Statistical thermodynamics - mean energy of a nonlinear oscillator

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Homework Statement



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

where x,p, and m have their usual meanings; the paramater, 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 [itex]\epsilon[/itex]= [itex]\frac{p^{2}}{2m}[/itex] + [itex]\frac{1}{2}[/itex]k[itex]x^{2}[/itex]+a[itex]x^{4}[/itex].

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

Answers and Replies

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vanhees71
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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 [itex]x^4[/itex] are not doable in closed form with elementary functions).
 

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