definition.
I simply shouldn't try to apply the theory to anything real! :)

That's one of the subtler points of stat. mech. The "thermodynamic limit" is not as simple as it looks!

 

[Philip Koeck]

Which Reif? The Berkeley physics course one or the more advanced standalone textbook? I'm fine with both.

I would say the Berkeley one.

 

[vanhees71]

That's a good choice. The other book is very detailed and sometimes you can get lost in these details, though it's a very good source if you want to study the issues in more depth from different points of view.

 

[Philip Koeck]

That's a good choice. The other book is very detailed and sometimes you can get lost in these details, though it's a very good source if you want to study the issues in more depth from different points of view.

How do we do this? Should I just start a thread called "Reading Reif together" or something like that, or should this come from an advisor?

 

[vanhees71]

I don't know. I'd say, just post such an "intro posting" and see what happens.

 

[Philip Koeck]

I don't know. I'd say, just post such an "intro posting" and see what happens.

It's started.

 

[Demystifier]

Then I'll go a step further and decide that the system should consist of a single atom.

It doesn't matter whether the number of particles is $N=1$ or $N=10^{23}$. One has to distinguish the number of particles in the system (which doesn't need to be large), from the number of systems in the ensemble (which must be large). In the Gibbs theory of ensembles, an ensemble is a fictitious ensemble; it's a thing we imagine to conceptualize the notion of probability. For example, you can say that the probability of a single coin to be in the heads state is 1/2, but to conceptualize this you can imagine that you flipped this single coin a 1000 times.

That being said, once you said that $N$ is fixed ($N=1$ in your case), you can talk about a canonical ensemble, but not about a grand-canonical ensemble.