# Elementary Principles in Statistical Mechanics/Chapter IX

Transcriber's note: The original text used an older notation for the factorial operation, similar in appearance to ${\underline {|n}}$ (see also Mathworld). This notation is difficult to transcribe faithfully in Wikisource, and so the factorials in this chapter have been convered to the modern notation, $n!$.
CHAPTER IX.
THE FUNCTION ϕ AND THE CANONICAL DISTRIBUTION.
In this chapter we shall return to the consideration of the canonical distribution, in order to investigate those properties which are especially related to the function of the energy which we have denoted by $\phi$.
If we denote by $N$, as usual, the total number of systems in the ensemble,
* $Ne^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon$
$\epsilon$
$\epsilon +d\epsilon$
| $Ne^{{\frac {\psi -\epsilon }{\Theta }}+\phi }$ | (317) |
$\epsilon =\infty$
| $\int \limits _{V=0}^{\epsilon =\infty }e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon =1$ | (318) |
$\epsilon =-\infty$
$n>2$
$e^{\phi }$
$n>2$
| ${\frac {d\phi }{d\epsilon }}={\frac {1}{\Theta }}.$ | (319) |
This value of $d\phi /d\epsilon$ is also, when $n>2$, its average value in the ensemble. For we have identically, by integration by parts,
| $\int \limits _{V=0}^{\epsilon =\infty }{\frac {d\phi }{d\epsilon }}e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon ={\bigge^{{\frac {\psi -\epsilon }{\Theta }}+\phi }{\bigg ]}_{V=0}^{\epsilon =\infty }+{\frac {1}{\Theta }}\int \limits _{V=0}^{\epsilon =\infty }e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon .$ | (320) |
$n>2$
$N$
| ${\overline {\frac {d\phi }{d\epsilon }}}={\frac {1}{\Theta }}.$ | (321) |
$d\phi /d\epsilon$
Hitherto, in our consideration of the quantities $V$, $V_{q}$, $V_{p}$, $\phi$, $\phi _{q}$, $\phi _{p}$, we have regarded the external coördinates as constant. It is evident, however, from their definitions that $V$ and $\phi$ are in general functions of the external coördinates and the energy ($\epsilon$), that $V_{q}$ and $\phi _{q}$ are in general functions of the external coördinates and the potential energy ($\epsilon _{q}$). $V_{p}$ and $\phi _{p}$ we have found to be functions of the kinetic energy ($\epsilon _{p}$) alone. In the equation
| $e^{-{\frac {\psi }{\Theta }}}=\int \limits _{V=0}^{\epsilon =\infty }e^{-{\frac {\epsilon }{\Theta }}+\phi }\,d\epsilon ,$ | (322) |
$\psi$
$\Theta$
$\phi$
$\psi$
$n>2$
| ${\begin{aligned}&e^{-{\frac {\psi }{\Theta }}}{\bigg-{\frac {1}{\Theta }}\,d\psi +{\frac {\psi }{\Theta ^{2}}}\,d\Theta {\bigg )}={\frac {1}{\Theta ^{2}}}\,d\Theta \int \limits _{V=0}^{\epsilon =\infty }\epsilon e^{-{\frac {\epsilon }{\Theta }}+\phi }\,d\epsilon \\&\qquad+da_{1}\int \limits _{V=0}^{\epsilon =\infty }{\frac {d\phi }{da_{1}}}e^{-{\frac {\epsilon }{\Theta }}+\phi }\,d\epsilon +da_{2}\int \limits _{V=0}^{\epsilon =\infty }{\frac {d\phi }{da_{2}}}e^{-{\frac {\epsilon }{\Theta }}+\phi }\,d\epsilon +\mathrm {etc.} \end{aligned}}$ | (323) |
$e^{\phi }$
$V$
$n>2$
| $-{\frac {1}{\Theta }}\,d\psi +{\frac {\psi }{\Theta ^{2}}}\,d\Theta ={\frac {\overline {\epsilon }}{\Theta ^{2}}}\,d\Theta +{\overline {\frac {d\phi }{da_{1}}}}\,da_{1}+{\overline {\frac {d\phi }{da_{2}}}}\,da_{2}+\mathrm {etc.} ,$ | (324) |
| ${\frac {\psi +{\overline {\epsilon }}}{\Theta }}={\overline {\eta }},$ | (325) |
| $d\psi ={\overline {\eta }}\,d\Theta -\Theta {\overline {\frac {d\phi }{da_{1}}}}\,da_{1}-\Theta {\overline {\frac {d\phi }{da_{2}}}}\,da_{2}-\mathrm {etc.} .$ | (326) |
| ${\overline {\frac {d\phi }{da_{1}}}}={\frac {\overline {A_{1}}}{\Theta }},\quad {\overline {\frac {d\phi }{da_{2}}}}={\frac {\overline {A_{2}}}{\Theta }},\quad \mathrm {etc.}$ | (327) |
| ${\overline {{\bigg{\frac {d\phi }{da_{1}}}{\bigg )}}}_{\epsilon ,a}=-{\overline {{\bigg{\frac {d\phi }{d\epsilon }}{\bigg )}}}_{a}{\overline {{\bigg{\frac {d\epsilon }{da_{1}}}{\bigg )}}}_{a,q}$ | (328) |
| ${\overline {{\bigg{\frac {d\phi }{da_{1}}}{\bigg )}}}_{\epsilon ,a}=-{\overline {{\bigg{\frac {d\phi }{d\epsilon }}{\bigg )}_{a}{\bigg{\frac {d\epsilon }{da_{1}}}{\bigg )}_{\phi ,a}}}$ | (329) |
| ${\bigg{\frac {d\phi }{da_{1}}}{\bigg )}_{\epsilon ,a}=-{\bigg{\frac {d\phi }{d\epsilon }}{\bigg )}_{a}{\bigg{\frac {d\epsilon }{da_{1}}}{\bigg )}_{\phi ,a}$ | (330) |
​Moreover, if we eliminate $d\psi$ from (326) by the equation
| $d\psi =\Theta \,d{\overline {\eta }}+{\overline {\eta }}\,d\Theta +d{\overline {\epsilon }},$ | (331) |
| $d{\overline {\epsilon }}=-\Theta \,d{\overline {\eta }}-\Theta {\overline {\frac {d\phi }{da_{1}}}}\,da_{1}-\Theta {\overline {\frac {d\phi }{da_{2}}}}\,da_{2}-\mathrm {etc.} ,$ | (332) |
| $-d{\overline {\eta }}={\overline {\frac {d\phi }{d\epsilon }}}\,d{\overline {\epsilon }}+{\overline {\frac {d\phi }{da_{1}}}}\,da_{1}+{\overline {\frac {d\phi }{da_{2}}}}\,da_{2}+\mathrm {etc.}$ | (333) |
| $d\phi ={\frac {d\phi }{d\epsilon }}\,d\epsilon +{\frac {d\phi }{da_{1}}}\,da_{1}+{\frac {d\phi }{da_{2}}}\,da_{2}+\mathrm {etc.}$ | (334) |
$\epsilon _{0}$
| $d\phi _{0}={\bigg{\frac {d\phi }{d\epsilon }}{\bigg )}_{0}\,d\epsilon _{0}+{\bigg{\frac {d\phi }{da_{1}}}{\bigg )}_{0}\,da_{1}+{\bigg{\frac {d\phi }{da_{2}}}{\bigg )}_{0}\,da_{2}+\mathrm {etc.}$ | (335) |
| ${\overline {\frac {d\phi }{d\epsilon }}}={\bigg{\frac {d\phi }{d\epsilon }}{\bigg )}_{0}={\frac {1}{\Theta }},$ | (336) |
$n>2$
$d\phi /da_{1}$
$d\phi /da_{2}$
$\epsilon$
$n$
* ${\overline {\frac {d\phi }{da_{1}}}},\quad {\overline {\frac {d\phi }{da_{2}}}},\quad \mathrm {etc.} ,$
* ${\bigg{\frac {d\phi }{da_{1}}}{\bigg )}_{0},\quad {\bigg{\frac {d\phi }{da_{2}}}{\bigg )}_{0},\quad \mathrm {etc.}$
$d{\overline {\epsilon }}$
$d\epsilon _{0}$
$n$
| $-d{\overline {\eta }}=d\phi _{0}$ | (337) |
$-{\overline {\eta }}$
$\phi _{0}$
${\overline {\eta }}+\phi _{0}$
$n$
$-{\overline {\eta }}$
$\phi$
${\overline {\eta }}+\phi _{0}$
${\overline {\eta }}$
$\phi _{0}$
Now we have already noticed a certain correspondence between the quantities $\Theta$ and ${\overline {\eta }}$ and those which in thermodynamics are called temperature and entropy. The property just demonstrated, with those expressed by equation (336), therefore suggests that the quantities $\phi$ and $d\epsilon /d\phi$ may also correspond to the thermodynamic notions of entropy and temperature. We leave the discussion of this point to a subsequent chapter, and only mention it here to justify the somewhat detailed investigation of the relations of these quantities.
We may get a clearer view of the limiting form of the relations when the number of degrees of freedom is indefinitely increased, if we expand the function $\phi$ in a series arranged according to ascending powers of $\epsilon -\epsilon _{0}$. This expansion may be written
| $\phi =\phi _{0}+{\bigg{\frac {d\phi }{d\epsilon }}{\bigg )}_{0}(\epsilon -\epsilon _{0})+{\bigg{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}{\frac {(\epsilon -\epsilon _{0})^{2}}{2}}+{\bigg{\frac {d^{3}\phi }{d\epsilon ^{3}}}{\bigg )}_{0}{\frac {(\epsilon -\epsilon _{0})^{3}}{3!}}+\mathrm {etc.}$ | (338) |
* ${\frac {\psi -\epsilon }{\Theta }}={\frac {\psi -\epsilon _{0}}{\Theta }}-{\frac {\epsilon -\epsilon _{0}}{\Theta }},$
| ${\frac {\psi -\epsilon }{\Theta }}+\phi ={\frac {\psi -\epsilon _{0}}{\Theta }}+\phi _{0}+{\bigg{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}{\frac {(\epsilon -\epsilon _{0})^{2}}{2}}+{\bigg{\frac {d^{3}\phi }{d\epsilon ^{3}}}{\bigg )}{\frac {(\epsilon -\epsilon _{0})^{3}}{3!}}+\mathrm {etc.}$ | (339) |
* $\int \limits _{\epsilon '}^{\epsilon ''}e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon ,$
$\epsilon '$
$\epsilon ''$
| $e^{{\frac {\psi -\epsilon _{0}}{\Theta }}+\phi _{0}}\int \limits _{\epsilon '}^{\epsilon ''}e^{{\big{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\big )}_{0}{\frac {(\epsilon -\epsilon _{0})^{2}}{2}}+{\big{\frac {d^{3}\phi }{d\epsilon ^{3}}}{\big )}_{0}{\frac {(\epsilon -\epsilon _{0})^{3}}{3!}}+\mathrm {etc.} }\,d\epsilon .$ | (340) |
$\epsilon -\epsilon _{0}$
| $e^{{\frac {\psi -\epsilon _{0}}{\Theta }}+\phi _{0}}\int \limits _{\epsilon '}^{\epsilon ''}e^{{\big{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\big )}_{0}{\frac {(\epsilon -\epsilon _{0})^{2}}{2}}}\,d\epsilon .$ | (341) |
This shows that for a very great number of degrees of freedom the probability of deviations of energy from the most probable value ($\epsilon _{0}$) approaches the form expressed by the 'law of errors.' With this approximate law, we get ​
| $e^{{\frac {\psi -\epsilon _{0}}{\Theta }}+\phi _{0}}\left({\frac {-2\pi }{{\Big{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\Big )}_{0}}}\right)^{\frac {1}{2}}=1,$ | (342) |
| ${\overline {\epsilon }}=\epsilon _{0},\qquad {\overline {(\epsilon -\epsilon _{0})^{2}}}=-{\bigg{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}^{-},$ | (343) |
| ${\frac {\psi -\epsilon _{0}}{\Theta }}+\phi _{0}={\tfrac {1}{2}}\log {\frac {{\Big{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\Big )}_{0}}{-2\pi }}=-{\tfrac {1}{2}}\log(2\pi {\overline {(\epsilon -{\overline {\epsilon }})^{2}}}).$ | (344) |
* ${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}={\frac {2}{n}}{\frac {d{\overline {\epsilon }}}{d{\overline {\epsilon }}_{p}}}{\overline {\epsilon }}_{p}^{2}.$
| ${\overline {\eta }}+\phi _{0}={\frac {\psi -\epsilon _{0}}{\Theta }}+\phi _{0}=-{\tfrac {1}{2}}\log(2\pi {\overline {(\epsilon -{\overline {\epsilon }})^{2}}})=-{\tfrac {1}{2}}\log {\bigg{\frac {4\pi }{n}}{\frac {d{\overline {\epsilon }}}{d{\overline {\epsilon }}_{p}}}{\overline {\epsilon }}_{p}^{2}{\bigg )}$ | (345) |
${\overline {\eta }}-\phi _{0}$
$\log n$
${\overline {\eta }}+\phi _{0}-{\tfrac {1}{2}}\log n$
$\phi _{0}$
$-{\overline {\eta }}$
$n$
Equation (338) gives for the first approximation
| ${\overline {\phi }}-\phi _{0}={\bigg{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}{\frac {\overline {(\epsilon -\epsilon _{0})^{2}}}{2}}=-{\tfrac {1}{2}},$ | (346) |
| ${\overline {(\phi -\phi _{0})(\epsilon -\epsilon _{0})}}={\frac {\overline {(\epsilon -\epsilon _{0})^{2}}}{\Theta }}={\frac {d{\overline {\epsilon }}}{d{\overline {\epsilon }}_{p}}}{\overline {\epsilon }}_{p},$ | (347) |
| ${\overline {(\phi -\phi _{0})^{2}}}={\frac {\overline {(\epsilon -\epsilon _{0})^{2}}}{\Theta ^{2}}}={\frac {n}{2}}{\frac {d{\overline {\epsilon }}}{d{\overline {\epsilon }}_{p}}}$ | (348) |
$n$
* ${\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}={\bigg{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}(\epsilon -\epsilon _{0}),$
| ${\overline {{\bigg{\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}{\bigg )}(\epsilon -\epsilon _{0})}}={\bigg{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}{\overline {(\epsilon -\epsilon _{0})^{2}}}=-1,$ | (349) |
| ${\overline {{\bigg{\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}{\bigg )}^{2}}}={\bigg{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}^{2}{\overline {(\epsilon -\epsilon _{0})^{2}}}={\frac {1}{\overline {(\epsilon -\epsilon _{0})^{2}}}}=-{\bigg{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}.$ | (350) |
$n$
It should be observed that the approximate distribution of the ensemble in energy according to the 'law of errors' is not dependent on the particular form of the function of the energy which we have assumed for the index of probability ($\eta$). In any case, we must have
| $\int \limits _{V=0}^{\epsilon =\infty }e^{\eta +\phi }\,d\epsilon =1,$ | (351) |
$e^{\eta +\phi }$
$\epsilon =\infty$
$\epsilon =-\infty$
$\epsilon$
$n>2$
$e^{\phi }$
$\epsilon$
$\eta +\phi$
$\epsilon _{0}$
| ${\bigg{\frac {d\eta }{d\epsilon }}{\bigg )}_{0}+{\bigg{\frac {d\phi }{d\epsilon }}{\bigg )}_{0}=0.$ | (352) |
$\epsilon '$
$\epsilon ''$
* $\int \limits _{\epsilon '}^{\epsilon ''}e^{\eta +\phi }\,d\epsilon .$
$\eta$
$\phi$
$\epsilon -\epsilon _{0}$
| $e^{\eta _{0}+\phi _{0}}\int \limits _{\epsilon '}^{\epsilon ''}e^{{\big{\big{\frac {d^{2}\eta }{d\epsilon ^{2}}}{\big )}_{0}+{\big{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\big )}_{0}{\big ]}{\frac {(\epsilon -\epsilon _{0})^{2}}{2}}}\,d\epsilon .$ | (353) |
| $\eta _{0}+\phi _{0}={\tfrac {1}{2}}\log {\bigg{\frac {-1}{2\pi }}{\bigg{\frac {d^{2}\eta }{d\epsilon ^{2}}}{\bigg )}_{0}-{\frac {1}{2\pi }}{\bigg{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}{\bigg ]},$ | (354) |
| ${\overline {(\epsilon -\epsilon _{0})^{2}}}={\bigg-{\bigg{\frac {d^{2}\eta }{d\epsilon ^{2}}}{\bigg )}_{0}-{\bigg{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}{\bigg ]}^{-1}.$ | (355) |
| $-{\bigg{\frac {d^{2}\eta }{d\epsilon ^{2}}}{\bigg )}_{0}-{\bigg{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}$ | (356) |
$n$
$-d^{2}\phi /d\epsilon ^{2}$
$\eta$
We may obtain other properties pertaining to average values in a canonical ensemble by the method used for the average of $d\phi /d\epsilon$. Let $u$ be any function of the energy, either alone or with $\Theta$ and the external coördinates. The average value of $u$ in the ensemble is determined by the equation
| ${\overline {u}}=\int \limits _{V=0}^{\epsilon =\infty }u\,e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon .$ | (357) |
| $\int \limits _{V=0}^{\epsilon =\infty }{\bigg{\frac {du}{d\epsilon }}-{\frac {u}{\Theta }}+u{\frac {d\phi }{d\epsilon }}{\bigg )}e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon ={\biggu\,e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }{\bigg ]}_{V=0}^{\epsilon =\infty }$ | (358) |
| ${\overline {\frac {du}{d\epsilon }}}-{\frac {\overline {u}}{\Theta }}+{\overline {u{\frac {d\phi }{d\epsilon }}}}={\biggu\,e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }{\bigg ]}_{V=0}^{\epsilon =\infty }$ | (359) |
If we set $u=1$, (a value which need not be excluded,) the second member of this equation vanishes, as shown on page 101, if $n>2$, and we get
| ${\overline {\frac {d\phi }{d\epsilon }}}={\frac {1}{\Theta }}$ | (360) |
$n>2$
$u$
​The impossibility of a canonical distribution occurs when the equation
| $e^{-{\frac {\psi }{\Theta }}}=\int \limits _{V=0}^{\epsilon =\infty }u\,e^{-{\frac {\epsilon }{\Theta }}+\phi }\,d\epsilon$ | (361) |
$\psi$
$\psi$
$\psi =-\infty$
* $d{\frac {\psi }{\Theta }}=-{\frac {\overline {\epsilon }}{\Theta ^{2}}}\,d\Theta .$
$\Theta$
$\Theta$
$-\psi$
${\overline {\epsilon }}$
$\Theta$
$-\psi$
${\overline {\epsilon }}$
We assume then that for any finite value of $\Theta$ the second member of (361) has a finite value.
When this condition is fulfilled, the second member of (359) will vanish for $u=e^{-\phi }V$. For, if we set $\Theta '=2\Theta$,
* $e^{-{\frac {\epsilon }{\Theta }}}V=e^{-{\frac {\epsilon }{\Theta }}}\int \limits _{V=0}^{\epsilon }e^{\phi }\,d\epsilon \leq e^{-{\frac {\epsilon }{\Theta '}}}\int \limits _{V=0}^{\epsilon }e^{-{\frac {\epsilon }{\Theta '}}+\phi }\,d\epsilon \leq e^{-{\frac {\epsilon '}{\Theta '}}}e^{-{\frac {\psi '}{\Theta '}}},$
$\psi '$
$\psi$
$\Theta '$
$\epsilon =\infty$
$\epsilon$
* $e^{-{\frac {\epsilon }{\Theta }}}V\leq \int \limits _{V=0}^{\epsilon }e^{-{\frac {\epsilon }{\Theta }}+\phi }\,d\epsilon .$
$\epsilon$
$V=0$
* $1-{\overline {e^{-\phi }V{\frac {d\phi }{d\epsilon }}}}-{\overline {e^{-\phi }{\frac {V}{\Theta }}}}+{\overline {e^{-\phi }V{\frac {d\phi }{d\epsilon }}}}=0,$
| ${\overline {e^{-\phi }V}}=\Theta .$ | (362) |
$n$
$e^{-\phi }V$
If $n>2$, the second member of (359) may easily be shown to vanish for any of the following values of $u$ viz.: $\phi$, $e^{\phi }$, $\epsilon$, $\epsilon ^{m}$, where $m$ denotes any positive number. It will also vanish, when $n>4$, for $u=d\phi /d\epsilon$, and when $n>2h$ for $u=e^{-\phi }\,d^{h}V/d\epsilon ^{h}$. When the second member of (359) vanishes, and $n>2$, we may write
| ${\overline {(u-{\overline {u}}){\bigg{\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}{\bigg )}}}={\overline {u{\frac {d\phi }{d\epsilon }}}}-{\frac {\overline {u}}{\Theta }}=-{\overline {\frac {du}{d\epsilon }}}.$ | (363) |
If $n>2$,
| ${\overline {(\phi -{\overline {\phi }}){\bigg{\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}{\bigg )}}}={\overline {\phi {\frac {d\phi }{d\epsilon }}}}-{\frac {\overline {\phi }}{\Theta }}=-{\frac {1}{\Theta }},$ | (364) |
| ${\overline {(e^{\phi }-{\overline {e^{\phi }}}){\bigg{\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}{\bigg )}}}={\overline {e^{\phi }{\frac {d\phi }{d\epsilon }}}}-{\frac {\overline {e^{\phi }}}{\Theta }}=-{\overline {e^{\phi }{\frac {d\phi }{d\epsilon }}}},$ | (365) |
| $2{\overline {e^{\phi }{\frac {d\phi }{d\epsilon }}}}={\overline {e^{\phi }}}{\overline {\frac {d\phi }{d\epsilon }}}={\frac {\overline {e^{\phi }}}{\Theta }},$ | (366) |
| ${\overline {(\epsilon -{\overline {\epsilon }}){\bigg{\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}{\bigg )}}}={\overline {\epsilon {\frac {d\phi }{d\epsilon }}}}-{\frac {\overline {\epsilon }}{\Theta }}=-1,$ | (367) |
| ${\overline {(\epsilon ^{m}-{\overline {\epsilon ^{m}}}){\bigg{\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}{\bigg )}}}={\overline {\epsilon ^{m}{\frac {d\phi }{d\epsilon }}}}-{\frac {\overline {\epsilon ^{m}}}{\Theta }}=-m{\overline {\epsilon ^{m-1}}}.$ | (368) |
$n>4$
| ${\overline {{\bigg{\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}{\bigg )}^{2}}}={\overline {{\bigg{\frac {d\phi }{d\epsilon }}{\bigg )}^{2}}}-{\frac {1}{\Theta ^{2}}}=-{\overline {{\bigg{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}^{2}}}.$ | (369) |
$n>2h$
* ${\overline {e^{-\phi }{\frac {d^{h}V}{d\epsilon ^{h}}}{\frac {d\phi }{d\epsilon }}}}-{\frac {1}{\Theta }}{\overline {e^{-\phi }{\frac {d^{h}V}{d\epsilon ^{h}}}}}={\overline {e^{-\phi }{\frac {d^{h}V}{d\epsilon ^{h}}}{\frac {d\phi }{d\epsilon }}}}-{\overline {e^{-\phi }{\frac {d^{h+1}V}{d\epsilon ^{h+1}}}}}$
| ${\overline {e^{-\phi }{\frac {d^{h+1}V}{d\epsilon ^{h+1}}}}}={\frac {1}{\Theta }}{\overline {e^{-\phi }{\frac {d^{h}V}{d\epsilon ^{h}}}}}$ | (370) |
| ${\overline {e^{-\phi }{\frac {d^{h+1}V}{d\epsilon ^{h+1}}}}}={\frac {1}{\Theta ^{h}}}.$ | (371) |
$h$
* ${\overline {\frac {d\phi }{d\epsilon }}}={\frac {1}{\Theta }},\quad \mathrm {if~} n>2,$
* ${\overline {\frac {d^{2}\phi }{d\epsilon ^{2}}}}+{\overline {{\bigg{\frac {d\phi }{d\epsilon }}{\bigg )}^{2}}}={\frac {1}{\Theta ^{2}}}\quad \mathrm {if~} n>4,$
| ${\overline {\frac {d^{3}\phi }{d\epsilon ^{3}}}}+3{\overline {{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\frac {d\phi }{d\epsilon }}}}+{\overline {{\bigg{\frac {d\phi }{d\epsilon }}{\bigg )}^{3}}}={\frac {1}{\Theta ^{3}}}\quad \mathrm {if~} n>6.$ | (372) |
​If $V_{q}$ is a continuous increasing function of $\epsilon _{q}$, commencing with $V_{q}=0$, the average value in a canonical ensemble of any function of $\epsilon _{q}$, either alone or with the modulus and the external coördinates, is given by equation (275), which is identical with (357) except that $\epsilon$, $\phi$, and $\psi$ have the suffix $(~)_{q}$. The equation may be transformed so as to give an equation identical with (359) except for the suffixes. If we add the same suffixes to equation (361), the finite value of its members will determine the possibility of the canonical distribution.
From these data, it is easy to derive equations similar to (360), (362)-(372), except that the conditions of their validity must be differently stated. The equation
* ${\overline {e^{-\phi _{q}}V_{q}}}=\Theta$
$V_{q}$
$n$
$V$
$V_{q}$
If $V_{q}$ satisfies the condition mentioned, and $e^{\phi _{q}}$ a similar condition, i. e., if $e^{\phi _{q}}$ is a continuous increasing function of $\epsilon _{q}$, commencing with the value $e^{\phi _{q}}=0$, equations will hold similar to those given for the case when $n>2$, viz., similar to (360), (364)-(368). Especially important is
* ${\overline {\frac {d\phi _{q}}{d\epsilon _{q}}}}={\frac {1}{\Theta }}.$
$V_{q}$
$e^{\phi _{q}}$
$dV_{q}/d\epsilon _{q}$
$d^{2}V_{q}/d\epsilon _{q}^{2}$
$n>4$
$d^{3}V_{q}/d\epsilon _{q}^{3}$
$n>6$
$V_{q}$
$h$
$n>2h$
​These conditions take the place of those given above relating to $n$. In fact, we might give conditions relating to the differential coefficients of $V$, similar to those given relating to the differential coefficients of $V_{q}$, instead of the conditions relating to $n$, for the validity of equations (360), (363)-(372). This would somewhat extend the application of the equations.
* ↑ See page 96.
* ↑ See equations (321) and (104). Suffixes are here added to the differential coefficients, to make the meaning perfectly distinct, although the same quantities may be written elsewhere without the suffixes, when it is believed that there is no danger of misapprehension. The suffixes indicate the quantities which are constant in the differentiation, the single letter $a$ standing for all the letters $a_{1}$, $a_{2}$, etc., or all except the one which is explicitly varied.
* ↑ If a higher degree of accuracy is desired than is afforded by this formula, it may be multiplied by the series obtained from $e^{{\big{\frac {d^{3}\phi }{d\epsilon ^{3}}}{\big )}_{0}{\frac {(\epsilon -\epsilon _{0})^{3}}{3!}}+\mathrm {etc.} }$ by the ordinary formula for the expansion in series of an exponential function. There would be no especial analytical difficulty in taking account of a moderate number of terms of such a series, which would commence $1+{\bigg{\frac {d^{3}\phi }{d\epsilon ^{3}}}{\bigg )}_{0}{\frac {(\epsilon -\epsilon _{0})^{3}}{3!}}+{\bigg{\frac {d^{4}\phi }{d\epsilon ^{4}}}{\bigg )}_{0}{\frac {(\epsilon -\epsilon _{0})^{4}}{4!}}+\mathrm {etc.}$
* ↑ Compare (289), (314).
* ↑ We shall find hereafter that the equation ${\overline {{\bigg{\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}{\bigg )}(\epsilon -{\overline {\epsilon }})}}=-1$ is exact for any value of $n$ greater than 2, and that the equation ${\overline {{\bigg{\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}{\bigg )}^{2}}}=-{\overline {\frac {d^{2}\phi }{d\epsilon ^{2}}}}$ is exact for any value of $n$ greater than 4.
* ↑ A more general equation, which is not limited to ensembles canonically distributed, is ${\overline {\frac {du}{d\epsilon }}}+{\overline {u{\frac {d\eta }{d\epsilon }}}}+{\overline {u{\frac {d\phi }{d\epsilon }}}}={\biggu\,e^{\eta +\phi }{\bigg ]}_{V=0}^{\epsilon =\infty }$ where $\eta$ denotes, as usual, the index of probability of phase.
* ↑ The term finite applied to the modulus is intended to exclude the value zero as well as infinity.
* ↑ This equation may also be obtained from equations (252) and (321). Compare also equation (349) which was derived by an approximative method.
* ↑ Compare equation (350), obtained by an approximative method.