the modulus (${\displaystyle \Theta }$), the external coördinates (${\displaystyle a_{1}}$, etc.), and the average values in the ensemble of the energy (${\displaystyle \epsilon }$), the index of probability (${\displaystyle \eta }$), and the external forces (${\displaystyle A_{1}}$, etc.) exerted by the systems, the following differential equation will hold:

${\displaystyle d{\overline {\epsilon }}=-\Theta d{\overline {\eta }}-{\overline {A}}_{1}da_{1}-{\overline {A}}_{2}da_{2}-\mathrm {etc.} }$

(483)

This equation, if we neglect the sign of averages, is identical in form with the thermodynamic equation (482), the modulus (${\displaystyle \Theta }$) corresponding to temperature, and the index of probability of phase with its sign reversed corresponding to entropy.^[1]

We have also shown that the average square of the anomalies of ${\displaystyle \epsilon }$, that is, of the deviations of the individual values from the average, is in general of the same order of magnitude as the reciprocal of the number of degrees of freedom, and therefore to human observation the individual values are indistinguishable from the average values when the number of degrees of freedom is very great.^[2] In this case also the anomalies of ${\displaystyle \eta }$ are practically insensible. The same is true of the anomalies of the external forces (${\displaystyle A_{1}}$, etc.), so far as these are the result of the anomalies of energy, so that when these forces are sensibly determined by the energy and the external coördinates, and the number of degrees of freedom is very great, the anomalies of these forces are insensible.

The mathematical operations by which the finite equation between ${\displaystyle {\overline {\epsilon }}}$, ${\displaystyle {\overline {\eta }}}$, and ${\displaystyle a_{1}}$, etc., is deduced from that which gives the energy (${\displaystyle \epsilon }$) of a system in terms of the momenta (${\displaystyle p_{1}\ldots p_{n}}$) and coördinates both internal (${\displaystyle q_{1}\ldots q_{n}}$) and external (${\displaystyle a_{1}}$, etc.), are indicated by the equation

${\displaystyle e^{-{\frac {\psi }{\Theta }}}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\,dq_{1}\ldots dq_{n}\,dp_{1}\ldots dp_{n},,}$

(484)

where

${\displaystyle \psi =-\Theta {\overline {\eta }}+{\overline {\epsilon }}.}$

We have also shown that when systems of different ensembles are brought into conditions analogous to thermal contact, the average result is a passage of energy from the ensemble

1. ↑ See Chapter IV, pages 44, 45.
2. ↑ See Chapter VII, pages 73-75.