1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Statistical thermodynamics - mean energy of a nonlinear oscillator

  1. Apr 15, 2014 #1
    1. The problem statement, all variables and given/known data

    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

    3. 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
  2. jcsd
  3. Apr 16, 2014 #2


    User Avatar
    Science Advisor
    Gold Member
    2017 Award

    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).
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted