Page:Elementary Principles in Statistical Mechanics (1902).djvu/71

the multiple integral may be resolved into the product of integrals

$\int \ldots \int e^{\frac {\psi -\epsilon _{q}}{\Theta }}d{\dot {x}}_{1}\ldots dz_{\nu }\int e^{-{\frac {m_{1}{\dot {x}}_{1}{}^{2}}{2\Theta }}}m_{1}\,d{\dot {x}}_{1}\ldots \int e^{-{\frac {m_{1}{\dot {z}}_{n}u{}^{2}}{2\Theta }}}m_{1}\,d{\dot {z}}_{\nu }.$

(121)

This shows that the probability that the configuration lies within any given limits is independent of the velocities, and that the probability that any component velocity lies within any given limits is independent of the other component velocities and of the configuration.

Since

$\int _{-\infty }^{+\infty }e^{-{\frac {m_{1}{\dot {x}}_{1}{}^{2}}{2\Theta }}}m_{1}\,d{\dot {x}}_{1}={\sqrt {2\pi m_{1}\Theta }},$

(122)

and

$\int _{-\infty }^{+\infty }m_{1}{\dot {x}}_{1}{}^{2}e^{-{\frac {m_{1}{\dot {x}}_{1}{}^{2}}{2\Theta }}}m_{1}\,d{\dot {x}}_{1}={\sqrt {{\tfrac {1}{2}}\pi m_{1}\Theta ^{3}}},$

(123)

the average value of the part of the kinetic energy due to the velocity ${\dot {x}}_{1}$, which is expressed by the quotient of these integrals, is ${\tfrac {1}{2}}\Theta$. This is true whether the average is taken for the whole ensemble or for any particular configuration, whether it is taken without reference to the other component velocities, or only those systems are considered in which the other component velocities have particular values or lie within specified limits.

The number of coördinates is $3\nu$ or $n$. We have, therefore, for the average value of the kinetic energy of a system

${\overline {\epsilon }}_{p}={\tfrac {3}{2}}\nu \Theta ={\tfrac {1}{2}}n\Theta .$

(124)

This is equally true whether we take the average for the whole ensemble, or limit the average to a single configuration. The distribution of the systems with respect to their component velocities follows the 'law of errors'; the probability that the value of any component velocity lies within any given limits being represented by the value of the corresponding integral in (121) for those limits, divided by $(2\pi m\Theta )^{\frac {1}{2}}$,