Statistical thermodynamics - mean energy of a nonlinear oscillator

[galactic]

Homework Statement

Consider a classical one-dimensional nonlinear oscillator whose energy is given by $\epsilon$=$\frac{p^{2}}{2m}$+a$x^{4}$

where x,p, and m have their usual meanings; the parameter, a, is a constant

a) If the oscillator is in equilibrium with a heat bath at temperature T, calculate its mean kinetic energy, the mean potential energy, and mean total energy (it is not necessary to evaluate any integrals explicitly)

b) Consider a classical one-dimensional oscillator whose energy is given by $\epsilon$= $\frac{p^{2}}{2m}$ + $\frac{1}{2}$k$x^{2}$+a$x^{4}$.

In this case the anharmonic contribution a$x^{4}$ is very small. What is the leading contribution of this term to the mean potential energy? (Recall that for small u, $e^{u}$~ 1 + u

The Attempt at a Solution

This relates to information in Gould and Tobochnik Chapter 6 (statistical and thermal physics). I have no idea how to approach this problem, and any guidance or thought provoking questions to help me get started would be appreciated

 

[vanhees71]

First think about, what's the phase-space distribution function in thermal equilibrium! Then it's pretty easy to evalute the mean values (although the integrals for the potential energy for the $x^4$ are not doable in closed form with elementary functions).