Derivation of Grand Canonical Ensemble from scratch?

[tim_lou]

I've been studying and thinking about statistical physics for a couple days now... and what bothers me is the grand canonical partition function. Namely that for a system with fixed chemical potential and energy [itex]\epsilon_i[/itex] the probability of having [itex]N_i[/itex] particle in that state is proportional to:

[tex]P\sim \exp[-\beta(\epsilon_i -\mu N_i)][/tex]

I completely understand the derivation from book, basically it invokes the thermodynamic identity. (the dS= bla bla and what not)

However, I really want to view statistical physics in a whole different view point. I want to be able to derive P, V, S, E, and chemical potential using basic combinatoric argument and prove that the chemical potential is indeed G/N (for non-interacting particles).

I've read some other books and understand where the Boltzmann statistics comes in. Basically, the books use combinatorics to show that for classical particles, the multiplicity is:
[tex]\Omega=\prod_i \frac{g_i^{N_i}}{N_i!}[/tex]

and from that, and a couple energy and particles constrain, Boltzmann statistics naturally comes in. However, is there a similar way to proceed for the grand canonical partition function?

Let me make this question precise:

Suppose that we have the universe at a fixed temperature, T, and there is only one species of particle and the university has N of these particles (N is not fixed). Further suppose that there are fixed number of states of energy, and for each energy [itex]\epsilon_i[/tex] we have degeneracy [itex]g_i[/itex] (both are fixed constants).

suppose the multiplicity is given by:
[tex]\Omega=\prod_i \frac{g_i^{N_i}}{N_i!}[/tex]

we look at the sub system (of the universe) that has energy [itex]\epsilon_i[/itex], what is the probability that this system has [tex]N_i[/tex] particles? how would the chemical potential come in?

I know that the energy of the universe must be constant:
[tex]E=\sum_i \epsilon_i N_i[/tex]

Do I need further constrains on the problem or what?

 

[marcusl]

For an alternate derivation of the Grand Canonical Assembly from the viewpoint of information theory and inference, including derivation of the quantities you are interested in, see

Jaynes, E. T., 1957, `Information Theory and Statistical Mechanics,' Phys. Rev., 106, 620.

http://bayes.wustl.edu/etj/articles/theory.1.pdf"

 

[tim_lou]

Thank you for the reply, marcusl, though I did not understand a lot of the things in the article you posted. However, I have found some answers from some statistical physics books I found.

the answer is very simple... basically instead of partition the whole universe under one parameter, one partitions the universe using [itex]\epsilon_i[/itex] and [itex]N_i[/tex]... and the result follows naturally.