# Proof of the general form of the equipartition theorem

1. Jan 28, 2013

### Homo Novus

1. The problem statement, all variables and given/known data

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

2. Relevant equations

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

Note that $\beta=\frac{1}{kT}$.

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. Jan 28, 2013

### TSny

Hello, Homo Novus.
Can you elaborate on what you did when you were plugging stuff in?

3. Jan 28, 2013

### Homo Novus

I tried messing around with the formulas, but honestly I don't really know what I'm doing. I tried to find $P(a_jp_j^2,b_jq_j^2)$ from $P(q,p)$, but I don't know what to do with the $d^{sN}q\,d^{sN}p$ left over... and then, assuming I do manage to find the correct distribution formula, I don't know how to find the average.

4. Jan 28, 2013

### TSny

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 $\overline{x^2}$?]