# Specific heat of solid of one dimensional quartic oscillators

## Homework Statement

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.

## Homework Equations

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.

## Answers and Replies

gabbagabbahey
Homework Helper
Gold Member
Try the substitution $t=\frac{\beta q x^4}{4}$ and make use of the gamma function:

$$\Gamma(z)\equiv \int_0^\infty t^{z-1}e^{-t}dt$$