# Linear degree of freedom - Equipartition theorem

## Homework Statement

Consider a classical 'degree of freedom' that is linear rather than quadratic: E = c|q| for some constant c. Derive the equipartition theorem using this energy and show that the average energy is Ebar = kT.

## Homework Equations

$Z = \sum e^{-\beta E(q)} = \sum e^{-\beta c|q|}$

$Z = \frac{1}{\Delta q} \int_{-\infty}^{+\infty} e^{-\beta c |q|}dq$

## The Attempt at a Solution

The question seems straight forward, but I'm having a hard time grasping it.

Using the second equation, If I carry out that integral I get:

$\frac{1}{\Delta q} \frac {-1}{\beta c} \left [ e^{-\beta cq} \right ]_{-\infty}^{+\infty} = 0$

Which doesn't help at all. I'm not sure if there is a trick to the integral or I have to use another method.

Any help will be much appreciated.

PS. This is coursework but not a homework question. I am just doing this question to study for a test.