# Statistical mechanics - Partition function of a system of N particles

1. Jan 2, 2013

### Jalo

1. The problem statement, all variables and given/known data

Imagine a system with N distinguishable particles. Each particle may be in two states of energy: -ε and +ε.

Find the the partition function of the system

2. Relevant equations

3. The attempt at a solution

I know that I have to find the partition function for a single function, Z, and my final result will be ZN. Now, I'll say that:

(Where it says ε it's meant to be ε(r) )

Z = Ʃr exp(-β(ε - ε) ) = Ʃr exp(-βε) * exp(βε) =
= Ʃr exp(-βε) * Ʃr exp(βε)

I'm sure this is incorrect. It doesn't make sense in my head.. E(r) is the energy associated with each microstate, therefore saying that E(r) = ε(r) - ε(r) can't make any sense! I know that the result is:

Z = ( exp(βε) + exp(-βε) )N

I have no idea how to get there tho. How did it became a sum? How do I get rid of the summatories?

Any help will be appreciated!
Thanks.

2. Jan 2, 2013

### klawlor419

The partition function is a summation over states. You simply are using the summation wrong. It is not a summation over the energy levels of within the exponent. It is a summation over e(-Es/T).

3. Jan 2, 2013

### klawlor419

Look at any example problem in a thermo book for a 2-state system

4. Jan 3, 2013

### Jalo

8ikmAm I not summing over the expoent of the energy of each microstate?

Do you know any good statistical mechanics book you'd advise me reading?

EDIT:
Is it a summation over all the states of energy instead of the energies of each microstate? Because then the solution would make sense!

Last edited: Jan 3, 2013
5. Jan 3, 2013

### klawlor419

No. You are summing over the exponential function raised to the -Es/T.

A good book is Thermal Physics by Kittel + Kroemer

6. Jan 3, 2013

### klawlor419

Where Es is the energy of the s-th state