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Proof of the general form of the equipartition theorem

  1. Jan 28, 2013 #1
    1. The problem statement, all variables and given/known data

    For a Hamiltonian of the form [itex]H=\sum_{j=1}^m a_j p_j^2 + \sum_{j=1}^n b_j q_j^2 + H'(q_{n+1}, \dots, q_{sN}, p_{m+1}, \dots, p_{sN})[/itex] (for a system with n coordinates and m momenta, s degrees of freedom and N particles), show that [itex]\overline{a_jp_j^2}=\frac{kT}{2}[/itex], for j=1,...,m and [itex]\overline{b_jq_j^2}=\frac{kT}{2}[/itex], for j=1,...,n.

    2. Relevant equations

    [itex]P(q,p)=\frac{exp[-\beta H(q,p)]}{Z_N}[/itex]
    [itex]Z_N=\frac{1}{N!h^{sN}}\int \int exp[-\beta H(q,p)]\,d^{sN}q\,d^{sN}p[/itex]
    [itex]P_x(x_1,...,x_N)dx_1...dx_n = P_y(y_1(x_1,...,x_N),...,y_N(x_1,...,x_N)) \left|J\left(\frac{y_1,...,y_N}{x_1,...,x_N}\right)\right| dx_1...dx_N[/itex]

    Note that [itex]\beta=\frac{1}{kT}[/itex].

    3. The attempt at a solution

    I honestly don't really know where to go with this... I've tried plugging stuff in, but I'm really at a loss. Any help would be greatly appreciated. Thanks so much in advance!
     
  2. jcsd
  3. Jan 28, 2013 #2

    TSny

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    Hello, Homo Novus.
    Can you elaborate on what you did when you were plugging stuff in?
     
  4. Jan 28, 2013 #3
    I tried messing around with the formulas, but honestly I don't really know what I'm doing. I tried to find [itex]P(a_jp_j^2,b_jq_j^2)[/itex] from [itex]P(q,p)[/itex], but I don't know what to do with the [itex]d^{sN}q\,d^{sN}p[/itex] left over... and then, assuming I do manage to find the correct distribution formula, I don't know how to find the average.
     
  5. Jan 28, 2013 #4

    TSny

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    Suppose you have a probability distribution function ##p(x)## for some continuous variable x. And suppose you have some function of x, ##f(x)##. Do you know how to find the average value of ##f(x)##: ##\overline{f(x)}##? [Edit: In particular, how would you find ##\overline{x^2}##?]
     
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