Page:Elementary Principles in Statistical Mechanics (1902).djvu/153

CHAPTER XI.

MAXIMUM AND MINIMUM PROPERTIES OF VARIOUS DISTRIBUTIONS IN PHASE.

In the following theorems we suppose, as always, that the systems forming an ensemble are identical in nature and in the values of the external coördinates, which are here regarded as constants.

Theorem I. If an ensemble of systems is so distributed in phase that the index of probability is a function of the energy, the average value of the index is less than for any other distribution in which the distribution in energy is unaltered.

Let us write $\eta$ for the index which is a function of the energy, and $\eta +\Delta \eta$ for any other which gives the same distribution in energy. It is to be proved that

$\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}(\eta +\Delta \eta )e^{\eta +\Delta \eta }\,dp_{1}\ldots dq_{n}>\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\eta e^{\eta }\,dp_{1}\ldots dq_{n},$

(419)

where $\eta$ is a function of the energy, and $\Delta \eta$ a function of the phase, which are subject to the conditions that

$\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{\eta +\Delta \eta }\,dp_{1}\ldots dq_{n}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{\eta }\,dp_{1}\ldots dq_{n}=1,$

(420)

and that for any value of the energy ($\epsilon '$)

$\mathop {\int \ldots \int } _{\epsilon =\epsilon '}^{\epsilon =\epsilon '+d\epsilon '}e^{\eta +\Delta \eta }\,dp_{1}\ldots dq_{n}=\mathop {\int \ldots \int } _{\epsilon =\epsilon '}^{\epsilon =\epsilon '+d\epsilon '}e^{\eta }\,dp_{1}\ldots dq_{n}.$

(421)

Equation (420) expresses the general relations which $\eta$ and $\eta +\Delta \eta$ must satisfy in order to be indices of any distributions, and (421) expresses the condition that they give the same distribution in energy.