Page:Elementary Principles in Statistical Mechanics (1902).djvu/78

the fractional part of the ensemble which lies within any given limits of configuration (136) may be written

$\int \ldots \int e^{\frac {\psi _{q}-\epsilon _{q}}{\Theta }}\,\Delta _{\dot {q}}{}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n},$

(142)

where the constant $\psi _{q}$ may be determined by the condition that the integral extended over all configurations has the value unity.^[1]

1. In the simple but important case in which $\Delta _{\dot {q}}$ is independent of the $q$'s, and $\epsilon _{q}$ a quadratic function of the $q$'s, if we write $\epsilon _{a}$ for the least value of $\epsilon _{q}$ (or of $\epsilon$) consistent with the given values of the external coördinates, the equation determining $\psi _{q}$ may be written

   $e^{\frac {\epsilon _{a}-\psi _{q}}{\Theta }}=\Delta _{\dot {q}}{}^{\frac {1}{2}}\int \limits _{-\infty }^{+\infty }\ldots \int \limits _{-\infty }^{+\infty }e^{-{\frac {\epsilon _{q}-\epsilon _{a}}{\Theta }}}\,dq_{1}\ldots dq_{n}.$

   If we denote by $q_{1}{}',\ldots q_{n}{}'$ the values of $q_{1},\ldots q_{n}$ which give $\epsilon _{q}$ its least value $\epsilon _{a}$, it is evident that $\epsilon _{q}-\epsilon _{a}$ is a homogenous quadratic function of the differences $q_{1}-q_{1}{}'$, etc., and that $dq_{1},\ldots dq_{n}$ may be regarded as the differentials of these differences. The evaluation of this integral is therefore analytically similar to that of the integral

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

   for which we have found the value $\Delta _{p}{}^{-{\frac {1}{2}}}(2\pi \Theta )^{\frac {n}{2}}$. By the same method, or by analogy, we get

   $e^{\frac {\epsilon _{a}-\psi _{q}}{\Theta }}={\bigg (}{\frac {\Delta _{\dot {q}}}{\Delta _{q}}}{\bigg )}^{\frac {1}{2}}(2\pi \Theta )^{\frac {n}{2}},$

   where $\Delta _{q}$ is the Hessian of the potential energy as function of the $q$'s. It will be observed that $\Delta _{q}$ depends on the forces of the system and is independent of the masses, while $\Delta _{\dot {q}}$ or its reciprocal $\Delta _{p}$ depends on the masses and is independent of the forces. While each Hessian depends on the system of coördinates employed, the ratio $\Delta _{q}/\Delta _{\dot {q}}$ is the same for all systems.

   Multiplying the last equation by (140), we have

   $e^{\frac {\epsilon _{a}-\psi }{\Theta }}={\bigg (}{\frac {\Delta _{\dot {q}}}{\Delta _{q}}}{\bigg )}^{\frac {1}{2}}(2\pi \Theta )^{n}.$

   For the average value of the potential energy, we have

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