Page:Elementary Principles in Statistical Mechanics (1902).djvu/32

will represent algebraically the decrease of the number of systems within the limits due to systems passing the limits $p_{1}'$ and $p_{1}''$.

The decrease in the number of systems within the limits due to systems passing the limits $q_{1}'$ and $q_{1}''$ may be found in the same way. This will give

${\bigg (}{\frac {d(D\,{\dot {p}}_{1})}{dp_{1}}}+{\frac {d(D\,{\dot {q}}_{1})}{dq_{1}}}{\bigg )}\,dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n}\,dt$

(15)

for the decrease due to passing the four limits $p_{1}'$, $p_{1}''$, $q_{1}'$, $q_{1}''$. But since the equations of motion (3) give

${\frac {d{\dot {p}}_{1}}{dp_{1}}}+{\frac {d{\dot {q}}_{1}}{dq_{1}}}=0,$

(16)

the expression reduces to

${\bigg (}{\frac {dD}{dp_{1}}}{\dot {p}}_{1}+{\frac {dD}{dq_{1}}}{\dot {q}}_{1}{\bigg )}\,dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n}\,dt$

(17)

If we prefix $\Sigma$ to denote summation relative to the suffixes $1\ldots n$, we get the total decrease in the number of systems within the limits in the time $dt$. That is,

${\begin{aligned}\Sigma {\Bigg (}{\frac {dD}{dp_{1}}}{\dot {p}}_{1}+{\frac {dD}{dq_{1}}}{\dot {q}}_{1}{\Bigg )}\,dp_{1}\ldots &dp_{n}\,dq_{1}\ldots dq_{n}\,dt=\\&-dD\,dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n},\end{aligned}}$

(18)

or

${\Bigg (}{\frac {dD}{dt}}{\Bigg )}_{p,q}=-\Sigma {\Bigg (}{\frac {dD}{dp_{1}}}{\dot {p}}_{1}+{\frac {dD}{dq_{1}}}{\dot {q}}_{1}{\Bigg )},$

(19)

where the suffix applied to the differential coefficient indicates that the $p$'s and $q$'s are to be regarded as constant in the differentiation. The condition of statistical equilibrium is therefore

$\Sigma {\Bigg (}{\frac {dD}{dp_{1}}}{\dot {p}}_{1}+{\frac {dD}{dq_{1}}}{\dot {q}}_{1}{\Bigg )}=0.$

(20)

If at any instant this condition is fulfilled for all values of the $p$'s and $q$'s, $(dD/dt)_{p,q}$ vanishes, and therefore the condition will continue to hold, and the distribution in phase will be permanent, so long as the external coördinates remain constant. But the statistical equilibrium would in general be disturbed by a change in the values of the external coördinates, which