Page:Elementary Principles in Statistical Mechanics (1902).djvu/160

of the probability-coefficients of the original ensembles, the average index of probability of the resulting ensemble cannot be greater than the same linear function of the average indices of the original ensembles. It can be equal to it only when the original ensembles are similarly distributed in phase.

Let $P_{1}$, $P_{2}$, etc. be the probability-coefficients of the original ensembles, and $P$ that of the ensemble formed by combining them; and let $N_{1}$, $N_{2}$, etc. be the numbers of systems in the original ensembles. It is evident that we shall have

$P=c_{1}P_{1}+c_{2}P_{2}+\mathrm {etc.} =\Sigma (c_{1}P_{1}),$

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where

$c_{1}={\frac {N_{1}}{\Sigma N_{1}}},\qquad c_{2}={\frac {N_{2}}{\Sigma N_{1}}},\qquad \mathrm {etc.}$

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The main proposition to be proved is that

$\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}P\log P\,dp_{1}\ldots dq_{n}\leq \Sigma {\Bigg [}c_{1}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}P_{1}\log P_{1}\,dp_{1}\ldots dq_{n}{\Bigg ]}$

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or

$\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}[\Sigma (c_{1}P_{1}\log P_{1})-P\log P]\,dp_{1}\ldots dq_{n}\geq 0.$

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If we set

$Q_{1}=P_{1}\log P_{1}-P_{1}\log P-P_{1}+P$

$Q_{1}$ will be positive, except when it vanishes for $P_{1}=P$. To prove this, we may regard $P_{1}$ and $P$ as any positive quantities. Then

${\bigg (}{\frac {dQ_{1}}{dP_{1}}}{\bigg )}_{P}=\log P_{1}-\log P,$

${\bigg (}{\frac {d^{2}Q_{1}}{dP_{1}{}^{2}}}{\bigg )}_{P}={\frac {1}{P_{1}}}.$

Since $Q_{1}$ and $dQ_{1}/dP_{1}$ vanish for $P_{1}=P$, and the second differential coefficient is always positive, $Q_{1}$ must be positive except when $P_{1}=P$. Therefore, if $Q_{2}$, etc. have similar definitions,

$\Sigma (c_{1}Q_{1})\geq 0.$

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