A system consists of N very weakly interacting particles at temperature T sufficiently high so that classical stat mech is applicable. Each particle has mass M and is free to perform one dimensional oscillations about its equilibrium position. Calculate the heat capacity of this system of particles if the restoring force is proportional to x^3.
spring constant = q
energy of one oscillator E = p^2 / 2m + qx^4 / 4
partition function: Z = integral ( exp(-BE ) dx dp
both integrals from -inf to +inf
where B = 1/kT
The Attempt at a Solution
Cv = N d/dT (average E)
average E = - d / dB [ ln Z ]
Z = integral [exp (-Bp^2/2m) dp ] * integral [exp (-Bqx^4/4) dx ]
the first integral is simply sqrt(pi * 2m / B )
I have no idea how to find the integral of exp(-x^4), so I can't find this partition function.