Classical harmonic oscillator in the canonical ensemble

Phase-space partition function and thermodynamics of a classical harmonic oscillator; a standard illustration of equipartition.

Classical harmonic oscillator in the canonical ensemble

Model

Consider a one-dimensional classical harmonic oscillator with coordinate $q\in\mathbb{R}$ and momentum $p\in\mathbb{R}$ , with Hamiltonian

H(p,q)=\frac{p^2}{2m}+\frac12 m\omega^2 q^2 .

We place the system in the canonical ensemble at inverse temperature $\beta=1/(k_B T)$ .

Using the classical phase-space definition of the canonical partition function , with the usual $1/h$ normalization,

Z_1(\beta)=\frac{1}{h}\int_{\mathbb{R}}\int_{\mathbb{R}} e^{-\beta H(p,q)}\,dp\,dq .

The Gaussian integrals factor:

\int_{\mathbb{R}} e^{-\beta p^2/(2m)}\,dp=\sqrt{\frac{2\pi m}{\beta}},\qquad \int_{\mathbb{R}} e^{-\beta m\omega^2 q^2/2}\,dq=\sqrt{\frac{2\pi}{\beta m\omega^2}}.

Hence

Z_1(\beta)=\frac{1}{h}\,\frac{2\pi}{\beta\omega}=\frac{1}{\beta\hbar\omega},

using $h=2\pi\hbar$ . (Changing the phase-space normalization shifts $F$ and $S$ by constants, but does not change $U$ or $C_V$ .)

F(\beta)=-\beta^{-1}\ln Z_1(\beta)=k_B T\,\ln(\beta\hbar\omega).

U(\beta)=-\frac{\partial}{\partial\beta}\ln Z_1(\beta)=\frac{1}{\beta}=k_B T .

The thermodynamic entropy (canonical identity $S=k_B(\ln Z+\beta U)$ ) is

S(\beta)=k_B\bigl(\ln Z_1(\beta)+\beta U(\beta)\bigr)=k_B\Bigl(1-\ln(\beta\hbar\omega)\Bigr).

The heat capacity at constant volume (see heat capacity at constant volume ) is

C_V=\left(\frac{\partial U}{\partial T}\right)=k_B .

As ensemble averages under the canonical Gibbs density, equipartition yields

\Bigl\langle \frac{p^2}{2m}\Bigr\rangle=\frac12 k_B T,\qquad \Bigl\langle \frac12 m\omega^2 q^2\Bigr\rangle=\frac12 k_B T,

\langle p^2\rangle = m k_B T,\qquad \langle q^2\rangle = \frac{k_B T}{m\omega^2}.

For $N$ independent (distinguishable) classical oscillators of the same frequency,

Z_N(\beta)=\bigl(Z_1(\beta)\bigr)^N,\qquad U_N = N k_B T,\qquad C_V = N k_B.