# Stat Mech : Particles in Box sort of

1. Sep 8, 2008

### K.J.Healey

1. The problem statement, all variables and given/known data
I don't want to do it word for word, but its like this:
Statistical Mechanics is the topic.

Given a volume V and N particles within it.
Assume no correlation in the location of the particles.

1. Find the probability that a region of volume "v" contains "n" exactly "n" particles.
2. Show for large N>>n its Poisson Dist.
3. Show otherwise (1 < <n> < N) its Gaussian.

2. Relevant equations

3. The attempt at a solution
I'm using mathematica to try to do this, so perhaps itll make it easier.
$$\Omega _1=v^n;\text{ }\text{(*Dist for n in v*)}$$
$$\Omega _2=(V-v)^{\text{NN}-n};\text{ }\text{(*}\text{Dist} \text{for} n \text{NOT} \text{in} v, \text{but} \text{in} V-v\text{*)}$$
$$\Omega [\text{v\_},\text{n\_}]=\alpha \Omega _1 \Omega _2;\text{ }\text{(*}\text{Total}\text{*)}$$
$$\text{nCr}[\text{N\_},\text{n\_}] = N!/(n!(N-n)!);\text{ }\text{(*}\text{Possible} \text{Comb}. \text{of} \text{choosing} n/N\text{*)}$$

Now IIRC the probability should be:
$$P[\text{v\_},\text{n\_}]=\sum _{n=0}^{\text{NN}} \Omega [v,n]\text{nCr}[\text{NN},n]$$

Where I sum over all possible combinations of states, and the robabilities of being in those states. (ignoring the proportionality constant)

Now so my first question as I work on this is, should I let Mathematica perform the sum? If it does I get an exact result, which then will not lead to a way to turn a sum of x^n/n! into exponential when I need to approximate N>>n to get the poisson distribution.

Does anyone have any insight at all?