Page:Elementary Principles in Statistical Mechanics (1902).djvu/108

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84

AVERAGE VALUES IN A CANONICAL

or

{\frac {d{\overline {u}}}{d\Theta }}={\overline {\frac {du}{d\Theta }}}-{\overline {\frac {u(\psi -\epsilon )}{\Theta ^{2}}}}+{\frac {\overline {u}}{\Theta }}{\frac {d\psi }{d\Theta }}.

(251)

Setting

u=1

in this equation, we get

{\frac {d\psi }{d\Theta }}={\frac {\psi -{\overline {\epsilon }}}{\Theta }},

and substituting this value, we have

{\frac {\overline {u}}{d\Theta }}={\overline {\frac {du}{d\Theta }}}+{\frac {\overline {u\epsilon }}{\Theta ^{2}}}-{\frac {{\overline {u}}\,{\overline {\epsilon }}}{\Theta ^{2}}}

or

\Theta ^{2}{\frac {\overline {u}}{d\Theta }}-\Theta ^{2}{\overline {\frac {du}{d\Theta }}}={\overline {u\epsilon }}-{\overline {u}}\,{\overline {\epsilon }}={\overline {(u-{\overline {u}})(\epsilon -{\overline {\epsilon }})}}.

(252)

If we differentiate equation (250) with respect to $a$ (which may represent any of the external coördinates), and write $A$ for the force $-{\frac {d\epsilon }{dA}}$ , we get

{\frac {\overline {u}}{da}}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg (}{\frac {du}{da}}+{\frac {u}{\Theta }}{\frac {d\psi }{da}}+{\frac {u}{\Theta }}A{\bigg )}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}

or

{\frac {\overline {u}}{da}}={\overline {\frac {du}{da}}}+{\frac {\overline {u}}{\Theta }}{\frac {d\psi }{da}}+{\frac {\overline {uA}}{\Theta }}

(253)

Setting

u=1

in this equation, we get

{\frac {d\psi }{da}}=-{\overline {A}}.

Substituting this value, we have

{\frac {\overline {u}}{da}}={\overline {\frac {du}{da}}}+{\frac {\overline {uA}}{\Theta }}-{\frac {{\overline {u}}\,{\overline {A}}}{\Theta }},

(254)

or

\Theta {\frac {\overline {u}}{da}}-\Theta {\overline {\frac {du}{da}}}={\overline {uA}}-{\overline {u}}\,{\overline {A}}={\overline {(u-{\overline {u}})(A-{\overline {A}})}}.

(255)

Repeated applications of the principles expressed by equations (252) and (255) are perhaps best made in the particular cases. Yet we may write (252) in this form