Page:Elementary Principles in Statistical Mechanics (1902).djvu/86

This page has been proofread, but needs to be validated.

62

EXTENSION IN CONFIGURATION

within certain infinitesimal limits of velocity. The second of these numbers divided by the first expresses the probability that a system which is only specified as falling within the infinitesimal limits of configuration shall also fall within the infinitesimal limits of velocity. If the limits with respect to velocity are expressed by the condition that the momenta shall fall between the limits $p_{1}$ and $p_{1}+dp_{1}$ ,... $p_{n}$ and $p_{n}+dp_{n}$ the extension-in-velocity within those limits will be

\Delta _{p}{}^{\frac {1}{2}}\,dp_{1}\ldots dp_{n},

and we may express the probability in question by

e^{\eta _{p}}\Delta _{p}{}^{\frac {1}{2}}\,dp_{1}\ldots dp_{n}.

(162)

This may be regarded as defining

\eta _{p}

.

The probability that a system which is only specified as having a configuration within certain infinitesimal limits shall also fall within any given limits of velocity will be expressed by the multiple integral

\int \ldots \int e^{\eta _{p}}\Delta _{p}{}^{\frac {1}{2}}\,dp_{1}\ldots dp_{n},

(163)

or its equivalent

\int \ldots \int e^{\eta _{p}}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,d{\dot {q}}_{1}\ldots d{\dot {q}}_{n},

(164)

taken within the given limits.

It follows that the probability that the system will fall within the limits of velocity, ${\dot {q}}_{1}$ and ${\dot {q}}_{1}+d{\dot {q}}_{1}$ ,... ${\dot {q}}_{n}$ and ${\dot {q}}_{n}+d{\dot {q}}_{n}$ is expressed by

e^{\eta _{p}}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,d{\dot {q}}_{1}\ldots d{\dot {q}}_{n}.

(165)

The value of the integrals (163), (164) is independent of the system of coördinates and momenta which is used, as is also the value of the same integrals without the factor $e^{\eta _{p}}$ ; therefore the value of $\eta _{p}$ must be independent of the system of coördinates and momenta. We may call $e^{\eta _{p}}$ the coefficient of probability of velocity, and $\eta _{p}$ the index of probability of velocity.