Proof of the general form of the equipartition theorem

[Homo Novus]

Homework Statement

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.

Homework 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 $\beta=\frac{1}{kT}$.

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!

 

[TSny]

Hello, Homo Novus.

The Attempt at a Solution

... I've tried plugging stuff in, but I'm really at a loss.

Can you elaborate on what you did when you were plugging stuff in?

 

[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.

 

[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 $\overline{x^2}$?]

 

[paranormal]

The equipartition theorem states that in thermal equilibrium, the average energy of each degree of freedom is equal to \frac{1}{2}kT, where k is the Boltzmann constant and T is the temperature.

To prove this for the given Hamiltonian, we start by expressing the partition function Z_N in terms of the Hamiltonian H:

Z_N=\frac{1}{N!h^{sN}}\int \int exp[-\beta H(q,p)]\,d^{sN}q\,d^{sN}p

Next, we use the definition of the average value of a quantity A:

\overline{A}=\frac{1}{Z_N}\int \int A(q,p)exp[-\beta H(q,p)]\,d^{sN}q\,d^{sN}p

Substituting the Hamiltonian given in the problem, we have:

\overline{a_jp_j^2}=\frac{1}{Z_N}\int \int a_jp_j^2 exp[-\beta (\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}))]\,d^{sN}q\,d^{sN}p

Using the property of exponential functions, we can expand the expression inside the integral as:

exp[-\beta (\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}))]=exp[-\beta \sum_{j=1}^m a_j p_j^2]exp[-\beta \sum_{j=1}^n b_j q_j^2]exp[-\beta H'(q_{n+1}, \dots, q_{sN}, p_{m+1}, \dots, p_{sN})]

Since we are only interested in the terms involving \overline{a_jp_j^2}, we can ignore the third exponential term and focus on the first two. Using the property of integr