Now the average value in the ensemble of any quantity (which we shall denote in general by a horizontal line above the proper symbol) is determined by the equation

${\displaystyle {\overline {u}}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}ue^{\frac {\phi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}.}$

(108)

Comparing this with the preceding equation, we have

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

(109)

Or, since

${\displaystyle {\frac {\psi -\epsilon }{\Theta }}=\eta ,}$

(110)

and

${\displaystyle {\frac {\psi -{\overline {\epsilon }}}{\Theta }}={\overline {\eta }},}$

(111)

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

(112)

Moreover, since (111) gives

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

(113)

we have also

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

(114)

This equation, if we neglect the sign of averages, is identical in form with the thermodynamic equation

${\displaystyle d\eta ={\frac {d\epsilon +A_{1}da_{1}+A_{2}da_{2}+\mathrm {etc.} }{T}},}$

(115)

or

${\displaystyle d\epsilon =Td\eta -A_{1}da_{1}-A_{2}da_{2}-\mathrm {etc.} ,}$

(116)

which expresses the relation between the energy, temperature, and entropy of a body in thermodynamic equilibrium, and the forces which it exerts on external bodies, — a relation which is the mathematical expression of the second law of thermodynamics for reversible changes. The modulus in the statistical equation corresponds to temperature in the thermodynamic equation, and the average index of probability with its sign reversed corresponds to entropy. But in the thermodynamic equation the entropy (${\displaystyle \eta }$) is a quantity which is