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Stat Mech : Particles in Box sort of

  1. Sep 8, 2008 #1
    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.
    [tex]
    \Omega _1=v^n;\text{ }\text{(*Dist for n in v*)}[/tex]
    [tex]
    \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{*)}[/tex]
    [tex]
    \Omega [\text{v$\_$},\text{n$\_$}]=\alpha \Omega _1 \Omega _2;\text{ }\text{(*}\text{Total}\text{*)}[/tex]
    [tex]
    \text{nCr}[\text{N$\_$},\text{n$\_$}] = N!/(n!(N-n)!);\text{ }\text{(*}\text{Possible} \text{Comb}. \text{of} \text{choosing} n/N\text{*)}
    [/tex]

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

    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?
     
  2. jcsd
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