Page:Elementary Principles in Statistical Mechanics (1902).djvu/76

the $u$'s. The integrals may always be taken from a less to a greater value of a $u$.

The general integral which expresses the fractional part of the ensemble which falls within any given limits of phase is thus reduced to the form

$\int \ldots \int e^{\frac {\psi -\epsilon _{q}}{\Theta }}{\frac {d(p_{1}\ldots p_{n})}{d(u_{1}\ldots u_{n})}}e^{-{\frac {u_{1}{}^{2}\ldots u_{n}{}^{2}}{2\Theta }}}\,du_{1}\ldots du_{n}\,dq_{1}\ldots dq_{n}.$

(134)

For the average value of the part of the kinetic energy which is represented by ${\tfrac {1}{2}}u_{1}{}^{2}$ whether the average is taken for the whole ensemble, or for a given configuration, we have therefore

${\tfrac {1}{2}}{\overline {u}}_{1}{}^{2}={\frac {\int _{-\infty }^{+\infty }{\tfrac {1}{2}}u_{1}{}^{2}e^{-{\frac {u_{1}{}^{2}}{2\Theta }}}\,du_{1}}{\int _{-\infty }^{+\infty }e^{-{\frac {u_{1}{}^{2}}{2\Theta }}}\,du_{1}}}={\frac {({\tfrac {1}{2}}\pi \Theta ^{3})^{\frac {1}{2}}}{(2\pi \Theta )^{\frac {1}{2}}}}={\frac {\Theta }{2}},$

(135)

and for the average of the whole kinetic energy, ${\tfrac {1}{2}}n\Theta$, as before.

The fractional part of the ensemble which lies within any given limits of configuration, is found by integrating (184) with respect to the $u$'s from $-\infty$ to $+\infty$. This gives

$(2\pi \Theta )^{\frac {n}{2}}\int \ldots \int e^{\frac {\psi -\epsilon _{q}}{\Theta }}{\frac {d(p_{1}\ldots p_{n})}{d(u_{1}\ldots u_{n})}}\,dq_{1}\ldots dq_{n}$

(136)

which shows that the value of the Jacobian is independent of the manner in which $2\epsilon _{p}$ is divided into a sum of squares. We may verify this directly, and at the same time obtain a more convenient expression for the Jacobian, as follows. It will be observed that since the $u$'s are linear functions of the $p$'s, and the $p$'s linear functions of the ${\dot {q}}$'s, the $u$'s will be linear functions of the ${\dot {q}}$'s, so that a differential coefficient of the form $du/d{\dot {q}}$ will be independent of the ${\dot {q}}$'s, and function of the $q$'s alone. Let us write $dp_{x}/du_{y}$ for the general element of the Jacobian determinant. We have