Page:Elementary Principles in Statistical Mechanics (1902).djvu/185

Hence

${\overline {\eta }}_{1}''+{\frac {{\overline {\epsilon }}_{2}'}{\Theta _{2}}}\leq {\overline {\eta }}_{1}'+{\frac {{\overline {\epsilon }}_{2}''}{\Theta _{2}}}$

(470)

which may be written

${\overline {\eta }}_{1}'-{\overline {\eta }}_{1}''\geq {\frac {{\overline {\epsilon }}_{2}'-{\overline {\epsilon }}_{2}''}{\Theta _{2}}}$

(471)

This may be compared with the thennodynamic principle that when a body (which need not be in thermal equilibrium) is brought into thermal contact with another of a given temperature, the increase of entropy of the first cannot be less (algebraically) than the loss of heat by the second divided by its temperature. Where ${\overline {W}}$ is negligible, we may write

${\overline {\eta }}_{1}''+{\frac {{\overline {\epsilon }}_{1}''}{\Theta _{2}}}\leq {\overline {\eta }}_{1}'+{\frac {{\overline {\epsilon }}_{1}'}{\Theta _{2}}}$

(472)

Now, by Theorem III of Chapter XI, the quantity

${\overline {\eta }}_{1}+{\frac {{\overline {\epsilon }}_{1}}{\Theta _{2}}}$

(473)

has a minimum value when the ensemble to which ${\overline {\eta }}_{1}$ and ${\overline {\epsilon }}_{1}$ relate is distributed canonically with the modulus $\Theta _{2}$. If the ensemble had originally this distribution, the sign $<$ in (472) would be impossible. In fact, in this case, it would be easy to show that the preceding formulae on which (472) is founded would all have the sign $=$. But when the two ensembles are not both originally distributed canonically with the same modulus, the formulae indicate that the quantity (473) may be diminished by bringing the ensemble to which $\epsilon _{1}$ and $\eta _{1}$ relate into connection with another which is canonically distributed with modulus $\Theta _{2}$, and therefore, that by repeated operations of this kind the ensemble of which the original distribution was entirely arbitrary might be brought approximately into a state of canonical distribution with the modulus $\Theta _{2}$. We may compare this with the thermodynamic principle that a body of which the original thermal state may be entirely arbitrary, may be brought approximately into a state of thermal equilibrium with any given temperature by repeated connections with other bodies of that temperature.