Page:Elementary Principles in Statistical Mechanics (1902).djvu/31

In the first place, let us consider the number of systems which in the time $dt$ pass into or out of the specified element by $p_{1}$ passing the limit $p_{1}'$. It will be convenient, and it is evidently allowable, to suppose $dt$ so small that the quantities ${\dot {p}}_{1}\,dt$, ${\dot {q}}_{1}\,dt$, etc., which represent the increments of $p_{1}$, $q_{1}$, etc., in the time $dt$ shall be infinitely small in comparison with the infinitesimal differences $p_{1}''-p_{1}'$, $q_{1}''-q_{1}'$, etc., which determine the magnitude of the element of extension-in-phase. The systems for which $p_{1}$ passes the limit $p_{1}'$ in the interval $dt$ are those for which at the commencement of this interval the value of $p_{1}$ lies between $p_{1}'$ and $p_{1}'-{\dot {p}}_{1}\,dt$, as is evident if we consider separately the cases in which ${\dot {p}}_{1}$ is positive and negative. Those systems for which $p_{1}$ lies between these limits, and the other $p$'s and $q$'s between the limits specified in (9), will therefore pass into or out of the element considered according as ${\dot {p}}$ is positive or negative, unless indeed they also pass some other limit specified in (9) during the same interval of time. But the number which pass any two of these limits will be represented by an expression containing the square of $dt$ as a factor, and is evidently negligible, when $dt$ is sufficiently small, compared with the number which we are seeking to evaluate, and which (with neglect of terms containing $dt^{2}$) may be found by substituting ${\dot {p}}_{1}\,dt$ for $p_{1}''-p_{1}'$ in (10) or for $dp_{1}$ in (11).

The expression

$D\,{\dot {p}}_{1}\,dt\,dp_{2}\ldots dp_{n}\,dq_{1}\ldots dq_{n}$

(13)

will therefore represent, according as it is positive or negative, the increase or decrease of the number of systems within the given limits which is due to systems passing the limit $p_{1}'$. A similar expression, in which however $D$ and ${\dot {p}}$ will have slightly different values (being determined for $p_{1}''$ instead of $p_{1}'$), will represent the decrease or increase of the number of systems due to the passing of the limit $p_{1}''$. The difference of the two expressions, or

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

(14)