Page:Elementary Principles in Statistical Mechanics (1902).djvu/40

It is the relative numbers of systems which fall within different limits, rather than the absolute numbers, with which we are most concerned. It is indeed only with regard to relative numbers that such discussions as the preceding will apply with literal precision, since the nature of our reasoning implies that the number of systems in the smallest element of space which we consider is very great. This is evidently inconsistent with a finite value of the total number of systems, or of the density-in-phase. Now if the value of $D$ is infinite, we cannot speak of any definite number of systems within any finite limits, since all such numbers are infinite. But the ratios of these infinite numbers may be perfectly definite. If we write $N$ for the total number of systems, and set

$P={\frac {D}{N}},$

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$P$ may remain finite, when $N$ and $D$ become infinite. The integral

$\int \ldots \int P\,dp_{1}\ldots dq_{n}$

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taken within any given limits, will evidently express the ratio of the number of systems falling within those limits to the whole number of systems. This is the same thing as the probability that an unspecified system of the ensemble (i. e. one of which we only know that it belongs to the ensemble) will lie within the given limits. The product

$P\,dp_{1}\ldots dq_{n}$

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expresses the probability that an unspecified system of the ensemble will be found in the element of extension-in-phase $dp_{1}\ldots dq_{n}$. We shall call $P$ the coefficient of probability of the phase considered. Its natural logarithm we shall call the index of probability of the phase, and denote it by the letter $\eta$.

If we substitute $NP$ and $Ne^{\eta }$ for $D$ in equation (19), we get

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

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and

${\bigg (}{\frac {d\eta }{dt}}{\bigg )}_{p,q}=-\Sigma {\bigg (}{\frac {d\eta }{dp_{1}}}{\dot {p}}_{1}+{\frac {d\eta }{dq_{1}}}d{\dot {q}}_{1}{\bigg )}.$

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