Elementary Boltzmann Statistics

[Phyisab****]

***SOLVED***

Homework Statement

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

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!

 

[ideasrule]

Seems right to me.

 

[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.

 

[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.

 

[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 separate one atom partion functions." Thanks for your help again.

 

[ideasrule]

Oh, the partition function for N particles is a function in classical mechanics. It is the second definition here:

http://en.wikipedia.org/wiki/Partition_function_(statistical_mechanics)#Definition

and is different from the partition function in this question.

 

[Phyisab****]

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