Page:Elementary Principles in Statistical Mechanics (1902).djvu/70

CHAPTER V.

AVERAGE VALUES IN A CANONICAL ENSEMBLE OF SYSTEMS.

In the simple but important case of a system of material points, if we use rectangular coördinates, we have for the product of the differentials of the coördinates

$dx_{1}\,dy_{1}\,dz_{1}\ldots dx_{\nu }\,dy_{\nu }\,dz_{\nu },$

and for the product of the differentials of the momenta

$m_{1}\,d{\dot {x}}_{1}\,m_{1}\,d{\dot {y}}_{1}\,m_{1}\,d{\dot {z}}_{1}\ldots m_{\nu }\,d{\dot {x}}_{\nu }\,m_{\nu }\,d{\dot {y}}_{\nu }\,m_{\nu }\,d{\dot {z}}_{\nu }.$

The product of these expressions, which represents an element of extension-in-phase, may be briefly written

$m_{1}\,d{\dot {x}}_{1}\ldots m_{\nu }\,d{\dot {z}}_{\nu }\,dx_{1}\ldots dz_{\nu };$

and the integral

$\int \ldots \int e^{\frac {\psi -\epsilon }{\Theta }}m_{1}\,d{\dot {x}}_{1}\ldots m_{\nu }\,d{\dot {z}}_{\nu }\,dx_{1}\ldots dz_{\nu }$

(118)

will represent the probability that a system taken at random from an ensemble canonically distributed will fall within any given limits of phase.

In this case

$\epsilon =\epsilon _{q}+{\tfrac {1}{2}}m_{1}{\dot {x}}_{1}{}^{2}\ldots {\tfrac {1}{2}}m_{\nu }{\dot {z}}_{\nu }{}^{2},$

(119)

and

$e^{\frac {\psi -\epsilon }{\Theta }}=e^{\frac {\psi -\epsilon _{q}}{\Theta }}e^{-{\frac {m_{1}{\dot {x}}_{1}{}^{2}}{2\Theta }}}\ldots e^{-{\frac {m_{\nu }{\dot {z}}_{\nu }{}^{2}}{2\Theta }}}.$

(120)

The potential energy ($\epsilon _{q}$) is independent of the velocities, and if the limits of integration for the coördinates are independent of the velocities, and the limits of the several velocities are independent of each other as well as of the coördinates,