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

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1. Apr 21, 2015

### Dazed&Confused

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

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

3. 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.

2. Apr 21, 2015

### gleem

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

3. Apr 21, 2015

### Dazed&Confused

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