# Elementary Principles in Statistical Mechanics/Chapter VIII

CHAPTER VIII.
ON CERTAIN IMPORTANT FUNCTIONS OF THE ENERGIES OF A SYSTEM.
In order to consider more particularly the distribution of a canonical ensemble in energy, and for other purposes, it will be convenient to use the following definitions and notations.
Let us denote by $V$ the extension-in-phase below a certain limit of energy which we shall call $\epsilon$. That is, let
| $V=\int \ldots \int dp_{1}\ldots dq_{n},$ | (265) |
$\epsilon$
* $\int \ldots \int e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n}$
* $e^{-{\frac {\epsilon }{\Theta }}}\int \ldots \int dp_{1}\ldots dq_{n}$
$\epsilon$
$\epsilon$
$V$
$\epsilon$
$\epsilon$
$\epsilon$
$\epsilon$
$\epsilon$
$\epsilon =-\infty$
Let us also set
| $\phi =\log {\frac {dV}{d\epsilon }}.$ | (266) |
$\epsilon '$
$\epsilon ''$
| $\int _{\epsilon '}^{\epsilon ''}e^{\phi }\,d\epsilon .$ | (267) |
$e^{\phi }\,d\epsilon$
$dp_{1}\ldots dq_{n}$
$2n$
In particular we observe that the probability that the energy of an unspecified system of a canonical ensemble lies between the limits $\epsilon '$ and $\epsilon ''$ will be represented by the integral
| $\int _{\epsilon '}^{\epsilon ''}e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon ,$ | (268) |
| ${\overline {u}}=\int _{V=0}^{\epsilon =\infty }ue^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon ,$ | (269) |
$\psi$
| $e^{-{\frac {\psi }{\Theta }}}=\int _{V=0}^{\epsilon =\infty }e^{-{\frac {\epsilon }{\Theta }}+\phi }\,d\epsilon ,$ | (270) |
$V=0$
$\epsilon$
​In like manner, let us denote by $V_{q}$ the extension-in-configuration below a certain limit of potential energy which we may call $\epsilon _{q}$. That is, let
| $V_{q}=\int \ldots \int \Delta _{\dot {q}}^{\frac {1}{2}}dq_{1}\ldots dq_{n},$ | (271) |
$\epsilon _{q}$
$V_{q}$
$\epsilon _{q}$
$\epsilon _{q}$
$\epsilon _{q}$
$\epsilon _{q}$
$\epsilon _{q}=-\infty$
$\epsilon _{q}$
$\epsilon _{q}$
$V_{q}$
$\epsilon _{q}$
$V_{q}$
Let us also set
| $\phi _{q}=\log {\frac {dV_{q}}{d\epsilon _{q}}}.$ | (272) |
$\epsilon _{q}'$
$\epsilon _{q}''$
| $\int _{\epsilon _{q}'}^{\epsilon _{q}''}e^{\phi _{q}}\,d\epsilon _{q}$ | (273) |
$V_{q}$
$\epsilon _{q}$
$e^{\phi _{q}}\,d\epsilon _{q}$
$\Delta _{\dot {q}}^{\frac {1}{2}}dq_{1}\ldots dq_{n},$
$n$
We may often avoid the inconvenience occasioned by formulae becoming illusory on account of discontinuities in the values of $V_{q}$ as function of $\epsilon _{q}$ by substituting for the given discontinuous function a continuous function which is practically equivalent to the given function for the purposes of the evaluations desired. It only requires infinitesimal changes of potential energy to destroy the finite extensions-in-configuration of constant potential energy which are the cause of the difficulty.
In the case of an ensemble of systems canonically distributed in configuration, when $V_{q}$ is, or may be regarded as, a continuous function of $\epsilon _{q}$ (within the limits considered), the probability that the potential energy of an unspecified system lies between the limits $\epsilon _{q}'$ and $\epsilon _{q}''$ is given by the integral
| $\int _{\epsilon _{q}'}^{\epsilon _{q}''}e^{{\frac {\psi _{q}-\epsilon _{q}}{\Theta }}+\phi _{q}}d\epsilon _{q},$ | (274) |
$\psi$
$\epsilon _{q}$
| ${\overline {u}}=\int _{V_{q}=0}^{\epsilon _{q}=\infty }ue^{{\frac {\psi -\epsilon _{q}}{\Theta }}+\phi _{q}}d\epsilon _{q}.$ | (275) |
$V_{q}$
$\epsilon _{q}$
$dV_{q}$
$e^{\phi _{q}}d\epsilon _{q}$
In like manner also, for any given configuration, let us denote by $V_{p}$ the extension-in-velocity below a certain limit of kinetic energy specified by $\epsilon _{p}$. That is, let
| $V_{p}=\int \ldots \int \Delta _{p}^{\frac {1}{2}}dp_{1}\ldots dp_{n},$ | (276) |
$\epsilon _{p}$
$V_{p}$
$\epsilon _{p}$
$\epsilon _{p}$
| $\phi _{p}=\log {\frac {V_{p}}{d\epsilon _{p}}}.$ | (277) |
$\epsilon _{p}'$
$\epsilon _{p}''$
| $\int _{\epsilon _{p}'}^{\epsilon _{p}''}e^{\phi _{p}}\,d\epsilon _{p}.$ | (278) |
$e^{\phi _{p}}\,d\epsilon _{p}$
$\Delta _{p}^{\frac {1}{2}}dp_{1}\ldots dp_{n}$
$\Delta _{\dot {q}}^{\frac {1}{2}}d{\dot {q}}_{1}\ldots d{\dot {q}}_{n}$
$n$
It is easy to express $V_{p}$ and $\phi _{p}$ in terms of $\epsilon _{p}$. Since $\Delta _{p}$ is function of the coördinates alone, we have by definition
| $V_{p}=\Delta _{p}^{\frac {1}{2}}\int \ldots \int dp_{1}\ldots dp_{n}$ | (279) |
$\epsilon _{p}$
| $\epsilon _{p}=F(p_{1},\ldots p_{n}),$ | (280) |
$\epsilon _{p}=1$
| $F(p_{1},\ldots p_{n})=1,$ | (281) |
$\epsilon _{p}=a^{2}$
| $F(p_{1},\ldots p_{n})=a^{2}.$ | (282) |
$F$
| $F{\bigg{\frac {p_{1}}{a}},\ldots {\frac {p_{n}}{a}}{\bigg )}=1$ | (283) |
$V_{p}$
| $V_{p}=a^{n}\Delta _{p}^{\frac {1}{2}}\int \ldots \int d{\frac {p_{1}}{a}}\ldots d{\frac {p_{n}}{a}}.$ | (284) |
$V_{p}$
$\epsilon _{p}=1$
$V_{p}$
$\epsilon _{p}=a^{2}$
| $(V_{p})_{\epsilon _{p}=a^{2}}=a^{n}(V_{p})_{\epsilon _{p}=1}$ | (285) |
$V_{p}$
$\epsilon _{p}^{\frac {n}{2}}$
| $V_{p}=C\epsilon _{p}^{\frac {n}{2}},\quad e^{\phi _{p}}={\frac {n}{2}}C\epsilon _{p}^{{\frac {n}{2}}-1}$ | (286) |
$C$
To determine this constant, let us consider the case of a canonical distribution, for which we have
* $\int _{0}^{\infty }e^{{\frac {\psi _{p}-\epsilon _{p}}{\Theta }}+\phi _{p}}\,d\epsilon _{p}=1,$
* $e^{\frac {\psi _{p}}{\Theta }}=(2\pi \Theta )^{-{\frac {n}{2}}}.$
Substituting this value, and that of $e^{\phi _{p}}$ from (286), we get
* ${\frac {n}{2}}C\int _{0}^{\infty }e^{-{\frac {\epsilon _{p}}{\Theta }}}\epsilon _{p}^{{\frac {n}{2}}-1}\,d\epsilon _{p}=(2\pi \Theta )^{\frac {n}{2}},$
* ${\frac {n}{2}}C\int _{0}^{\infty }e^{-{\frac {\epsilon _{p}}{\Theta }}}{\bigg{\frac {\epsilon _{p}}{\Theta }}{\bigg )}^{{\frac {n}{2}}-1}\,d{\bigg{\frac {\epsilon _{p}}{\Theta }}{\bigg )}=(2\pi )^{\frac {n}{2}},$
| ${\frac {n}{2}}C\Gamma {\bigg{\frac {n}{2}}{\bigg )}=(2\pi )^{\frac {n}{2}},$ | (287) |
* $C={\frac {(2\pi )^{\frac {n}{2}}}{\Gamma ({\tfrac {1}{2}}n+1)}}.$
$C$
| $V_{p}={\frac {(2\pi \epsilon _{p})^{\frac {n}{2}}}{\Gamma ({\tfrac {1}{2}}n+1)}}$ | (288) |
| $e^{\phi _{p}}={\frac {(2\pi )^{\frac {n}{2}}\epsilon _{p}^{{\frac {n}{2}}-1}}{\Gamma ({\tfrac {1}{2}}n)}}$ | (289) |
It will be observed that the values of $V_{p}$ and $\phi _{p}$ for any given $\epsilon _{p}$ are independent of the configuration, and even of the nature of the system considered, except with respect to its number of degrees of freedom.
Returning to the canonical ensemble, we may express the probability that the kinetic energy of a system of a given configuration, but otherwise unspecified, falls within given limits, by either member of the following equation
| $\int e^{{\frac {\psi _{p}-\epsilon _{p}}{\Theta }}+\phi _{p}}\,d\epsilon _{p}={\frac {1}{\Gamma ({\tfrac {1}{2}}n)}}\int e^{-{\frac {\epsilon _{p}}{\Theta }}}{\bigg{\frac {\epsilon _{p}}{\Theta }}{\bigg )}^{{\tfrac {n}{2}}-1}\,d{\bigg{\frac {\epsilon _{p}}{\Theta }}{\bigg )}.$ | (290) |
The average value of any function of the kinetic energy, either for the whole ensemble, or for any particular configuration, is given by
| ${\overline {u}}={\frac {1}{\Theta ^{\frac {n}{2}}\Gamma ({\tfrac {1}{2}}n)}}\int _{0}^{\infty }u\,e^{-{\frac {\epsilon _{p}}{\Theta }}}{\epsilon _{p}}^{{\tfrac {n}{2}}-1}\,d\epsilon _{p}$ | (291) |
Thus:
| ${\overline {{\epsilon _{p}}^{m}}}={\frac {\Gamma (m+{\tfrac {1}{2}}n)}{\Gamma ({\tfrac {1}{2}}n)}}\Theta ^{m},\quad \mathrm {if} \quad m+{\tfrac {1}{2}}n>0;$ | (292) |
| ${\overline {V_{p}}}={\frac {\Gamma (n)}{\Gamma ({\tfrac {1}{2}}n+1)\Gamma ({\tfrac {1}{2}}n)}}(2\pi \Theta )^{\tfrac {n}{2}};$ | (293) |
| ${\overline {e^{\phi _{p}}}}={\frac {\Gamma (n-1)}{[\Gamma ({\tfrac {1}{2}}n)]^{2}}}(2\pi )^{\tfrac {n}{2}}\Theta ^{{\tfrac {n}{2}}-1},\quad \mathrm {if} \quad n>1;$ | (294) |
| ${\overline {\frac {d\phi _{p}}{d\epsilon _{p}}}}={\frac {1}{\Theta }},\quad \mathrm {if} \quad n>2;$ | (295) |
| ${\overline {e^{-\phi _{p}}V_{p}}}=\Theta ,.$ | (296) |
$n=2$
$e^{\phi _{p}}=2\pi$
$d\phi _{p}/d\epsilon _{p}=0$
$\epsilon _{p}$
The definitions of $V$, $V_{q}$, and $V_{p}$ give
| $V=\int \int dV_{p}\,dV_{q}$ | (297) |
$\epsilon$
$V$
| $V=\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }V_{p}\,dV_{q},$ | (298) |
| $e^{\phi }={\frac {dV}{d\epsilon }}=\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }e^{\phi _{p}}\,dV_{q},$ | (299) |
$V_{p}$
$e^{\phi _{p}}$
$V_{q}$
| $\epsilon _{p}+\epsilon _{q}=\mathrm {constant} =\epsilon .$ | (300) |
If $n>2$, $e^{\phi _{p}}$ vanishes at the upper limit, i. e., for $\epsilon _{p}=0$, and we get by another differentiation
| $e^{\phi }{\frac {d\phi }{d\epsilon }}=\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }e^{\phi _{p}}{\frac {d\phi _{p}}{d\epsilon _{p}}}\,dV_{q}.$ | (301) |
| $V=\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }V_{p}e^{\phi _{q}}\,d\epsilon _{q},$ | (302) |
| $e^{\phi }=\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }e^{\phi _{p}+\phi _{q}}\,d\epsilon _{q},$ | (303) |
$V_{q}$
$\epsilon _{q}$
$V_{q}=0$
$V_{q}$
If we substitute in these equations the values of $V_{p}$ and $e^{\phi _{p}}$ which we have found, we get
| $V={\frac {(2\pi )^{\tfrac {n}{2}}}{\Gamma ({\tfrac {1}{2}}n+1)}}\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }(\epsilon -\epsilon _{q})^{\tfrac {n}{2}}dV_{q},$ | (304) |
| $e^{\phi }={\frac {(2\pi )^{\tfrac {n}{2}}}{\Gamma ({\tfrac {1}{2}}n)}}\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }(\epsilon -\epsilon _{q})^{{\tfrac {n}{2}}-1}dV_{q},$ | (305) |
$e^{\phi _{q}}d\epsilon _{q}$
$dV_{q}$
$n$
$V_{q}$
$\epsilon _{q}$
$V$
$e^{\phi }$
It appears from these equations that $V$ is always a continuous increasing function of $\epsilon$, commencing with the value $V=0$, even when this is not the case with respect to $V_{q}$ and $\epsilon _{q}$. The same is true of $e^{\phi }$, when $n>2$, or when $n=2$ if $V_{q}$ increases continuously with $\epsilon _{q}$ from the value $V_{q}=0$.
The last equation may be derived from the preceding by differentiation with respect to $\epsilon$. Successive differentiations give, if $h<{\tfrac {1}{2}}n+1$,
| ${\frac {d^{h}V}{d\epsilon ^{h}}}=\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }{\frac {d^{h}V_{p}}{d\epsilon _{p}^{h}}}\,dV_{q}={\frac {(2\pi )^{\tfrac {n}{2}}}{\Gamma ({\tfrac {1}{2}}n+1-h)}}\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }(\epsilon -\epsilon _{q})^{{\tfrac {n}{2}}-h}dV_{q}.$ | (306) |
$d^{h}V/d\epsilon ^{h}$
$h<{\tfrac {1}{2}}n+1$
$\epsilon$
$h<{\tfrac {1}{2}}n$
$\epsilon$
$d^{h}V/d\epsilon ^{h}$
$\epsilon$
$h<{\tfrac {1}{2}}n$
$n$
| ${\frac {d^{\tfrac {n}{2}}V}{d\epsilon ^{\tfrac {n}{2}}}}=(2\pi )^{\tfrac {n}{2}}(V_{q})_{\epsilon _{q}=\epsilon }$ | (307) |
$V_{q}$
$\epsilon +q$
${\frac {1}{(2\pi )^{\tfrac {n}{2}}}}{\frac {d^{\tfrac {n}{2}}V}{d\epsilon ^{\tfrac {n}{2}}}}$
$\epsilon$
When $n$ is large, approximate formulae will be more available. It will be sufficient to indicate the method proposed, without precise discussion of the limits of its applicability or of the degree of its approximation. For the value of $e^{\phi }$ corresponding to any given $\epsilon$, we have
| $e^{\phi }=\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }e^{\phi _{p}+\phi _{q}}\,d\epsilon _{q}=\int \limits _{0}^{\epsilon }e^{\phi _{p}+\phi _{q}}\,d\epsilon _{p},$ | (308) |
$\phi _{p}+\phi _{q}$
| ${\frac {d\phi _{p}}{d\epsilon _{p}}}={\frac {d\phi _{q}}{d\epsilon _{q}}}.$ | (309) |
$\epsilon _{p}$
$\epsilon _{q}$
$\epsilon _{p}$
$\epsilon _{q}$
| $\phi _{p}=\phi _{p}'+{\bigg{\frac {d\phi _{p}}{d\epsilon _{p}}}{\bigg )}'(\epsilon _{p}-\epsilon _{p}')+{\bigg{\frac {d^{2}\phi _{p}}{d\epsilon _{p}^{2}}}{\bigg )}'{\frac {(\epsilon _{p}-\epsilon _{p}')^{2}}{2}}+\mathrm {etc.}$ | (310) |
| $\phi _{q}=\phi _{q}'+{\bigg{\frac {d\phi _{q}}{d\epsilon _{q}}}{\bigg )}'(\epsilon _{q}-\epsilon _{q}')+{\bigg{\frac {d^{2}\phi _{q}}{d\epsilon _{q}^{2}}}{\bigg )}'{\frac {(\epsilon _{q}-\epsilon _{q}')}{2}}+\mathrm {etc.}$ | (311) |
* $\epsilon _{p}-\epsilon _{p}'=-(\epsilon _{q}-\epsilon _{q}'),$
| $e^{\phi }=e^{\phi _{p}'+\phi _{q}'}\int \limits _{-\infty }^{+\infty }e^{{\Big{\bigg{\frac {d^{2}\phi _{p}}{d\epsilon _{p}^{2}}}{\bigg )}'+{\bigg{\frac {d^{2}\phi _{q}}{d\epsilon _{q}^{2}}}{\bigg )}'{\Big ]}{\frac {(\epsilon _{q}-\epsilon _{q}')^{2}}{2}}}\,d\epsilon _{q},$ | (312) |
$\pm \infty$
$\epsilon _{q}-\epsilon _{q}'$
This gives
| $e^{\phi }=e^{\phi _{p}'+\phi _{q}'}\left({\frac {-2\pi }{{\Big{\frac {d^{2}\phi _{p}}{d\epsilon _{p}^{2}}}{\Big )}'+{\Big{\frac {d^{2}\phi _{q}}{d\epsilon _{q}^{2}}}{\Big )}'}}\right)^{\frac {1}{2}}$ | (313) |
| $\phi =\phi _{p}'+\phi _{q}'+{\tfrac {1}{2}}\log(2\pi )-{\tfrac {1}{2}}\log {\bigg-{\Big{\frac {d^{2}\phi _{p}}{d\epsilon _{p}^{2}}}{\Big )}'-{\Big{\frac {d^{2}\phi _{q}}{d\epsilon _{q}^{2}}}{\Big )}'{\bigg ]}.$ | (314) |
$\phi$
$\epsilon$
$\phi _{q}$
$\epsilon _{q}$
Any two systems may be regarded as together forming a third system. If we have $V$ or $\phi$ given as function of $\epsilon$ for any two systems, we may express by quadratures $V$ and $\phi$ for the system formed by combining the two. If we distinguish by the suffixes $(~)_{1}$, $(~)_{2}$, $(~)_{12}$ the quantities relating to the three systems, we have easily from the definitions of these quantities
| $V_{12}=\int \int dV_{1}\,dV_{2}=\int V_{2}\,dV_{1}=\int V_{1}\,dV_{2}=\int V_{1}e^{\phi _{2}}\,d\epsilon _{2},$ | (315) |
| $e^{\phi _{12}}=\int e^{\phi _{2}}\,dV_{1}=\int e^{\phi _{1}}\,dV_{2}=\int e^{\phi _{1}+\phi _{2}}\,d\epsilon _{2},$ | (316) |
* $V_{1}=0,V_{2}=0,\mathrm {and} \epsilon _{1}+\epsilon _{2}=\epsilon _{12},$
$\epsilon _{1}$
$\epsilon _{2}$
It will be observed that these equations are identical in form with those by which $V$ and $\phi$ are derived from $V_{p}$ or $\phi _{p}$ and $V_{q}$ or $\phi _{q}$, except that they do not admit in the general case those transformations which result from substituting for $V_{p}$ or $\phi _{p}$ the particular functions which these symbols always represent.
​Similar formulae may be used to derive $V_{q}$ or $\phi _{q}$ for the compound system, when one of these quantities is known as function of the potential energy in each of the systems combined.
The operation represented by such an equation as
* $e^{\phi _{12}}=\int e^{\phi _{1}}e^{\phi _{2}}\,d\epsilon _{1}$
We may take a horizontal line as an axis of abscissas, and lay off $\epsilon _{1}$ as an abscissa measured to the right of any origin, and erect $e^{\phi _{1}}$ as a corresponding ordinate, thus determining a certain curve. Again, taking a different origin, we may lay off $\epsilon _{2}$ as abscissas measured to the left, and determine a second curve by erecting the ordinates $e^{\phi _{2}}$. We may suppose the distance between the origins to be $\epsilon _{12}$, the second origin being to the right if $\epsilon _{12}$ is positive. We may determine a third curve by erecting at every point in the line (between the least values of $\epsilon _{1}$ and $\epsilon _{2}$) an ordinate which represents the product of the two ordinates belonging to the curves already described. The area between this third curve and the axis of abscissas will represent the value of $e^{\phi _{12}}$. To get the value of this quantity for varying values of $\epsilon _{12}$, we may suppose the first two curves to be rigidly constructed, and to be capable of being moved independently. We may increase or diminish $\epsilon _{12}$ by moving one of these curves to the right or left. The third curve must be constructed anew for each different value of $\epsilon _{12}$.
* ↑ This is a necessary condition of the canonical distribution. See Chapter IV, p. 35.
* ↑ Compare equation (93).
* ↑ Compare equation (108).
* ↑ Compare equation (92).
* ↑ If $V_{q}$ were infinite for finite values of $\epsilon _{q}$, $V$ would evidently be infinite for finite values of $\epsilon$.
* ↑ Very similar values for $V_{q}$, $e^{\phi _{q}}$, $V$, and $e^{\phi }$ may be found in the same way in the case discussed in the preceding foot-notes (see pages 54, 72, 77, and 79), in which $\epsilon _{q}$ is a quadratic function of the $q$'s, and $\Delta _{\dot {q}}$ independent of the $q$'s. In this case we have $V_{q}={\bigg{\frac {\Delta _{\dot {q}}}{\Delta _{q}}}{\bigg )}^{\frac {1}{2}}{\frac {(2\pi )^{\frac {n}{2}}(\epsilon _{q}-\epsilon _{a})^{\frac {n}{2}}}{\Gamma ({\tfrac {1}{2}}n+1)}},$ $e^{\phi _{q}}={\bigg{\frac {\Delta _{\dot {q}}}{\Delta _{q}}}{\bigg )}^{\frac {1}{2}}{\frac {(2\pi )^{\frac {n}{2}}(\epsilon _{q}-\epsilon _{a})^{{\frac {n}{2}}-1}}{\Gamma ({\tfrac {1}{2}}n)}},$ $V={\bigg{\frac {\Delta _{\dot {q}}}{\Delta _{q}}}{\bigg )}^{\frac {1}{2}}{\frac {(2\pi )^{n}(\epsilon _{q}-\epsilon _{a})^{n}}{\Gamma (n+1)}},$ $e^{\phi }={\bigg{\frac {\Delta _{\dot {q}}}{\Delta _{q}}}{\bigg )}^{\frac {1}{2}}{\frac {(2\pi )^{n}(\epsilon _{q}-\epsilon _{a})^{n-1}}{\Gamma (n)}}.$
* ↑ The corresponding equation for the average value of any function of the potential energy, when this is a quadratic function of the $q$'s, and $\Delta _{\dot {q}}$ is independent of the $q$'s, is ${\overline {u}}={\frac {1}{\Theta ^{\frac {n}{2}}\Gamma ({\tfrac {1}{2}}n)}}\int _{\epsilon _{a}}^{\infty }u\,e^{-{\frac {\epsilon _{q}-\epsilon _{a}}{\Theta }}}(\epsilon _{q}-\epsilon _{a})^{{\tfrac {n}{2}}-1}\,d\epsilon _{q}.$ In the same case, the average value of any function of the (total) energy is given by the equation ${\overline {u}}={\frac {1}{\Theta ^{n}\Gamma (n)}}\int _{0}^{\infty }u\,e^{-{\frac {\epsilon -\epsilon _{a}}{\Theta }}}(\epsilon -\epsilon _{a})^{{\tfrac {n}{2}}-1}\,d\epsilon .$ Hence in this case ${\overline {(\epsilon _{q}-\epsilon _{a})^{m}}}={\frac {\Gamma (m+{\tfrac {1}{2}}n)}{\Gamma ({\tfrac {1}{2}}n)}}\Theta ^{m},\quad \mathrm {if} \quad m+{\tfrac {1}{2}}n>0.$ ${\overline {(\epsilon -\epsilon _{a})^{m}}}={\frac {\Gamma (m+n)}{\Gamma (n)}}\Theta ^{m},\quad \mathrm {if} \quad m+n>0.$ ${\overline {e^{-\phi _{q}}V_{q}}}={\overline {e^{-\phi }V}}=\Theta ,$ ${\overline {\frac {d\phi _{q}}{d\epsilon _{q}}}}={\frac {1}{\Theta }},\quad \mathrm {if} \quad n>2,$ and ${\overline {\frac {d\phi }{d\epsilon }}}={\frac {1}{\Theta }},\quad \mathrm {if} \quad n>1.$ If $n=1$, $e^{\phi }=2\pi$ and $d\phi /d\epsilon =0$ for any value of $\epsilon$. If $n=2$, the case is the same with respect to $\phi _{q}$.
* ↑ This equation has already been proved for positive integral powers of the kinetic energy. See page 77.