- #1
tim_lou
- 682
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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 \epsilon_i the probability of having N_i particle in that state is proportional to:
P\sim \exp[-\beta(\epsilon_i -\mu N_i)]
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:
\Omega=\prod_i \frac{g_i^{N_i}}{N_i!}
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 \epsilon_i[/tex] we have degeneracy g_i (both are fixed constants).<br /> <br /> suppose the multiplicity is given by:<br /> \Omega=\prod_i \frac{g_i^{N_i}}{N_i!}<br /> <br /> we look at the sub system (of the universe) that has energy \epsilon_i, what is the probability that this system has N_i particles? how would the chemical potential come in?<br /> <br /> I know that the energy of the universe must be constant:<br /> E=\sum_i \epsilon_i N_i<br /> <br /> Do I need further constrains on the problem or what?
P\sim \exp[-\beta(\epsilon_i -\mu N_i)]
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:
\Omega=\prod_i \frac{g_i^{N_i}}{N_i!}
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 \epsilon_i[/tex] we have degeneracy g_i (both are fixed constants).<br /> <br /> suppose the multiplicity is given by:<br /> \Omega=\prod_i \frac{g_i^{N_i}}{N_i!}<br /> <br /> we look at the sub system (of the universe) that has energy \epsilon_i, what is the probability that this system has N_i particles? how would the chemical potential come in?<br /> <br /> I know that the energy of the universe must be constant:<br /> E=\sum_i \epsilon_i N_i<br /> <br /> Do I need further constrains on the problem or what?
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