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}+ax^{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}kx^{2}+ax^{4}.

In this case the anharmonic contribution ax^{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).