Mean energy of system with ## E = \alpha |x|^n ##

[Dazed&Confused]

Homework Statement

If the energy $E$ of a system behaves like $E = \alpha |x|^n$, where $n =1, 2, 3, \dots$ and $\alpha > 0$, show that $\langle E \rangle = \xi k_B T$, where $\xi$ is a numerical constant.

Homework Equations

$$ \langle E \rangle = \frac{ \int_{- \infty}^{ \infty} Ee^{-\beta E}}{\int_{-\infty}^{ \infty} e^{-\beta E}},$$ where $\beta = \frac{1}{k_BT}.$

The Attempt at a Solution

Since the integral is even, it can be written as $$\frac{ \int_{0}^{ \infty} \alpha x^n e^{-\beta \alpha x^n}}{\int_{0}^{ \infty} e^{-\beta \alpha x^n}},$$

It can also be written as

$$ \frac{ \frac{ \partial}{ \partial \beta} \left ( \int_{0}^{ \infty} - e^{-\beta \alpha x^n} \right ) } {\int_{0}^{ \infty} e^{-\beta \alpha x^n}}$$

where the partial derivative was taken outside the integral.

I have no idea how to solve the integral. Mathematica didn't draw up anything useful.

 

[gleem]

Have you tried a change of variable e.g. y=xⁿ?

 

[Dazed&Confused]

I think I did try it, but it didn't look like it could result in anything useful.