The equipartition law of classical thermodynamics

5.9 The equipartition law of classical thermodynamics

We now will investigate systems for which the one particle energy is written as

\begin{equation*} E(\vec{\xi}) = \sum_{i,j=1}^{s} a_{ij}\xi_i\xi_j \qquad \mbox{(bilinear form)} \qquad . \label{Eq:bilin_energy} \end{equation*}

(5.73)

This function is homogeneous of second order, i.e.

\begin{equation*} \sum_{j=1}^{s} \xi_j \frac{\partial E}{\partial \xi_j} = 2 E \qquad . \end{equation*}

(5.74)

Nearly all types of kinetic energy and many types of potential energy like that of a harmonic oscillator are written as a bilinear form of Eq. (5.73), i.e. are homogeneous of second order.
For classical particles the Boltzmann approximation holds:

\begin{equation*} f(E,T) = \frac{\exp\left(-\frac{E}{k T} \right)}{Z} \qquad . \end{equation*}

(5.75)

Thus we find for the inner energy

\begin{equation*} U = \left\langle E\right\rangle = \iiint E(\vec{\xi}) \frac{\exp\left(-\frac{E(\vec{\xi})}{k T} \right)}{Z} d\xi_1 ... d\xi_s = \frac{1}{2} \sum_{j=1}^s \iiint \xi_j \frac{\partial E(\vec{\xi})}{\partial \xi_j} \frac{\exp\left(-\frac{E(\vec{\xi})}{k T} \right)}{Z} d\xi_1 ... d\xi_s \qquad . \end{equation*}

(5.76)

For the norm we find:

\begin{equation*} 1 = \iiint \frac{\exp\left(-\frac{E(\vec{\xi})}{k T} \right)}{Z} d\xi_1 ... d\xi_s \qquad . \end{equation*}

(5.77)

Therefore we get finally

\begin{equation*} U = \frac{-k T}{2 Z} \sum_{j=1}^{s} \iiint \xi_j \frac{\partial}{\partial \xi_j} \left[ \exp\left(-\frac{E}{k T} \right)\right] d\xi_1 \cdots d\xi_s \qquad , \end{equation*}

(5.78)

and after partial integration:

\begin{equation*} U = \frac{-k T}{2 Z} \sum_{j=1}^{s}\left[\xi_j \exp\left(-\frac{E(\vec{\xi})}{k T}\right)_{boundaries} - \iiint \exp\left(-\frac{E(\vec{\xi})}{k T} \right) d\xi_1 \cdots d\xi_s \right]\qquad . \end{equation*}

(5.79)

The first term vanishes at the boundaries, the second one is the partition function; thus we find

\begin{equation*} U = \frac{k T}{2} s \qquad . \end{equation*}

(5.80)

Independent of the special form of the energy function each degree of freedom adds \(0.5 kT\) to the inner energy of the system.
The specific heat capacity is

\begin{equation*} C = \frac{k}{2} s \qquad , \end{equation*}

(5.81)

independent of the temperature. This is the famous equipartition law of classical thermodynamics.