Elementary Principles in Statistical Mechanics/Chapter VIII

by Josiah Willard Gibbs

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)

the integration being extended (with constant values of the external coördinates) over all phases for which the energy is less than the limit $\epsilon$. We shall suppose that the value of this integral is not infinite, except for an infinite value of the limiting energy. This will not exclude any kind of system to which the canonical distribution is applicable. For if

$\int \ldots \int e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n}$

taken without limits has a finite value,^[1] the less value represented by

$e^{-{\frac {\epsilon }{\Theta }}}\int \ldots \int dp_{1}\ldots dq_{n}$

taken below a limiting value of $\epsilon$, and with the $\epsilon$ before the integral sign representing that limiting value, will also be finite. Therefore the value of $V$, which differs only by a constant factor, will also be finite, for finite $\epsilon$. It is a function of $\epsilon$ and the external coördinates, a continuous increasing function of $\epsilon$, which becomes infinite with $\epsilon$, and vanishes for the smallest possible value of $\epsilon$, or for $\epsilon =-\infty$, if the energy may be diminished without limit.

Let us also set

$\phi =\log {\frac {dV}{d\epsilon }}.$

(266)

The extension in phase between any two limits of energy, $\epsilon '$ and $\epsilon ''$, will be represented by the integral

$\int _{\epsilon '}^{\epsilon ''}e^{\phi }\,d\epsilon .$

(267)

And in general, we may substitute $e^{\phi }\,d\epsilon$ for $dp_{1}\ldots dq_{n}$ in a $2n$-fold integral, reducing it to a simple integral, whenever the limits can be expressed by the energy alone, and the other factor under the integral sign is a function of the energy alone, or with quantities which are constant in the integration.

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^[2]

$\int _{\epsilon '}^{\epsilon ''}e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon ,$

(268)

and that the average value in the ensemble of any quantity which only varies with the energy is given by the equation^[3]

${\overline {u}}=\int _{V=0}^{\epsilon =\infty }ue^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon ,$

(269)

where we may regard the constant $\psi$ as determined by the equation^[4]

$e^{-{\frac {\psi }{\Theta }}}=\int _{V=0}^{\epsilon =\infty }e^{-{\frac {\epsilon }{\Theta }}+\phi }\,d\epsilon ,$

(270)

In regard to the lower limit in these integrals, it will be observed that $V=0$ is equivalent to the condition that the value of $\epsilon$ is the least possible.

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)

the integration being extended (with constant values of the external coördinates) over all configurations for which the potential energy is less than $\epsilon _{q}$. $V_{q}$ will be a function of $\epsilon _{q}$ with the external coördinates, an increasing function of $\epsilon _{q}$, which does not become infinite (in such cases as we shall consider^[5]) for any finite value of $\epsilon _{q}$. It vanishes for the least possible value of $\epsilon _{q}$, or for $\epsilon _{q}=-\infty$, if $\epsilon _{q}$ can be diminished without limit. It is not always a continuous function of $\epsilon _{q}$. In fact, if there is a finite extension-in-configuration of constant potential energy, the corresponding value of $V_{q}$ will not include that extension-in-configuration, but if $\epsilon _{q}$ be increased infinitesimally, the corresponding value of $V_{q}$ will be increased by that finite extension-in-configuration.

Let us also set

$\phi _{q}=\log {\frac {dV_{q}}{d\epsilon _{q}}}.$

(272)

The extension-in-configuration between any two limits of potential energy $\epsilon _{q}'$ and $\epsilon _{q}''$ may be represented by the integral

$\int _{\epsilon _{q}'}^{\epsilon _{q}''}e^{\phi _{q}}\,d\epsilon _{q}$

(273)

whenever there is no discontinuity in the value of $V_{q}$ as function of $\epsilon _{q}$ between or at those limits, that is, whenever there is no finite extension-in-configuration of constant potential energy between or at the limits. And in general, with the restriction mentioned, we may substitute $e^{\phi _{q}}\,d\epsilon _{q}$ for $\Delta _{\dot {q}}^{\frac {1}{2}}dq_{1}\ldots dq_{n},$ in an $n$-fold integral, reducing it to a simple integral, when the limits are expressed by the potential energy, and the other factor under the integral sign is a function of the potential energy, either alone or with quantities which are constant in the integration.

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)

where $\psi$ may be determined by the condition that the value of the integral is unity, when the limits include all possible values of $\epsilon _{q}$. In the same case, the average value in the ensemble of any function of the potential energy is given by the equation

${\overline {u}}=\int _{V_{q}=0}^{\epsilon _{q}=\infty }ue^{{\frac {\psi -\epsilon _{q}}{\Theta }}+\phi _{q}}d\epsilon _{q}.$

(275)

When $V_{q}$ is not a continuous function of $\epsilon _{q}$, we may write $dV_{q}$ for $e^{\phi _{q}}d\epsilon _{q}$ in these formulae.

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)

the integration being extended, with constant values of the coördinates, both internal and external, over all values of the momenta for which the kinetic energy is less than the limit $\epsilon _{p}$. $V_{p}$ will evidently be a continuous increasing function of $\epsilon _{p}$ which vanishes and becomes infinite with $\epsilon _{p}$. Let us set

$\phi _{p}=\log {\frac {V_{p}}{d\epsilon _{p}}}.$

(277)

The extension-in-velocity between any two limits of kinetic energy $\epsilon _{p}'$ and $\epsilon _{p}''$ may be represented by the integral

$\int _{\epsilon _{p}'}^{\epsilon _{p}''}e^{\phi _{p}}\,d\epsilon _{p}.$

(278)

And in general, we may substitute $e^{\phi _{p}}\,d\epsilon _{p}$ for $\Delta _{p}^{\frac {1}{2}}dp_{1}\ldots dp_{n}$ or $\Delta _{\dot {q}}^{\frac {1}{2}}d{\dot {q}}_{1}\ldots d{\dot {q}}_{n}$ in an $n$-fold integral in which the coördinates are constant, reducing it to a simple integral, when the limits are expressed by the kinetic energy, and the other factor under the integral sign is a function of the kinetic energy, either alone or with quantities which are constant in the integration.

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)

the limits of the integral being given by $\epsilon _{p}$. That is, if

$\epsilon _{p}=F(p_{1},\ldots p_{n}),$

(280)

the limits of the integral for $\epsilon _{p}=1$, are given by the equation

$F(p_{1},\ldots p_{n})=1,$

(281)

and the limits of the integral for $\epsilon _{p}=a^{2}$, are given by the equation

$F(p_{1},\ldots p_{n})=a^{2}.$

(282)

But since $F$ represents a quadratic function, this equation may be written

$F{\bigg (}{\frac {p_{1}}{a}},\ldots {\frac {p_{n}}{a}}{\bigg )}=1$

(283)

The value of $V_{p}$ may also be put in the form

$V_{p}=a^{n}\Delta _{p}^{\frac {1}{2}}\int \ldots \int d{\frac {p_{1}}{a}}\ldots d{\frac {p_{n}}{a}}.$

(284)

Now we may determine $V_{p}$ for $\epsilon _{p}=1$ from (279) where the limits are expressed by (281), and $V_{p}$ for $\epsilon _{p}=a^{2}$ from (284) taking the limits from (283). The two integrals thus determined are evidently identical, and we have

$(V_{p})_{\epsilon _{p}=a^{2}}=a^{n}(V_{p})_{\epsilon _{p}=1}$

(285)

i. e., $V_{p}$ varies as $\epsilon _{p}^{\frac {n}{2}}$. We may therefore set

$V_{p}=C\epsilon _{p}^{\frac {n}{2}},\quad e^{\phi _{p}}={\frac {n}{2}}C\epsilon _{p}^{{\frac {n}{2}}-1}$

(286)

where $C$ is a constant, at least for fixed values of the internal coördinates.

To determine this constant, let us consider the case of a canonical distribution, for which we have

$\int _{0}^{\infty }e^{{\frac {\psi _{p}-\epsilon _{p}}{\Theta }}+\phi _{p}}\,d\epsilon _{p}=1,$

where

$e^{\frac {\psi _{p}}{\Theta }}=(2\pi \Theta )^{-{\frac {n}{2}}}.$

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)

$C={\frac {(2\pi )^{\frac {n}{2}}}{\Gamma ({\tfrac {1}{2}}n+1)}}.$

Having thus determined the value of the constant $C$, we may substitute it in the general expressions (286), and obtain the following values, which are perfectly general:

$V_{p}={\frac {(2\pi \epsilon _{p})^{\frac {n}{2}}}{\Gamma ({\tfrac {1}{2}}n+1)}}$

(288)

$e^{\phi _{p}}={\frac {(2\pi )^{\frac {n}{2}}\epsilon _{p}^{{\frac {n}{2}}-1}}{\Gamma ({\tfrac {1}{2}}n)}}$

[6](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)

Since this value is independent of the coördinates it also represents the probability that the kinetic energy of an unspecified system of a canonical ensemble falls within the limits. The form of the last integral also shows that the probability that the ratio of the kinetic energy to the modulus falls within given limits is independent also of the value of the modulus, being determined entirely by the number of degrees of freedom of the system and the limiting values of the ratio.

The average value of any function of the kinetic energy, either for the whole ensemble, or for any particular configuration, is given by

${\overline {u}}={\frac {1}{\Theta ^{\frac {n}{2}}\Gamma ({\tfrac {1}{2}}n)}}\int _{0}^{\infty }u\,e^{-{\frac {\epsilon _{p}}{\Theta }}}{\epsilon _{p}}^{{\tfrac {n}{2}}-1}\,d\epsilon _{p}$

[7](291)

Thus:

${\overline {{\epsilon _{p}}^{m}}}={\frac {\Gamma (m+{\tfrac {1}{2}}n)}{\Gamma ({\tfrac {1}{2}}n)}}\Theta ^{m},\quad \mathrm {if} \quad m+{\tfrac {1}{2}}n>0;$

[8](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)

If $n=2$, $e^{\phi _{p}}=2\pi$, and $d\phi _{p}/d\epsilon _{p}=0$, for any value of $\epsilon _{p}$.

The definitions of $V$, $V_{q}$, and $V_{p}$ give

$V=\int \int dV_{p}\,dV_{q}$

(297)

where the integrations cover all phases for which the energy is less than the limit $\epsilon$, for which the value of $V$ is sought. This gives

$V=\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }V_{p}\,dV_{q},$

(298)

and

$e^{\phi }={\frac {dV}{d\epsilon }}=\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }e^{\phi _{p}}\,dV_{q},$

(299)

where $V_{p}$ and $e^{\phi _{p}}$ are connected with $V_{q}$ by the equation

$\epsilon _{p}+\epsilon _{q}=\mathrm {constant} =\epsilon .$

(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)

We may also write

$V=\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }V_{p}e^{\phi _{q}}\,d\epsilon _{q},$

(302)

$e^{\phi }=\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }e^{\phi _{p}+\phi _{q}}\,d\epsilon _{q},$

(303)

etc., when $V_{q}$ is a continuous function of $\epsilon _{q}$ commencing with the value $V_{q}=0$, or when we choose to attribute to $V_{q}$ a fictitious continuity commencing with the value zero, as described on page 90.

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)

where $e^{\phi _{q}}d\epsilon _{q}$ may be substituted for $dV_{q}$ in the cases above described. If, therefore, $n$ is known, and $V_{q}$ as function of $\epsilon _{q}$, $V$ and $e^{\phi }$ may be found by quadratures.

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)

$d^{h}V/d\epsilon ^{h}$ is therefore positive if $h<{\tfrac {1}{2}}n+1$. It is an increasing function of $\epsilon$, if $h<{\tfrac {1}{2}}n$. If $\epsilon$ is not capable of being diminished without limit, $d^{h}V/d\epsilon ^{h}$ vanishes for the least possible value of $\epsilon$, if $h<{\tfrac {1}{2}}n$. If $n$ is even,

${\frac {d^{\tfrac {n}{2}}V}{d\epsilon ^{\tfrac {n}{2}}}}=(2\pi )^{\tfrac {n}{2}}(V_{q})_{\epsilon _{q}=\epsilon }$

(307)

That is, $V_{q}$ is the same function of $\epsilon +q$, as ${\frac {1}{(2\pi )^{\tfrac {n}{2}}}}{\frac {d^{\tfrac {n}{2}}V}{d\epsilon ^{\tfrac {n}{2}}}}$ of $\epsilon$.

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)

where the variables are connected by the equation (300). The maximum value of $\phi _{p}+\phi _{q}$ is therefore characterized by the equation

${\frac {d\phi _{p}}{d\epsilon _{p}}}={\frac {d\phi _{q}}{d\epsilon _{q}}}.$

(309)

The values of $\epsilon _{p}$ and $\epsilon _{q}$ determined by this maximum we shall distinguish by accents, and mark the corresponding values of functions of $\epsilon _{p}$ and $\epsilon _{q}$ in the same way. Now we have by Taylor's theorem

$\phi _{p}=\phi _{p}'+{\bigg (}{\frac {d\phi _{p}}{d\epsilon _{p}}}{\bigg )}'(\epsilon _{p}-\epsilon _{p}')+{\bigg (}{\frac {d^{2}\phi _{p}}{d\epsilon _{p}^{2}}}{\bigg )}'{\frac {(\epsilon _{p}-\epsilon _{p}')^{2}}{2}}+\mathrm {etc.}$

(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)

If the approximation is sufficient without going beyond the quadratic terms, since by (300)

$\epsilon _{p}-\epsilon _{p}'=-(\epsilon _{q}-\epsilon _{q}'),$

we may write

$e^{\phi }=e^{\phi _{p}'+\phi _{q}'}\int \limits _{-\infty }^{+\infty }e^{{\Big [}{\bigg (}{\frac {d^{2}\phi _{p}}{d\epsilon _{p}^{2}}}{\bigg )}'+{\bigg (}{\frac {d^{2}\phi _{q}}{d\epsilon _{q}^{2}}}{\bigg )}'{\Big ]}{\frac {(\epsilon _{q}-\epsilon _{q}')^{2}}{2}}}\,d\epsilon _{q},$

(312)

where the limits have been made $\pm \infty$ for analytical simplicity. This is allowable when the quantity in the square brackets has a very large negative value, since the part of the integral corresponding to other than very small values of $\epsilon _{q}-\epsilon _{q}'$ may be regarded as a vanishing quantity.

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)

or

$\phi =\phi _{p}'+\phi _{q}'+{\tfrac {1}{2}}\log(2\pi )-{\tfrac {1}{2}}\log {\bigg [}-{\Big (}{\frac {d^{2}\phi _{p}}{d\epsilon _{p}^{2}}}{\Big )}'-{\Big (}{\frac {d^{2}\phi _{q}}{d\epsilon _{q}^{2}}}{\Big )}'{\bigg ]}.$

(314)

From this equation, with (289), (300) and (309), we may determine the value of $\phi$ corresponding to any given value of $\epsilon$, when $\phi _{q}$ is known as function of $\epsilon _{q}$.

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)

where the double integral is to be taken within the limits

$V_{1}=0,V_{2}=0,\mathrm {and} \epsilon _{1}+\epsilon _{2}=\epsilon _{12},$

and the variables in the single integrals are connected by the last of these equations, while the limits are given by the first two, which characterize the least possible values of $\epsilon _{1}$ and $\epsilon _{2}$ respectively.

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

$e^{\phi _{12}}=\int e^{\phi _{1}}e^{\phi _{2}}\,d\epsilon _{1}$

is identical with one of the fundamental operations of the theory of errors, viz., that of finding the probability of an error from the probabilities of partial errors of which it is made up. It admits a simple geometrical illustration.

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 $\epsilon _{12}$.

---

1. This is a necessary condition of the canonical distribution. See Chapter IV, p. 35.
2. Compare equation (93).
3. Compare equation (108).
4. Compare equation (92).
5. If $V_{q}$ were infinite for finite values of $\epsilon _{q}$, $V$ would evidently be infinite for finite values of $\epsilon$.
6. 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

   $V_{q}={\bigg (}{\frac {\Delta _{\dot {q}}}{\Delta _{q}}}{\bigg )}^{\frac {1}{2}}{\frac {(2\pi )^{\frac {n}{2}}(\epsilon _{q}-\epsilon _{a})^{\frac {n}{2}}}{\Gamma ({\tfrac {1}{2}}n+1)}},$

   $e^{\phi _{q}}={\bigg (}{\frac {\Delta _{\dot {q}}}{\Delta _{q}}}{\bigg )}^{\frac {1}{2}}{\frac {(2\pi )^{\frac {n}{2}}(\epsilon _{q}-\epsilon _{a})^{{\frac {n}{2}}-1}}{\Gamma ({\tfrac {1}{2}}n)}},$

   $V={\bigg (}{\frac {\Delta _{\dot {q}}}{\Delta _{q}}}{\bigg )}^{\frac {1}{2}}{\frac {(2\pi )^{n}(\epsilon _{q}-\epsilon _{a})^{n}}{\Gamma (n+1)}},$

   $e^{\phi }={\bigg (}{\frac {\Delta _{\dot {q}}}{\Delta _{q}}}{\bigg )}^{\frac {1}{2}}{\frac {(2\pi )^{n}(\epsilon _{q}-\epsilon _{a})^{n-1}}{\Gamma (n)}}.$

7. 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 $\phi _{q}$.
8. This equation has already been proved for positive integral powers of the kinetic energy. See page 77.