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

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

(142)

${\displaystyle \psi _{q}}$

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

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

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

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

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

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

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

   Multiplying the last equation by (140), we have

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

   For the average value of the potential energy, we have

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