# Homework Help: Elementary Boltzmann Statistics

1. Jan 17, 2010

### Phyisab****

***SOLVED***

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

A system of particles is at equilibrium. What is the proportion of particles in the ground state?

3. The attempt at a solution

$$P(\epsilon_{s})=\frac{exp(-0/\tau)}{1+exp(\frac{-\epsilon_{s}}{\tau})+exp(\frac{-2\epsilon_{s}}{\tau})}$$

Is that right? This is a really basic problem but I haven't dealt with this in a while. Thanks!

Last edited: Jan 17, 2010
2. Jan 17, 2010

### ideasrule

Seems right to me.

3. Jan 17, 2010

### Phyisab****

So what threw me for a little bit of a loop about this was that the denominator of the equation I posted is the partition function.
$$Z_{1}= 1+exp(\frac{-\epsilon_{s}}{\tau})+exp(\frac{-2\epsilon_{s}}{\tau})$$

Then later I am asked what the partition function is. I guess the point is that Z above is the partition function for a single particle, not the whole system which would be

$$Z_{1}^{N}$$

does that seem right? Edit: The particles are not indistinguishable.

4. Jan 17, 2010

### ideasrule

I don't quite understand what you're saying. The partition function is exactly what you posted. You seem to think that there's another partition function for the whole system, which isn't the case.

5. Jan 17, 2010

### Phyisab****

Well then we are both confused :). I am asked

1. what is the partition function of a single particle?
2. what is the partition function for N particles?

Thanks for your time I really appreciate it.

Edit: Quoting Kittel & Kroemer, "If we have one atom in each of N distinct boxes (which is equivalent to N in one box, because they are non-interacting I assume), the partition function is the product of the seperate one atom partion functions." Thanks for your help again.

Last edited: Jan 17, 2010
6. Jan 17, 2010

### ideasrule

7. Jan 17, 2010

### Phyisab****

So in other words for N particles we are talking about the grand canonical partition function.