Page:Elementary Principles in Statistical Mechanics (1902).djvu/79

When an ensemble of systems is distributed in configuration in the manner indicated in this formula, i. e., when its distribution in configuration is the same as that of an ensemble canonically distributed in phase, we shall say, without any reference to its velocities, that it is canonically distributed in configuration.

For any given configuration, the fractional part of the systems which lie within any given limits of velocity is represented by the quotient of the multiple integral

$\int \ldots \int e^{-{\frac {\epsilon _{p}}{\Theta }}}\,dp_{1}\ldots dp_{n},$

or its equivalent

$\int \ldots \int e^{\frac {-u_{1}{}^{2}\ldots -u_{n}{}^{2}}{2\Theta }}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,du_{1}\ldots du_{n},$

taken within those limits divided by the value of the same integral for the limits $\pm \infty$. But the value, of the second multiple integral for the limits $\pm \infty$ is evidently

$\Delta _{\dot {q}}^{\frac {1}{2}}(2\pi \Theta )^{\frac {n}{2}}.$

We may therefore write

$\int \ldots \int e^{\frac {\psi _{p}-\epsilon _{p}}{\Theta }}\,du_{1}\ldots du_{n},$

(143)

The evaluation of this expression is similar to that of

${\overline {\epsilon }}_{q}-\epsilon _{a}={\frac {\int \limits _{-\infty }^{+\infty }\ldots \int \limits _{-\infty }^{+\infty }\epsilon _{p}e^{-{\frac {\epsilon _{p}}{\Theta }}}\,dp_{1}\ldots dp_{n}}{\int \limits _{-\infty }^{+\infty }\ldots \int \limits _{-\infty }^{+\infty }e^{-{\frac {\epsilon _{p}}{\Theta }}}\,dp_{1}\ldots dp_{n}}}.$

which expresses the average value of the kinetic energy, and for which we have found the value ${\tfrac {1}{2}}n\Theta$. We have accordingly

${\overline {\epsilon }}_{q}-\epsilon _{a}={\frac {1}{2}}n\Theta .$

Adding the equation

${\overline {\epsilon }}_{p}={\frac {1}{2}}n\Theta ,$

we have

${\overline {\epsilon }}-\epsilon _{a}=n\Theta .$