What is the Behavior of Particle Distribution Near Maximum in Thermal Physics?

[Gungator]

Homework Statement

There is a box of volume V that is divided into two equal parts: Left side = V/2 = right side.
The problem is:
Assume that there are n = (N_a/2) + k particles in the left hand partition where k is a small integer ( k << N_a). Show that the behavior near the maximum where n = N_a/2 is gaussian (i.e., quadratic in k).

Homework Equations

N_a = 6.022x10²³

This is what I don't know. I don't know if I'm supposed to use the binomial distribution or the gaussian(normal) distribution.

I'm in Thermal Physics because statistics was not a pre-req but we're using a lot of statistics and I've never had a course on it. So if anyone could help point me in the right direction of how to go about showing this, I would greatly appreciate it.

 

[AlexCdeP]

I feel like information is missing. I'm assuming that the particles in the box are ideal gas particles? If so I think you need to find the partition function. To give you a hint the partition function for an ideal gas is found using an integral (wikipedia it and hopefully you might have seen it before). If there are (Na/2)+k in the left hand part then there are (Na/2)-k in the right hand part. Each partition function is independent so if you found the partition function for either side, to get the total partition function for the whole box you could just multiply the two together. I haven't worked through it but I can see ((Na/2)+k)((Na/2)-k) is going to give you a quadratic.