the thermal contact of two bodies may increase but cannot diminish the sum of their entropies.

Let us especially consider the case in which the two original ensembles were both canonically distributed in phase with the respective moduli ${\displaystyle \Theta _{1}}$ and ${\displaystyle \Theta _{2}}$. We have then, by Theorem III of Chapter XI,

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

(464)

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

(465)

Whence with (463) we have

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

(466)

or

${\displaystyle {\frac {{\overline {\epsilon }}_{1}''-{\overline {\epsilon }}_{1}'}{\Theta _{1}}}+{\frac {{\overline {\epsilon }}_{2}''-{\overline {\epsilon }}_{2}'}{\Theta _{2}}}\geq 0.}$

(467)

If we write ${\displaystyle {\overline {W}}}$ for the average work done by the combined systems on the external bodies, we have by the principle of the conservation of energy

${\displaystyle {\overline {W}}={\overline {\epsilon }}_{12}'-{\overline {\epsilon }}_{12}''={\overline {\epsilon }}_{1}'-{\overline {\epsilon }}_{1}''+{\overline {\epsilon }}_{2}'-{\overline {\epsilon }}_{2}''.}$

(468)

Now if ${\displaystyle {\overline {W}}}$ is negligible, we have

${\displaystyle {\overline {\epsilon }}_{1}''-{\overline {\epsilon }}_{1}'=-({\overline {\epsilon }}_{2}''-{\overline {\epsilon }}_{2}')}$

(469)

and (467) shows that the ensemble which has the greater modulus must lose energy. This result may be compared to the thermodynamic principle, that when two bodies of different temperatures are brought together, that which has the higher temperature will lose energy.

Let us next suppose that the ensemble ${\displaystyle E_{2}}$ is originally canonically distributed with the modulus ${\displaystyle \Theta _{2}}$, but leave the distribution of the other arbitrary. We have, to determine the result of a similar process,

${\displaystyle {\overline {\eta }}_{1}''+{\overline {\eta }}_{2}''\leq {\overline {\eta }}_{1}'+{\overline {\eta }}_{2}'}$

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