Stat Mech : Particles in Box sort of

[K.J.Healey]

Homework Statement

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.

Homework Equations

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?

 

[Jhero]

Thank you for raising this interesting topic on statistical mechanics. I am a scientist with a background in physics and I would like to offer some insights on your questions.

To address your first question, whether you should let Mathematica perform the sum or not, I would suggest that you let it perform the sum and then compare the result with the expected theoretical result. This will not only help you check the validity of your calculations, but also provide a better understanding of the problem.

Moving on to your second question, let me clarify the concept of the Poisson distribution. It is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, if these events occur with a known average rate and independently of the time since the last event. In your problem, the number of particles in a given region of volume "v" follows a Poisson distribution when the number of particles is large (N>>n). This means that the probability of finding a certain number of particles in the region depends only on the average number of particles in the region and is independent of the specific configuration of the particles.

On the other hand, when the number of particles is not very large (1 < <n> < N), the distribution of particles follows a Gaussian distribution. This is because in this case, the particles are not distributed randomly and their positions are correlated, leading to a Gaussian distribution instead of a Poisson distribution.

I hope this helps in your understanding of the problem. Please let me know if you have any further questions. Good luck with your calculations!