Fall 2007 17 Sm

A system is composed of N identical classical oscillators, each of mass m, defined on a one-dimensional lattice. The potential for the oscillators has the form

\begin{align} U(x)=\epsilon \left|x/a\right|^n,\;\; \epsilon > 0, \; n>0. \end{align}

(Thus, the oscillators are harmonic for n = 2 and anharmonic otherwise).
Find the average thermal energy at temperature T.

Hint: An integral that appears in the course of evaluating the partition function cannot be computed in terms of elementary functions. Fortunately,it amounts only to an unimportant overall coefficient.




Classical partition function for a single oscillator is

\begin{align} \zeta=\int_{-\infty}^{\infty}dp\exp{\left(-\beta p^2/2m \right)}\int_{-\infty}^{\infty}dx\exp{\left( -\beta\epsilon\left|x/a \right|^n \right)} \end{align}

By change of variables, we bring this to the form

\begin{align} \zeta=\Gamma(1/2)\Gamma(1/n)\frac{2a}{n}\left(\frac{2m}{\beta} \right)^{1/2}\left( \frac{1}{\beta\epsilon}\right)^{1/n} \end{align}


\begin{align} \Gamma(z)\equiv\int_{0}^{\infty}dt\; t^{z-1}\exp{(-t)}. \end{align}

A learned reader would recognize this as the Euler Gamma-function. However, knowing this is not necessary. The product $\Gamma(1/2)\Gamma(1/n)$ is just a numerical coefficient, which will disappear from the final result.

The average energy is given by

\begin{align} E=-N\frac{\partial}{\partial \beta}\ln{\zeta}=\left(\frac{n+2}{2n}\right)N kT. \end{align}

This result resembles the equipartition theorem in the sense that material constants do not enter.

Add a New Comment
or Sign in as Wikidot user
(will not be published)
- +
Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-Share Alike 2.5 License.