# Elementary Principles in Statistical Mechanics/Chapter VII

CHAPTER VII.
FARTHER DISCUSSION OF AVERAGES IN A CANONICAL ENSEMBLE OF SYSTEMS.
Returning to the case of a canonical distribution, we have for the index of probability of configuration
| $\eta _{q}={\frac {\psi _{q}-\epsilon _{q}}{\Theta }}$ | (178) |
$u$
| ${\overline {u}}=\mathop {\int \ldots \int } _{\rm {config.}}^{\rm {all}}ue^{\frac {\psi _{q}-\epsilon _{q}}{\Theta }}\Delta _{\dot {q}}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n},$ | (179) |
$\psi _{q}$
| $e^{-{\frac {\psi _{q}}{\Theta }}}=\mathop {\int \ldots \int } _{\rm {config.}}^{\rm {all}}e^{-{\frac {\epsilon _{q}}{\Theta }}}\Delta _{\dot {q}}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n}.$ | (180) |
By differentiating the last equation we may obtain results analogous to those obtained in Chapter IV from the equation
* $e^{-{\frac {\psi }{\Theta }}}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n}.$
| $d\psi _{q}={\overline {\eta }}_{q}\,d\Theta -{\overline {A}}_{1}\,da_{1}-{\overline {A}}_{2}\,da_{2}-\mathrm {etc.} ,$ | (181) |
| $\psi _{q}={\overline {\epsilon }}_{q}+\Theta {\overline {\eta }}_{q},$ | (183) |
| $d\psi _{q}=d{\overline {\eta }}_{q}+{\overline {\eta }}_{q}\,d\Theta +\Theta \,d{\overline {\eta }}_{q},$ | (183) |
| $d{\overline {\epsilon }}_{q}=-\Theta d{\overline {\eta }}_{q}-{\overline {A}}_{1}\,da_{1}-{\overline {A}}_{2}\,da_{2}-\mathrm {etc.}$ | (184) |
For the index of probability of velocity, in the case of canonical distribution, we have by comparison of (144) and (163), or of (145) and (164),
| $\eta _{p}={\frac {\psi _{p}-\epsilon _{p}}{\Theta }}$ | (185) |
| ${\overline {\eta }}_{p}={\frac {\psi _{p}-{\overline {\epsilon }}_{p}}{\Theta }};$ | (186) |
| ${\overline {\epsilon }}_{p}={\tfrac {1}{2}}n\Theta ,$ | (187) |
| $\psi _{p}=-{\tfrac {1}{2}}n\Theta \log(2\pi \Theta ).$ | (188) |
| $d\psi _{p}={\overline {\eta }}_{p}\,d\Theta ,$ | (189) |
| $d{\overline {\epsilon }}_{p}=-\Theta \,d{\overline {\eta }}_{p}.$ | (190) |
* ${\overline {\epsilon }}={\overline {\epsilon }}_{q}+{\overline {\epsilon }}_{p},\quad {\overline {\eta }}={\overline {\eta }}_{q}+{\overline {\eta }}_{p}$
Equations (112) and (181) show that if $\psi$ or $\psi _{q}$ is known as function of $\Theta$ and $a_{1}$, $a_{2}$, etc., we can obtain by differentiation ${\overline {\epsilon }}$ or ${\overline {\epsilon }}_{q}$, and ${\overline {A}}_{1}$, ${\overline {A}}_{2}$ etc. as functions of the same variables. We have in fact
| ${\overline {\epsilon }}=\psi -\Theta {\overline {\eta }}=\psi -\Theta {\frac {d\psi }{d\Theta }}.$ | (191) |
| ${\overline {\epsilon }}_{q}=\psi _{q}-\Theta {\overline {\eta }}_{q}=\psi _{q}-\Theta {\frac {d\psi _{q}}{d\Theta }}.$ | (192) |
| ${\overline {\epsilon }}_{p}=\psi _{p}-\Theta \eta _{p}=\psi _{p}-\Theta {\frac {d\psi _{p}}{d\Theta }}.$ | (193) |
| ${\overline {A}}_{1}=-{\frac {d\psi }{da_{1}}}=-{\frac {d\psi _{q}}{da_{1}}},$ | (194) |
$\psi$
$\psi _{q}$
The average values of the squares or higher powers of the energies (total, potential, or kinetic) may easily be obtained by repeated differentiations of $\psi$, $\psi _{q}$, $\psi _{p}$, or ${\overline {\epsilon }}$, ${\overline {\epsilon }}_{q}$, ${\overline {\epsilon }}_{p}$, with respect to $\Theta$. By equation (108) we have
| ${\overline {\epsilon }}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\epsilon e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n},$ | (195) |
$\Theta$
| ${\frac {d{\overline {\epsilon }}}{d\Theta }}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg{\frac {\epsilon ^{2}-\psi \epsilon }{\Theta ^{2}}}+{\frac {\epsilon }{\Theta }}{\frac {d\psi }{d\Theta }}{\bigg )}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n},$ | (196) |
* ${\frac {d{\overline {\epsilon }}}{d\Theta }}={\frac {{\overline {\epsilon ^{2}}}-\psi {\overline {\epsilon }}}{\Theta ^{2}}}+{\frac {\overline {\epsilon }}{\Theta }}{\frac {d\psi }{d\Theta }},$
| ${\overline {\epsilon ^{2}}}=\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }}+{\overline {\epsilon }}{\bigg\psi -\Theta {\frac {d\psi }{d\Theta }}{\bigg )}.$ | (197) |
| ${\overline {\epsilon ^{2}}}={\overline {\epsilon }}^{2}+\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }}={\bigg\psi -\Theta {\frac {d\psi }{d\Theta }}{\bigg )}^{2}-\Theta ^{3}{\frac {d^{2}\psi }{d\Theta ^{2}}}.$ | (198) |
In precisely the same way, from the equation
| ${\overline {\epsilon }}_{q}=\mathop {\int \ldots \int } _{\rm {config.}}^{\rm {all}}\epsilon _{q}e^{\frac {\psi _{q}-\epsilon _{q}}{\Theta }}\Delta _{\dot {q}}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n},$ | (199) |
| ${\overline {\epsilon _{q}^{2}}}={\overline {\epsilon _{q}}}^{2}+\Theta ^{2}{\frac {d{\overline {\epsilon _{q}}}}{d\Theta }}={\bigg\psi _{q}-\Theta {\frac {d\psi _{q}}{d\Theta }}{\bigg )}^{2}-\Theta ^{3}{\frac {d^{2}\psi _{q}}{d\Theta ^{2}}}.$ | (200) |
In the same way also, if we confine ourselves to a particular configuration, from the equation
| ${\overline {\epsilon }}_{p}=\mathop {\int \ldots \int } _{\rm {veloc.}}^{\rm {all}}\epsilon _{p}e^{\frac {\psi _{p}-\epsilon _{p}}{\Theta }}\Delta _{p}^{\frac {1}{2}}\,dp_{1}\ldots dp_{n},$ | (201) |
| ${\overline {\epsilon _{p}^{2}}}={\overline {\epsilon _{p}}}^{2}+\Theta ^{2}{\frac {d{\epsilon _{p}}}{d\Theta }}={\bigg\psi _{p}-\Theta {\frac {d\psi _{p}}{d\Theta }}{\bigg )}^{2}-\Theta ^{3}{\frac {d^{2}\psi _{p}}{d\Theta ^{2}}},$ | (202) |
| ${\overline {\epsilon _{p}^{2}}}=({\tfrac {1}{4}}n^{2}+{\tfrac {1}{2}}n)\Theta ^{2}.$ | (203) |
${\overline {\epsilon _{p}^{2}}}$
Of especial importance are the anomalies of the energies, or their deviations from their average values. The average value ​of these anomalies is of course zero. The natural measure of such anomalies is the square root of their average square. Now
| ${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}={\overline {\epsilon ^{2}}}-{\overline {\epsilon }}^{2},$ | (204) |
| ${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}=\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }}=-\Theta ^{3}{\frac {d^{2}\psi }{d\Theta ^{2}}}.$ | (205) |
| ${\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{2}}}=\Theta ^{2}{\frac {d{\overline {\epsilon }}_{q}}{d\Theta }}=-\Theta ^{3}{\frac {d^{2}\psi _{q}}{d\Theta ^{2}}}.$ | (206) |
| ${\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{2}}}=\Theta ^{2}{\frac {d{\overline {\epsilon }}_{p}}{d\Theta }}=-\Theta ^{3}{\frac {d^{2}\psi _{p}}{d\Theta ^{2}}}={\tfrac {1}{2}}n\Theta ^{2}.$ | (207) |
| ${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}={\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{2}}}+{\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{2}}}.$ | (208) |
$d{\overline {\epsilon }}_{q}/d\Theta$
$d^{2}\psi _{q}/d\Theta ^{2}$
$d{\overline {\eta }}_{q}/d\Theta$
To get an idea of the order of magnitude of these quantities, we may use the average kinetic energy as a term of comparison, this quantity being independent of the arbitrary constant involved in the definition of the potential energy. Since ​
* ${\overline {\epsilon }}_{p}={\tfrac {1}{2}}n\Theta ,$
| ${\frac {\overline {(\epsilon _{p}-{\overline {\epsilon _{p}}})^{2}}}{{\overline {\epsilon _{p}}}^{2}}}={\frac {2}{n}},$ | (209) |
| ${\frac {\overline {(\epsilon _{q}-{\overline {\epsilon _{q}}})^{2}}}{{\overline {\epsilon _{p}}}^{2}}}={\frac {2}{n}}{\frac {d{\overline {\epsilon _{q}}}}{d{\overline {\epsilon _{p}}}}},$ | (210) |
| ${\frac {\overline {(\epsilon -{\overline {\epsilon }})^{2}}}{{\overline {\epsilon _{p}}}^{2}}}={\frac {2}{n}}{\frac {d{\overline {\epsilon }}}{d{\overline {\epsilon _{p}}}}}={\frac {2}{n}}+{\frac {2}{n}}{\frac {d{\overline {\epsilon _{q}}}}{d{\overline {\epsilon _{p}}}}}.$ | (211) |
These equations show that when the number of degrees of freedom of the systems is very great, the mean squares of the anomalies of the energies (total, potential, and kinetic) are very small in comparison with the mean square of the kinetic energy, unless indeed the differential coefficient $d{\overline {\epsilon }}_{q}/d{\overline {\epsilon }}_{p}$ is of the same order of magnitude as $n$. Such values of $d{\overline {\epsilon }}_{q}/d{\overline {\epsilon }}_{p}$ can only occur within intervals (${\overline {\epsilon }}_{p}''-{\overline {\epsilon }}_{p}'$) which are of the order of magnitude of $n^{-1}$ unless it be in cases in which ${\overline {\epsilon }}_{q}$ is in general of an order of magnitude higher than ${\overline {\epsilon }}_{p}$. Postponing for the moment the consideration of such cases, it will be interesting to examine more closely the case of large values of $d{\overline {\epsilon }}_{q}/d{\overline {\epsilon }}_{p}$ within narrow limits. Let us suppose that for ${\overline {\epsilon }}_{p}'$ and ${\overline {\epsilon }}_{p}''$ the value of $d{\overline {\epsilon }}_{q}/d{\overline {\epsilon }}_{p}$ is of the order of magnitude of unity, but between these values of ${\overline {\epsilon }}_{p}$ very great values of the differential coefficient occur. Then in the ensemble having modulus $\Theta ''$ and average energies ${\overline {\epsilon }}_{p}''$ and ${\overline {\epsilon }}_{q}''$, values of $\epsilon _{q}$ sensibly greater than ${\overline {\epsilon }}_{q}''$ will be so rare that we may call them practically negligible. They will be still more rare in an ensemble of less modulus. For if we differentiate the equation
* $\eta _{q}={\frac {\psi _{q}-\epsilon _{q}}{\Theta }}$
$\epsilon _{q}$
$\Theta$
$\psi _{q}$
| ${\bigg{\frac {d\eta _{q}}{d\Theta }}{\bigg )}_{\epsilon _{q}}={\frac {1}{\Theta }}{\frac {d\psi _{q}}{d\Theta }}-{\frac {\psi _{q}-\epsilon _{q}}{\Theta }},$ | (212) |
| ${\bigg{\frac {d\eta _{q}}{d\Theta }}{\bigg )}_{\epsilon _{q}}={\frac {\epsilon _{q}-{\overline {\epsilon }}_{q}}{\Theta ^{2}}}.$ | (213) |
$\Theta '$
${\overline {\epsilon }}_{p}'$
${\overline {\epsilon }}_{q}'$
${\overline {\epsilon }}_{q}$
${\overline {\epsilon }}_{q}'$
$\Theta$
$\Theta '$
$\Theta ''$
${\overline {\epsilon }}_{p}$
${\overline {\epsilon }}_{p}'$
${\overline {\epsilon }}_{p}''$
${\overline {\epsilon }}_{q}$
${\overline {\epsilon }}_{q}'$
${\overline {\epsilon }}_{q}''$
In the cases which remain to be considered, viz., when $d{\overline {\epsilon }}_{q}/d{\overline {\epsilon }}_{p}$ has very large values not confined to narrow limits, and consequently the differences of the mean potential energies in ensembles of different moduli are in general very large compared with the differences of the mean kinetic energies, it appears by (210) that the anomalies of mean square of potential energy, if not small in comparison with the mean kinetic energy, will yet in general be very small in comparison with differences of mean potential energy in ensembles having moderate differences of mean kinetic energy,—the exceptions being of the same character as described for the case when $d{\overline {\epsilon }}_{q}/d{\overline {\epsilon }}_{p}$ is not in general large.
It follows that to human experience and observation with respect to such an ensemble as we are considering, or with respect to systems which may be regarded as taken at random from such an ensemble, when the number of degrees of freedom is of such order of magnitude as the number of molecules in the bodies subject to our observation and experiment, $\epsilon -{\overline {\epsilon }}$, $\epsilon _{p}-{\overline {\epsilon }}_{p}$, $\epsilon _{q}-{\overline {\epsilon }}_{q}$ would be in general vanishing quantities, since such experience would not be wide enough to embrace the more considerable divergencies from the mean values, and such observation not nice enough to distinguish the ordinary divergencies. In other words, such ensembles would appear to human observation as ensembles of systems of uniform energy, and in which the potential and kinetic energies ( ​supposing that there were means of measuring these quantities separately) had each separately uniform values. Exceptions might occur when for particular values of the modulus the differential coefficient $d{\overline {\epsilon }}_{q}/d{\overline {\epsilon }}_{p}$ takes a very large value. To human observation the effect would be, that in ensembles in which $\Theta$ and ${\overline {\epsilon }}_{p}$ had certain critical values, ${\overline {\epsilon }}_{q}$ would be indeterminate within certain limits, viz., the values which would correspond to values of $\Theta$ and $\epsilon _{p}$ slightly less and slightly greater than the critical values. Such indeterminateness corresponds precisely to what we observe in experiments on the bodies which nature presents to us.
To obtain general formulae for the average values of powers of the energies, we may proceed as follows. If $h$ is any positive whole number, we have identically
| $\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\epsilon ^{h}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n}=\Theta ^{2}{\frac {d}{d\Theta }}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\epsilon ^{h-1}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n},$ | (214) |
| ${\overline {\epsilon ^{h}}}e^{-{\frac {\psi }{\Theta }}}=\Theta ^{2}{\frac {d}{d\Theta }}{\bigg{\overline {\epsilon ^{h-1}}}e^{-{\frac {\psi }{\Theta }}}{\bigg )}.$ | (215) |
| ${\overline {\epsilon ^{h}}}e^{-{\frac {\psi }{\Theta }}}={\bigg\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h}e^{-{\frac {\psi }{\Theta }}},$ | (216) |
| ${\overline {\epsilon ^{h}}}=e^{\frac {\psi }{\Theta }}{\bigg\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h}e^{-{\frac {\psi }{\Theta }}}.$ | (217) |
$h=1$
| ${\overline {\epsilon }}=-\Theta ^{2}{\frac {d}{d\Theta }}{\bigg{\frac {\psi }{\Theta }}{\bigg )}$ | (218) |
| ${\overline {\epsilon ^{h}}}={\overline {\epsilon }}{\overline {\epsilon ^{h-1}}}+\Theta ^{2}{\frac {d{\overline {\epsilon ^{h-1}}}}{d\Theta }}={\bigg{\overline {\epsilon }}+\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}\epsilon ^{h-1},$ | (219) |
| ${\overline {\epsilon ^{h}}}={\bigg{\overline {\epsilon }}+\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h-1}{\overline {\epsilon }}.$ | (220) |
In like manner from the identical equation
| $\mathop {\int \ldots \int } _{\rm {config.}}^{\rm {all}}\epsilon _{q}^{h}e^{-{\frac {\epsilon _{q}}{\Theta }}}\Delta _{\dot {q}}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n}=\Theta ^{2}{\frac {d}{d\Theta }}\mathop {\int \ldots \int } _{\rm {config.}}^{\rm {all}}\epsilon _{q}^{h-1}e^{-{\frac {\epsilon _{q}}{\Theta }}}\Delta _{\dot {q}}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n},$ | (221) |
| ${\overline {\epsilon _{q}^{h}}}=e^{\frac {\psi _{q}}{\Theta }}{\bigg\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h}e^{-{\frac {\psi _{q}}{\Theta }}}.,$ | (222) |
| ${\overline {\epsilon _{q}^{h}}}={\bigg{\overline {\epsilon }}_{q}+\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h-1}{\overline {\epsilon }}_{q}.$ | (223) |
With respect to the kinetic energy similar equations will hold for averages taken for any particular configuration, or for the whole ensemble. But since
* ${\overline {\epsilon }}_{p}={\frac {n}{2}}\Theta ,$
| ${\overline {\epsilon _{p}^{h}}}={\bigg{\overline {\epsilon }}_{p}+\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}{\overline {\epsilon _{p}^{h-1}}}$ | (224) |
| ${\overline {\epsilon _{p}^{h}}}={\bigg{\frac {n}{2}}\Theta +^{2}\Theta {\frac {d}{d\Theta }}{\bigg )}{\overline {\epsilon _{p}^{h-1}}}={\frac {n}{2}}{\bigg{\frac {n}{2}}\Theta +\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h-1}\Theta .$ | (225) |
| ${\overline {\epsilon _{p}^{2}}}={\bigg{\frac {n}{2}}+1{\bigg )}{\frac {n}{2}}\Theta ^{2}.$ | (226) |
| ${\overline {\epsilon _{p}^{3}}}={\bigg{\frac {n}{2}}+2{\bigg )}{\bigg{\frac {n}{2}}+1{\bigg )}{\frac {n}{2}}\Theta ^{2}.$ | (227) |
| ${\overline {\epsilon _{p}^{h}}}={\frac {\Gamma ({\tfrac {1}{2}}n+h)}{\Gamma ({\tfrac {1}{2}}n)}}\Theta ^{h}.$ | (228) |
The average values of the powers of the anomalies of the energies are perhaps most easily found as follows. We have identically, since ${\overline {\epsilon }}$ is a function of $\Theta$, while $\epsilon$ is a function of the $p$'s and $q$'s,
| ${\begin{aligned}&\Theta ^{2}{\frac {d}{d\Theta }}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}(\epsilon -{\overline {\epsilon }})^{h}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1},\ldots dq_{n}=\\&\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg\epsilon (\epsilon -{\overline {\epsilon }})^{h}-h(\epsilon -{\overline {\epsilon }})^{h-1}\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }}{\bigg ]}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1},\ldots dq_{n}\end{aligned}}$ | (229) |
| $\Theta ^{2}{\frac {d}{d\Theta }}{\bigg{\overline {(\epsilon -{\overline {\epsilon }})^{h}}}e^{-{\frac {\psi }{\Theta }}}{\bigg ]}={\bigg{\overline {\epsilon (\epsilon -{\overline {\epsilon }})^{h}}}-h{\overline {(\epsilon -{\overline {\epsilon }})^{h-1}}}\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }}{\bigg ]}e^{-{\frac {\psi }{\Theta }}},$ | (230) |
* $\Theta ^{2}{\frac {d}{d\Theta }}e^{-{\frac {\psi }{\Theta }}}={\overline {\epsilon }}e^{-{\frac {\psi }{\Theta }}},$
* $\Theta ^{2}{\frac {d}{d\Theta }}{\overline {(\epsilon -{\overline {\epsilon }})^{h}}}+{\overline {(\epsilon -{\overline {\epsilon }})^{h}}}{\overline {\epsilon }}={\overline {\epsilon (\epsilon -{\overline {\epsilon }})^{h}}}-h{\overline {(\epsilon -{\overline {\epsilon }})^{h-1}}}\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }},$
| ${\overline {(\epsilon -{\overline {\epsilon }})^{h+1}}}=\Theta ^{2}{\frac {d}{d\Theta }}{\overline {(\epsilon -{\overline {\epsilon }})^{h}}}+h{\overline {(\epsilon -{\overline {\epsilon }})^{h-1}}}\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }}.$ | (231) |
In precisely the same way we may obtain for the potential energy
| ${\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{h+1}}}=\Theta ^{2}{\frac {d}{d\Theta }}{\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{h}}}+h{\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{h-1}}}\Theta ^{2}{\frac {d{\overline {\epsilon }}_{q}}{d\Theta }}.$ | (232) |
By successive applications of (231) we obtain
* ${\begin{aligned}{\overline {(\epsilon -{\overline {\epsilon }})^{2}}}&=D{\overline {\epsilon }}\\{\overline {(\epsilon -{\overline {\epsilon }})^{3}}}&=D^{2}{\overline {\epsilon }}\\{\overline {(\epsilon -{\overline {\epsilon }})^{4}}}&=D^{3}{\overline {\epsilon }}+3(D{\overline {\epsilon }})^{2}\\{\overline {(\epsilon -{\overline {\epsilon }})^{5}}}&=D^{4}{\overline {\epsilon }}+10D{\overline {\epsilon }}D^{2}{\overline {\epsilon }}\\{\overline {(\epsilon -{\overline {\epsilon }})^{6}}}&=D^{5}{\overline {\epsilon }}+15D{\overline {\epsilon }}D^{3}{\overline {\epsilon }}+10(D^{2}{\overline {\epsilon }})^{2}+15(D{\overline {\epsilon }})^{3}\mathrm {etc.} \end{aligned}}$
$D$
$\Theta ^{2}\,d/d\Theta$
For the kinetic energy we may write similar equations in which the averages may be taken either for a single configuration or for the whole ensemble. But since
* ${\frac {d{\overline {\epsilon }}_{p}}{d\Theta }}={\frac {n}{2}}$
| ${\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h+1}}}=\Theta ^{2}{\frac {d}{d\Theta }}{\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h}}}+{\tfrac {1}{2}}nh\Theta ^{2}{\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h-1}}}$ | (233) |
| ${\frac {\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h+1}}}{{\overline {\epsilon }}_{p}^{h+1}}}={\frac {2\Theta }{n}}{\frac {d}{d\Theta }}{\frac {\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h}}}{{\overline {\epsilon }}_{p}^{h}}}+{\frac {2h}{n}}{\frac {\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h}}}{{\overline {\epsilon }}_{p}^{h}}}+{\frac {2h}{n}}{\frac {\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h-1}}}{{\overline {\epsilon }}_{p}^{h-1}}}$ | (234) |
* ${\frac {\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{0}}}{{\overline {\epsilon }}_{p}^{0}}}=1,\quad {\frac {\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{1}}}{{\overline {\epsilon }}_{p}^{1}}}=0$
$\Theta$
| ${\overline {{\bigg{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{h+1}}}={\frac {2h}{n}}{\overline {{\bigg{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{h}}}+{\frac {2h}{n}}{\overline {{\bigg{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{h-1}}}$ | (235) |
* ${\overline {{\bigg{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{0}}}=1,\quad {\overline {{\bigg{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{3}}}={\frac {8}{n^{2}}},$
* ${\overline {{\bigg{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{1}}}=0,\quad {\overline {{\bigg{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{4}}}={\frac {48}{n^{3}}}+{\frac {12}{n^{2}}},$
* ${\overline {{\bigg{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{2}}}={\frac {2}{n}},\quad \mathrm {etc.}$
It will be observed that when $\psi$ or ${\overline {\epsilon }}$ is given as function of $\Theta$, all averages of the form ${\overline {\epsilon ^{h}}}$ or ${\overline {(\epsilon -{\overline {\epsilon }})^{h}}}$ are thereby
​determined. So also if $\psi _{q}$ or ${\overline {\epsilon }}_{q}$ is given as a function of $\Theta$, all averages of the form ${\overline {\epsilon _{q}^{h}}}$ or ${\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{h})}}$ are determined. But
* ${\overline {\epsilon }}_{q}={\overline {\epsilon }}-{\tfrac {1}{2}}n\Theta .$
$\psi$
$\psi _{q}$
${\overline {\epsilon }}$
${\overline {\epsilon }}_{q}$
$\Theta$
$n$
* ${\overline {{\bigg{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{h}}}$
$n$
If we differentiate the equation
| $\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=1$ | (236) |
$a_{1}$
$\Theta$
| $\int \ldots \int {\bigg{\frac {d\psi }{da_{1}}}-{\frac {d\epsilon }{da_{1}}}{\bigg ]}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=0.$ | (237) |
$a_{1}$
$a_{2}$
$\Theta$
| $\int \ldots \int {\bigg{\frac {d^{2}\psi }{da_{1}^{2}}}-{\frac {d^{2}\epsilon }{da_{1}^{2}}}+{\frac {1}{\Theta }}{\bigg{\frac {d\psi }{da_{1}}}-{\frac {d\epsilon }{da_{1}}}{\bigg )}^{2}{\bigg ]}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=0,$ | (238) |
| ${\begin{aligned}\int \ldots \int {\bigg{\frac {d^{2}\psi }{da_{1}\,da_{2}}}-{\frac {d^{2}\epsilon }{da_{1}\,da_{2}}}&+{\frac {1}{\Theta }}{\bigg{\frac {d\psi }{da_{1}}}-{\frac {d\epsilon }{da_{1}}}{\bigg )}{\bigg{\frac {d\psi }{da_{2}}}-{\frac {d\epsilon }{da_{2}}}{\bigg )}{\bigg ]}\\&\qquad \qquad \qquad e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=0,\end{aligned}}$ | (239) |
| ${\begin{aligned}\int \ldots \int {\bigg{\frac {d^{2}\psi }{da_{1}\,d\Theta }}+{\bigg{\frac {d\psi }{da_{1}}}-{\frac {d\epsilon }{da_{1}}}{\bigg )}&{\bigg{\frac {1}{\Theta }}{\frac {d\psi }{d\Theta }}-{\frac {\psi -\epsilon }{\Theta ^{2}}}{\bigg )}{\bigg ]}\\&\qquad e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=0.\end{aligned}}$ | (240) |
| ${\frac {d\psi }{da_{1}}}={\overline {\frac {d\epsilon }{da_{1}}}}=-{\overline {A}}_{1},$ | (241) |
| ${\overline {(A_{1}-{\overline {A}}_{1})^{2}}}=\Theta {\bigg{\overline {\frac {d^{2}\epsilon }{da_{1}^{2}}}}-{\frac {d^{2}\psi }{da_{1}^{2}}}{\bigg )}=\Theta {\bigg{\frac {d{\overline {A}}_{1}}{da_{1}}}-{\overline {\frac {dA_{1}}{da_{1}}}}{\bigg )}$ | (242) |
| ${\begin{aligned}{\overline {(A_{1}-{\overline {A}}_{1})(A_{2}-{\overline {A}}_{2})}}&=\Theta {\bigg{\overline {\frac {d^{2}\epsilon }{da_{1}\,da_{2}}}}-{\frac {d^{2}\psi }{da_{1}\,da_{2}}}{\bigg )}\\&=\Theta {\bigg{\frac {d{\overline {A}}_{1}}{da_{2}}}-{\overline {\frac {dA_{1}}{da_{2}}}}{\bigg )}=\Theta {\bigg{\frac {d{\overline {A}}_{2}}{da_{1}}}-{\overline {\frac {dA_{2}}{da_{1}}}}{\bigg )}\end{aligned}}$ | (243) |
* ${\overline {(A_{1}-{\overline {A}}_{1})(\epsilon -{\overline {\epsilon }})}}=-\Theta ^{2}{\frac {d^{2}\psi }{da_{1}\,d\Theta }}=\Theta ^{2}{\frac {d{\overline {A}}_{1}}{d\Theta }}=-\Theta ^{2}{\frac {d{\overline {\eta }}}{da_{1}}}.$
$\Theta$
| ${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}=-\Theta ^{2}{\frac {d^{2}\psi }{d\Theta ^{2}}}=\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }}.$ | (244) |
| ${\overline {(A_{1}-{\overline {A}}_{1})(\epsilon -{\overline {\epsilon }})}}={\frac {d{\overline {A}}}{d{\overline {\epsilon }}}}{\overline {(\epsilon -{\overline {\epsilon }})^{2}}}.$ | (245) |
If $\psi$ or ${\overline {\epsilon }}$ is known as function of $\Theta$, $a_{1}$, $a_{2}$, etc., ${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}$ may be obtained by differentiation as function of the same variables. And if $\psi$, or ${\overline {A}}_{1}$, or ${\overline {\eta }}$ is known as function of $\Theta$, $a_{1}$, etc., ${\overline {(A_{1}-{\overline {A}}_{1})(\epsilon -{\overline {\epsilon }})}}$ may be obtained by differentiation. But ${\overline {(A_{1}-{\overline {A}}_{1})^{2}}}$ and ${\overline {(A_{1}-{\overline {A}}_{1})(A_{2}-{\overline {A}}_{2})}}$ cannot be obtained in any similar manner. We have seen that ${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}$ is in general a vanishing quantity for very great values of $n$, which we may regard as contained implicitly in $\Theta$ as a divisor. The same is true of ${\overline {(A_{1}-{\overline {A}}_{1})(\epsilon -{\overline {\epsilon }})}}$. It does not appear that we can assert the same of ${\overline {(A_{1}-{\overline {A}}_{1})^{2}}}$ or ${\overline {(A_{1}-{\overline {A}}_{1})(A_{2}-{\overline {A}}_{2})}}$, since ​$d^{2}\epsilon /da_{1}^{2}$ may be very great. The quantities $d^{2}\epsilon /da_{1}^{2}$ and $d^{2}\psi /da_{1}^{2}$ belong to the class called elasticities. The former expression represents an elasticity measured under the condition that while $a_{1}$ is varied the internal coördinates $q_{1},\ldots q_{n}$ all remain fixed. The latter is an elasticity measured under the condition that when $a_{1}$ is varied the ensemble remains canonically distributed within the same modulus. This corresponds to an elasticity in physics measured under the condition of constant temperature. It is evident that the former is greater than the latter, and it may be enormously greater.
The divergences of the force $A_{1}$ from its average value are due in part to the differences of energy in the systems of the ensemble, and in part to the differences in the value of the forces which exist in systems of the same energy. If we write ${\overline {A_{1}}}|_{\epsilon }$ for the average value of $A_{1}$ in systems of the ensemble which have any same energy, it will be determined by the equation
| ${\overline {A_{1}}}|_{\epsilon }={\frac {\int \ldots \int -{\frac {d\epsilon }{da_{1}}}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}}{\int \ldots \int e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}}}$ | (246) |
$\epsilon$
$\epsilon +d\epsilon$
$e^{\frac {\psi -\epsilon }{\Theta }}$
| ${\overline {A_{1}}}|_{\epsilon }={\frac {\int \ldots \int -{\frac {d\epsilon }{da_{1}}}\,dp_{1}\ldots dq_{n}}{\int \ldots \int \,dp_{1}\ldots dq_{n}}}$ | (247) |
$\epsilon$
$\epsilon +d\epsilon$
${\overline {A_{1}}}|_{\epsilon }$
$\Theta$
​Now we have identically
* $A_{1}-{\overline {A}}_{1}=(A_{1}-{\overline {A_{1}}}|_{\epsilon })+({\overline {A_{1}}}|_{\epsilon }-{\overline {A}}_{1}),$
$A_{1}-{\overline {A_{1}}}|_{\epsilon }$
$a_{1}$
* ${\overline {(A_{1}-{\overline {A}}_{1})^{2}}}={\overline {(A_{1}-{\overline {A_{1}}}|_{\epsilon })^{2}}}+2{\overline {(A_{1}-{\overline {A_{1}}}|_{\epsilon })({\overline {A_{1}}}|_{\epsilon }-{\overline {A}}_{1})}}+{\overline {({\overline {A_{1}}}|_{\epsilon }-{\overline {A}}_{1})^{2}}}.$
$(A_{1}-{\overline {A_{1}}}|_{\epsilon })({\overline {A_{1}}}|_{\epsilon }-{\overline {A}}_{1})$
| ${\overline {(A_{1}-{\overline {A}}_{1})^{2}}}={\overline {(A_{1}-{\overline {A_{1}}}|_{\epsilon })^{2}}}+{\overline {({\overline {A_{1}}}|_{\epsilon }-{\overline {A}}_{1})^{2}}}.$ | (248) |
| ${\overline {(A_{1}-{\overline {A}}_{1})(\epsilon -{\overline {\epsilon }})}}={\overline {({\overline {A_{1}}}|_{\epsilon }-{\overline {A}}_{1})(\epsilon -{\overline {\epsilon }})}}.$ | (249) |
$\epsilon -{\overline {\epsilon }}$
${\overline {A_{1}}}|_{\epsilon }-{\overline {A}}_{1}$
The properties of quantities of the form ${\overline {A_{1}}}|_{\epsilon }$ will be farther considered in Chapter X, which will be devoted to ensembles of constant energy.
It may not be without interest to consider some general formulae relating to averages in a canonical ensemble, which embrace many of the results which have been given in this chapter.
Let $u$ be any function of the internal and external coördinates with the momenta and modulus. We have by definition
| ${\overline {u}}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}ue^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}$ | (250) |
$\Theta$
* ${\frac {d{\overline {u}}}{d\Theta }}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg{\frac {du}{d\Theta }}-{\frac {u}{\Theta ^{2}}}(\psi -\epsilon )+{\frac {u}{\Theta }}{\frac {d\psi }{d\Theta }}{\bigg )}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n},$
| ${\frac {d{\overline {u}}}{d\Theta }}={\overline {\frac {du}{d\Theta }}}-{\overline {\frac {u(\psi -\epsilon )}{\Theta ^{2}}}}+{\frac {\overline {u}}{\Theta }}{\frac {d\psi }{d\Theta }}.$ | (251) |
$u=1$
* ${\frac {d\psi }{d\Theta }}={\frac {\psi -{\overline {\epsilon }}}{\Theta }},$
* ${\frac {\overline {u}}{d\Theta }}={\overline {\frac {du}{d\Theta }}}+{\frac {\overline {u\epsilon }}{\Theta ^{2}}}-{\frac {{\overline {u}}\,{\overline {\epsilon }}}{\Theta ^{2}}}$
| $\Theta ^{2}{\frac {\overline {u}}{d\Theta }}-\Theta ^{2}{\overline {\frac {du}{d\Theta }}}={\overline {u\epsilon }}-{\overline {u}}\,{\overline {\epsilon }}={\overline {(u-{\overline {u}})(\epsilon -{\overline {\epsilon }})}}.$ | (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
* ${\frac {\overline {u}}{da}}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg{\frac {du}{da}}+{\frac {u}{\Theta }}{\frac {d\psi }{da}}+{\frac {u}{\Theta }}A{\bigg )}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}$
| ${\frac {\overline {u}}{da}}={\overline {\frac {du}{da}}}+{\frac {\overline {u}}{\Theta }}{\frac {d\psi }{da}}+{\frac {\overline {uA}}{\Theta }}$ | (253) |
$u=1$
* ${\frac {d\psi }{da}}=-{\overline {A}}.$
| ${\frac {\overline {u}}{da}}={\overline {\frac {du}{da}}}+{\frac {\overline {uA}}{\Theta }}-{\frac {{\overline {u}}\,{\overline {A}}}{\Theta }},$ | (254) |
| $\Theta {\frac {\overline {u}}{da}}-\Theta {\overline {\frac {du}{da}}}={\overline {uA}}-{\overline {u}}\,{\overline {A}}={\overline {(u-{\overline {u}})(A-{\overline {A}})}}.$ | (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 ​
| ${\overline {(\epsilon +D)(u-{\overline {u}})}}=0,$ | (256) |
$D$
$\Theta ^{2}\,d/d\Theta$
| ${\overline {(\epsilon +D)^{h}(u-{\overline {u}})}}=0,$ | (257) |
$h$
$\epsilon$
$\Theta$
$(\epsilon +D)^{h}$
| ${\overline {(\epsilon +D)u}}=({\overline {\epsilon }}+D){\overline {u}},$ | (258) |
| ${\overline {(\epsilon +D)^{h}u}}=({\overline {\epsilon }}+D)^{h}{\overline {u}}.$ | (259) |
But the operator $({\overline {\epsilon }}+D)^{h}$, although in some respects more simple than the operator without the average sign on the $\epsilon$, cannot be expanded by the binomial theorem, since ${\overline {\epsilon }}$ is a function of $\Theta$ with the external coördinates.
So from equation (254) we have
| ${\overline {{\bigg{\frac {A}{\Theta }}+{\frac {d}{da}}{\bigg )}(u-{\overline {u}})}}=0,$ | (260) |
| ${\overline {{\bigg{\frac {A}{\Theta }}+{\frac {d}{da}}{\bigg )}^{h}(u-{\overline {u}})}}=0;$ | (261) |
| ${\overline {{\bigg{\frac {A}{\Theta }}+{\frac {d}{da}}{\bigg )}u}}={\bigg{\frac {\overline {A}}{\Theta }}+{\frac {d}{da}}{\bigg )}{\overline {u}},$ | (262) |
| ${\overline {{\bigg{\frac {A}{\Theta }}+{\frac {d}{da}}{\bigg )}^{h}u}}={\bigg{\frac {\overline {A}}{\Theta }}+{\frac {d}{da}}{\bigg )}^{h}{\overline {u}}.$ | (263) |
Again, if we now distinguish, as usual, the several external coördinates by suffixes, we may apply successively to the expression $u-{\overline {u}}$ any or all of the operators
| $\epsilon +\Theta ^{2}{\frac {d}{d\Theta }},\quad A_{1}+\Theta {\frac {d}{da_{1}}},\quad A_{2}+\Theta {\frac {d}{da_{2}}},\quad \mathrm {etc.}$ | (264) |
$u$
$u$
$\epsilon$
$A_{1}$
$A_{2}$
If $u$ is independent of the momenta, formulae similar to the preceding, but having $\epsilon _{q}$ in place of $\epsilon$, may be derived from equation (179).
* ↑ Pogg. Ann., Bd. CXVI, S. 73, (1862); ibid., Bd. CXXV, S. 353, (1865). See also Boltzmann, Sitzb. der Wiener Akad., Bd. LXIII, S. 728, (1871).
* ↑ Verwandlungswerth des Wärmeinhaltes.
* ↑ In the case discussed in the note on page 54, in which the potential energy is a quadratic function of the $q$'s, and $\Delta _{\dot {q}}$ independent of the $q$'s, we should get for the potential energy ${\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{2}}}={\frac {1}{2}}n\Theta ^{2},$ and for the total energy ${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}=n\Theta ^{2}.$ We may also write in this case, ${\frac {\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{2}}}{({\overline {\epsilon }}_{q}-\epsilon _{a})^{2}}}={\frac {2}{n}},$ ${\frac {\overline {(\epsilon -{\overline {\epsilon }})^{2}}}{({\overline {\epsilon }}-\epsilon _{a})^{2}}}={\frac {1}{n}}.$
* ↑ This implies that the kinetic and potential energies of individual systems would each separately have values sensibly constant in time.
* ↑ As an example, we may take a system consisting of a fluid in a cylinder under a weighted piston, with a vacuum between the piston and the top of the cylinder, which is closed. The weighted piston is to be regarded as a part of the system. (This is formally necessary in order to satisfy the condition of the invariability of the external coördinates.) It is evident that at a certain temperature, viz., when the pressure of saturated vapor balances the weight of the piston, there is an indeterminateness in the values of the potential and total energies as functions of the temperature.
* ↑ In the case discussed in the note on page 54 we may easily get ${\overline {(\epsilon _{q}-\epsilon _{a})^{h}}}={\bigg{\overline {\epsilon }}_{q}-\epsilon _{a}+\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}{\overline {(\epsilon _{q}-\epsilon _{a})^{h-1}}},$ which, with ${\overline {\epsilon }}_{q}-\epsilon _{a}={\frac {n}{2}}\Theta ,$ gives ${\overline {(\epsilon _{q}-\epsilon _{a})^{h}}}={\bigg{\frac {n}{2}}\Theta +\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}{\overline {(\epsilon _{q}-\epsilon _{a})^{h-1}}}={\frac {n}{2}}{\bigg{\frac {n}{2}}\Theta +\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h-1}\Theta .$ Hence ${\overline {(\epsilon _{q}-\epsilon _{a})^{h}}}={\overline {\epsilon _{p}^{h}}}.$ Again ${\overline {(\epsilon -\epsilon _{a})^{h}}}={\bigg{\overline {\epsilon }}-\epsilon _{a}+\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}{\overline {(\epsilon -\epsilon _{a})^{h-1}}},$ which with ${\overline {\epsilon }}-\epsilon _{a}=n\Theta$ gives ${\overline {(\epsilon -\epsilon _{a})^{h}}}={\biggn\Theta +\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}{\overline {(\epsilon -\epsilon _{a})^{h-1}}}=n{\biggn\Theta +\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h-1}\Theta ,$ hence ${\overline {(\epsilon -\epsilon _{a})^{h}}}={\frac {\Gamma (n+h)}{\Gamma (n)}}\Theta ^{h}.$
* ↑ In the case discussed in the preceding foot-notes we get easily ${\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{h}}}={\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h}}},$ and ${\overline {{\bigg{\frac {\epsilon _{q}-{\overline {\epsilon }}_{q}}{{\overline {\epsilon }}_{q}-\epsilon _{a}}}{\bigg )}^{h}}}={\overline {{\bigg{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{h}}}.$ For the total energy we have in this case ${\overline {{\bigg{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{h+1}}}={\frac {h}{n}}{\overline {{\bigg{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{h}}}+{\overline {{\frac {h}{n}}{\bigg{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{h-1}}}.$ ${\overline {{\bigg{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{2}}}={\frac {1}{n}}.\quad {\overline {{\bigg{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{4}}}={\frac {3}{n^{2}}}+{\frac {6}{n^{3}}},$ ${\overline {{\bigg{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{3}}}={\frac {2}{n^{2}}},\quad \mathrm {etc.}$