Page:Elementary Principles in Statistical Mechanics (1902).djvu/202

when

${\bigg (}{\frac {d\epsilon }{d\phi }}{\bigg )}_{a}\,d\phi +{\bigg (}{\frac {d\epsilon }{da_{1}}}{\bigg )}_{\phi ,a}\,da_{1}-A_{1}'da_{1}+\mathrm {etc.} +m_{1}{\dot {a}}_{1}d{\dot {a}}_{1}+\mathrm {etc.} =0.$

This requires

${\dot {a}}_{1}=0,\quad {\dot {a}}_{2}=0,\quad \mathrm {etc.} ,$

and

${\bigg (}{\frac {d\epsilon }{da_{1}}}{\bigg )}_{\phi ,a}=A_{1}',\quad {\bigg (}{\frac {d\epsilon }{da_{2}}}{\bigg )}_{\phi ,a}=A_{2}',\quad \mathrm {etc.}$

This shows that for any given values of $E$, $a_{1}$, $a_{2}$, etc. ${\Big (}{\frac {d\epsilon }{da_{1}}}{\Big )}_{\phi ,a}$, ${\Big (}{\frac {d\epsilon }{da_{2}}}{\Big )}_{\phi ,a}$, etc., represent the forces (in the generalized sense) which the external bodies would have to exert to make these values of $a_{1}$, $a+2$, etc., the most probable under the conditions specified. When the differences of the external forces which are exerted by the different systems are negligible,—$(d\epsilon /da_{1})_{\phi ,a}$, etc., represent these forces.

It is certainly in the quantities relating to a canonical ensemble, ${\overline {\epsilon }}$, $\Theta$, ${\overline {\eta }}$, ${\overline {A}}_{1}$, etc., $a_{1}$, etc. that we find the most complete correspondence with the quantities of the thermodynamic equation (482). Yet the conception itself of the canonical ensemble may seem to some artificial, and hardly germane to a natural exposition of the subject; and the quantities $\epsilon$, ${\frac {d\epsilon }{d\log V}}$, $\log V$, ${\overline {A_{1}}}|_{\epsilon }$, etc., $a_{1}$, etc., or $\epsilon$, ${\frac {d\epsilon }{d\phi }}$, $\phi$, ${\Big (}{\frac {d\epsilon }{da_{1}}}{\Big )}_{\phi ,a}$, etc., $a_{1}$, etc., which are closely related to ensembles of constant energy, and to average and most probable values in such ensembles, and most of which are defined without reference to any ensemble, may appear the most natural analogues of the thermodynamic quantities.

In regard to the naturalness of seeking analogies with the thermodynamic behavior of bodies in canonical or microcanonical ensembles of systems, much will depend upon how we approach the subject, especially upon the question whether we regard energy or temperature as an independent variable.

It is very natural to take energy for an independent variable rather than temperature, because ordinary mechanics furnishes us with a perfectly defined conception of energy, whereas the idea of something relating to a mechanical system and corre-