Elementary Principles in Statistical Mechanics/Chapter IX

by Josiah Willard Gibbs

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,

$Ne^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon$

will represent the number having energies between the limits $\epsilon$ and $\epsilon +d\epsilon$. The expression

$Ne^{{\frac {\psi -\epsilon }{\Theta }}+\phi }$

(317)

represents what may be called the density-in-energy. This vanishes for $\epsilon =\infty$, for otherwise the necessary equation

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

(318)

could not be fulfilled. For the same reason the density-in-energy will vanish for $\epsilon =-\infty$, if that is a possible value of the energy. Generally, however, the least possible value of the energy will be a finite value, for which, if $n>2$, $e^{\phi }$ will vanish,^[1] and therefore the density-in-energy. Now the density-in-energy is necessarily positive, and since it vanishes for extreme values of the energy if $n>2$, it must have a maximum in such cases, in which the energy may be said to have its most common or most probable value, and which is determined by the equation

${\frac {d\phi }{d\epsilon }}={\frac {1}{\Theta }}.$

(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,

$\int \limits _{V=0}^{\epsilon =\infty }{\frac {d\phi }{d\epsilon }}e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon ={\bigg [}e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }{\bigg ]}_{V=0}^{\epsilon =\infty }+{\frac {1}{\Theta }}\int \limits _{V=0}^{\epsilon =\infty }e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon .$

(320)

If $n>2$, the expression in the brackets, which multiplied by $N$ would represent the density-in-energy, vanishes at the limits, and we have by (269) and (318)

${\overline {\frac {d\phi }{d\epsilon }}}={\frac {1}{\Theta }}.$

(321)

It appears, therefore, that for systems of more than two degrees of freedom, the average value of $d\phi /d\epsilon$ in an ensemble canonically distributed is identical with the value of the same differential coefficient as calculated for the most common energy in the ensemble, both values being reciprocals of the modulus.

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)

by which $\psi$ may be determined, $\Theta$ and the external coördinates (contained implicitly in $\phi$) are constant in the integration. The equation shows that $\psi$ is a function of these constants. If their values are varied, we shall have by differentiation, if $n>2$,

${\begin{aligned}&e^{-{\frac {\psi }{\Theta }}}{\bigg (}-{\frac {1}{\Theta }}\,d\psi +{\frac {\psi }{\Theta ^{2}}}\,d\Theta {\bigg )}={\frac {1}{\Theta ^{2}}}\,d\Theta \int \limits _{V=0}^{\epsilon =\infty }\epsilon e^{-{\frac {\epsilon }{\Theta }}+\phi }\,d\epsilon \\&\qquad {}+da_{1}\int \limits _{V=0}^{\epsilon =\infty }{\frac {d\phi }{da_{1}}}e^{-{\frac {\epsilon }{\Theta }}+\phi }\,d\epsilon +da_{2}\int \limits _{V=0}^{\epsilon =\infty }{\frac {d\phi }{da_{2}}}e^{-{\frac {\epsilon }{\Theta }}+\phi }\,d\epsilon +\mathrm {etc.} \end{aligned}}$

(323)

(Since $e^{\phi }$ vanishes with $V$, when $n>2$, there are no terms due to the variations of the limits.) Hence by (269)

$-{\frac {1}{\Theta }}\,d\psi +{\frac {\psi }{\Theta ^{2}}}\,d\Theta ={\frac {\overline {\epsilon }}{\Theta ^{2}}}\,d\Theta +{\overline {\frac {d\phi }{da_{1}}}}\,da_{1}+{\overline {\frac {d\phi }{da_{2}}}}\,da_{2}+\mathrm {etc.} ,$

(324)

or, since

${\frac {\psi +{\overline {\epsilon }}}{\Theta }}={\overline {\eta }},$

(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)

Comparing this with (112), we get

${\overline {\frac {d\phi }{da_{1}}}}={\frac {\overline {A_{1}}}{\Theta }},\quad {\overline {\frac {d\phi }{da_{2}}}}={\frac {\overline {A_{2}}}{\Theta }},\quad \mathrm {etc.}$

(327)

The first of these equations might be written^[2]

${\overline {{\bigg (}{\frac {d\phi }{da_{1}}}{\bigg )}}}_{\epsilon ,a}=-{\overline {{\bigg (}{\frac {d\phi }{d\epsilon }}{\bigg )}}}_{a}{\overline {{\bigg (}{\frac {d\epsilon }{da_{1}}}{\bigg )}}}_{a,q}$

(328)

but must not be confounded with the equation

${\overline {{\bigg (}{\frac {d\phi }{da_{1}}}{\bigg )}}}_{\epsilon ,a}=-{\overline {{\bigg (}{\frac {d\phi }{d\epsilon }}{\bigg )}_{a}{\bigg (}{\frac {d\epsilon }{da_{1}}}{\bigg )}_{\phi ,a}}}$

(329)

which is derived immediately from the identity

${\bigg (}{\frac {d\phi }{da_{1}}}{\bigg )}_{\epsilon ,a}=-{\bigg (}{\frac {d\phi }{d\epsilon }}{\bigg )}_{a}{\bigg (}{\frac {d\epsilon }{da_{1}}}{\bigg )}_{\phi ,a}$

(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)

obtained by differentiating (325), we get

$d{\overline {\epsilon }}=-\Theta \,d{\overline {\eta }}-\Theta {\overline {\frac {d\phi }{da_{1}}}}\,da_{1}-\Theta {\overline {\frac {d\phi }{da_{2}}}}\,da_{2}-\mathrm {etc.} ,$

(332)

or by (321),

$-d{\overline {\eta }}={\overline {\frac {d\phi }{d\epsilon }}}\,d{\overline {\epsilon }}+{\overline {\frac {d\phi }{da_{1}}}}\,da_{1}+{\overline {\frac {d\phi }{da_{2}}}}\,da_{2}+\mathrm {etc.}$

(333)

Except for the signs of average, the second member of this equation is the same as that of the identity

$d\phi ={\frac {d\phi }{d\epsilon }}\,d\epsilon +{\frac {d\phi }{da_{1}}}\,da_{1}+{\frac {d\phi }{da_{2}}}\,da_{2}+\mathrm {etc.}$

(334)

For the more precise comparison of these equations, we may suppose that the energy in the last equation is some definite and fairly representative energy in the ensemble. For this purpose we might choose the average energy. It will perhaps be more convenient to choose the most common energy, which we shall denote by $\epsilon _{0}$. The same suffix will be applied to functions of the energy determined for this value. Our identity then becomes

$d\phi _{0}={\bigg (}{\frac {d\phi }{d\epsilon }}{\bigg )}_{0}\,d\epsilon _{0}+{\bigg (}{\frac {d\phi }{da_{1}}}{\bigg )}_{0}\,da_{1}+{\bigg (}{\frac {d\phi }{da_{2}}}{\bigg )}_{0}\,da_{2}+\mathrm {etc.}$

(335)

It has been shown that

${\overline {\frac {d\phi }{d\epsilon }}}={\bigg (}{\frac {d\phi }{d\epsilon }}{\bigg )}_{0}={\frac {1}{\Theta }},$

(336)

when $n>2$. Moreover, since the external coördinates have constant values throughout the ensemble, the values of $d\phi /da_{1}$, $d\phi /da_{2}$, etc. vary in the ensemble only on account of the variations of the energy ($\epsilon$), which, as we have seen, may be regarded as sensibly constant throughout the ensemble, when $n$ is very great. In this case, therefore, we may regard the average values

${\overline {\frac {d\phi }{da_{1}}}},\quad {\overline {\frac {d\phi }{da_{2}}}},\quad \mathrm {etc.} ,$

as practically equivalent to the values relating to the most common energy

${\bigg (}{\frac {d\phi }{da_{1}}}{\bigg )}_{0},\quad {\bigg (}{\frac {d\phi }{da_{2}}}{\bigg )}_{0},\quad \mathrm {etc.}$

In this case also $d{\overline {\epsilon }}$ is practically equivalent to $d\epsilon _{0}$. We have therefore, for very large values of $n$,

$-d{\overline {\eta }}=d\phi _{0}$

(337)

approximately. That is, except for an additive constant, $-{\overline {\eta }}$ may be regarded as practically equivalent to $\phi _{0}$, when the number of degrees of freedom of the system is very great. It is not meant by this that the variable part of ${\overline {\eta }}+\phi _{0}$ is numerically of a lower order of magnitude than unity. For when $n$ is very great, $-{\overline {\eta }}$ and $\phi$ are very great, and we can only conclude that the variable part of ${\overline {\eta }}+\phi _{0}$ is insignificant compared with the variable part of ${\overline {\eta }}$ or of $\phi _{0}$, taken separately.

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)

Adding the identical equation

${\frac {\psi -\epsilon }{\Theta }}={\frac {\psi -\epsilon _{0}}{\Theta }}-{\frac {\epsilon -\epsilon _{0}}{\Theta }},$

we get by (336)

${\frac {\psi -\epsilon }{\Theta }}+\phi ={\frac {\psi -\epsilon _{0}}{\Theta }}+\phi _{0}+{\bigg (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}{\frac {(\epsilon -\epsilon _{0})^{2}}{2}}+{\bigg (}{\frac {d^{3}\phi }{d\epsilon ^{3}}}{\bigg )}{\frac {(\epsilon -\epsilon _{0})^{3}}{3!}}+\mathrm {etc.}$

(339)

Substituting this value in

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

which expresses the probability that the energy of an unspecified system of the ensemble lies between the limits $\epsilon '$ and $\epsilon ''$, we get

$e^{{\frac {\psi -\epsilon _{0}}{\Theta }}+\phi _{0}}\int \limits _{\epsilon '}^{\epsilon ''}e^{{\big (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\big )}_{0}{\frac {(\epsilon -\epsilon _{0})^{2}}{2}}+{\big (}{\frac {d^{3}\phi }{d\epsilon ^{3}}}{\big )}_{0}{\frac {(\epsilon -\epsilon _{0})^{3}}{3!}}+\mathrm {etc.} }\,d\epsilon .$

(340)

When the number of degrees of freedom is very great, and $\epsilon -\epsilon _{0}$ in consequence very small, we may neglect the higher powers and write^[3]

$e^{{\frac {\psi -\epsilon _{0}}{\Theta }}+\phi _{0}}\int \limits _{\epsilon '}^{\epsilon ''}e^{{\big (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\big )}_{0}{\frac {(\epsilon -\epsilon _{0})^{2}}{2}}}\,d\epsilon .$

(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)

whence

${\frac {\psi -\epsilon _{0}}{\Theta }}+\phi _{0}={\tfrac {1}{2}}\log {\frac {{\Big (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\Big )}_{0}}{-2\pi }}=-{\tfrac {1}{2}}\log(2\pi {\overline {(\epsilon -{\overline {\epsilon }})^{2}}}).$

(344)

Now it has been proved in Chapter VII that

${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}={\frac {2}{n}}{\frac {d{\overline {\epsilon }}}{d{\overline {\epsilon }}_{p}}}{\overline {\epsilon }}_{p}{}^{2}.$

We have therefore

${\overline {\eta }}+\phi _{0}={\frac {\psi -\epsilon _{0}}{\Theta }}+\phi _{0}=-{\tfrac {1}{2}}\log(2\pi {\overline {(\epsilon -{\overline {\epsilon }})^{2}}})=-{\tfrac {1}{2}}\log {\bigg (}{\frac {4\pi }{n}}{\frac {d{\overline {\epsilon }}}{d{\overline {\epsilon }}_{p}}}{\overline {\epsilon }}_{p}{}^{2}{\bigg )}$

(345)

approximately. The order of magnitude of ${\overline {\eta }}-\phi _{0}$ is therefore that of $\log n$. This magnitude is mainly constant. The order of magnitude of ${\overline {\eta }}+\phi _{0}-{\tfrac {1}{2}}\log n$ is that of unity. The order of magnitude of $\phi _{0}$, and therefore of $-{\overline {\eta }}$, is that of $n$.^[4]

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)

The members of the last equation have the order of magnitude of $n$. Equation (338) gives also for the first approximation

${\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}={\bigg (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}(\epsilon -\epsilon _{0}),$

whence

${\overline {{\bigg (}{\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}{\bigg )}(\epsilon -\epsilon _{0})}}={\bigg (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}{\overline {(\epsilon -\epsilon _{0})^{2}}}=-1,$

(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)

This is of the order of magnitude of $n$.^[5]

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)

where $e^{\eta +\phi }$ is necessarily positive. This requires that it shall vanish for $\epsilon =\infty$, and also for $\epsilon =-\infty$, if this is a possible value. It has been shown in the last chapter that if $\epsilon$ has a (finite) least possible value (which is the usual case) and $n>2$, $e^{\phi }$ will vanish for that least value of $\epsilon$. In general therefore $\eta +\phi$ will have a maximum, which determines the most probable value of the energy. If we denote this value by $\epsilon _{0}$ and distinguish the corresponding values of the functions of the energy by the same suffix, we shall have

${\bigg (}{\frac {d\eta }{d\epsilon }}{\bigg )}_{0}+{\bigg (}{\frac {d\phi }{d\epsilon }}{\bigg )}_{0}=0.$

(352)

The probability that an unspecified system of the ensemble falls within any given limits of energy ($\epsilon '$ and $\epsilon ''$) is represented by

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

If we expand $\eta$ and $\phi$ in ascending powers of $\epsilon -\epsilon _{0}$, without going beyond the squares, the probability that the energy falls within the given limits takes the form of the 'law of errors'—

$e^{\eta _{0}+\phi _{0}}\int \limits _{\epsilon '}^{\epsilon ''}e^{{\big [}{\big (}{\frac {d^{2}\eta }{d\epsilon ^{2}}}{\big )}_{0}+{\big (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\big )}_{0}{\big ]}{\frac {(\epsilon -\epsilon _{0})^{2}}{2}}}\,d\epsilon .$

(353)

This gives

$\eta _{0}+\phi _{0}={\tfrac {1}{2}}\log {\bigg [}{\frac {-1}{2\pi }}{\bigg (}{\frac {d^{2}\eta }{d\epsilon ^{2}}}{\bigg )}_{0}-{\frac {1}{2\pi }}{\bigg (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}{\bigg ]},$

(354)

and

${\overline {(\epsilon -\epsilon _{0})^{2}}}={\bigg [}-{\bigg (}{\frac {d^{2}\eta }{d\epsilon ^{2}}}{\bigg )}_{0}-{\bigg (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}{\bigg ]}^{-1}.$

(355)

We shall have a close approximation in general when the quantities equated in (355) are very small, i. e., when

$-{\bigg (}{\frac {d^{2}\eta }{d\epsilon ^{2}}}{\bigg )}_{0}-{\bigg (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}$

(356)

is very great. Now when $n$ is very great, $-d^{2}\phi /d\epsilon ^{2}$ is of the same order of magnitude, and the condition that (356) shall be very great does not restrict very much the nature of the function $\eta$.

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)

Now we have identically

$\int \limits _{V=0}^{\epsilon =\infty }{\bigg (}{\frac {du}{d\epsilon }}-{\frac {u}{\Theta }}+u{\frac {d\phi }{d\epsilon }}{\bigg )}e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon ={\bigg [}u\,e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }{\bigg ]}_{V=0}^{\epsilon =\infty }$

(358)

Therefore, by the preceding equation

${\overline {\frac {du}{d\epsilon }}}-{\frac {\overline {u}}{\Theta }}+{\overline {u{\frac {d\phi }{d\epsilon }}}}={\bigg [}u\,e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }{\bigg ]}_{V=0}^{\epsilon =\infty }$

[6](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)

as before. It is evident from the same considerations that the second member of (359) will always vanish if $n>2$, unless $u$ becomes infinite at one of the limits, in which case a more careful examination of the value of the expression will be necessary. To facilitate the discussion of such cases, it will be convenient to introduce a certain limitation in regard to the nature of the system considered. We have necessarily supposed, in all our treatment of systems canonically distributed, that the system considered was such as to be capable of the canonical distribution with the given value of the modulus. We shall now suppose that the system is such as to be capable of a canonical distribution with any (finite)^[7] modulus. Let us see what cases we exclude by this last limitation.

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)

fails to determine a finite value for $\psi$. Evidently the equation cannot make $\psi$ an infinite positive quantity, the impossibility therefore occurs when the equation makes $\psi =-\infty$. Now we get easily from (191)

$d{\frac {\psi }{\Theta }}=-{\frac {\overline {\epsilon }}{\Theta ^{2}}}\,d\Theta .$

If the canonical distribution is possible for any values of $\Theta$, we can apply this equation so long as the canonical distribution is possible. The equation shows that as $\Theta$ is increased (without becoming infinite) $-\psi$ cannot become infinite unless ${\overline {\epsilon }}$ simultaneously becomes infinite, and that as $\Theta$ is decreased (without becoming zero) $-\psi$ cannot become infinite unless simultaneously ${\overline {\epsilon }}$ becomes an infinite negative quantity. The corresponding cases in thermodynamics would be bodies which could absorb or give out an infinite amount of heat without passing certain limits of temperature, when no external work is done in the positive or negative sense. Such infinite values present no analytical difficulties, and do not contradict the general laws of mechanics or of thermodynamics, but they are quite foreign to our ordinary experience of nature. In excluding such cases (which are certainly not entirely devoid of interest) we do not exclude any which are analogous to any actual cases in thermodynamics.

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 $\Theta '=2\Theta$,

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

where $\psi '$ denotes the value of $\psi$ for the modulus $\Theta '$. Since the last member of this formula vanishes for $\epsilon =\infty$, the less value represented by the first member must also vanish for the same value of $\epsilon$. Therefore the second member of (359), which differs only by a constant factor, vanishes at the upper limit. The case of the lower limit remains to be considered. Now

$e^{-{\frac {\epsilon }{\Theta }}}V\leq \int \limits _{V=0}^{\epsilon }e^{-{\frac {\epsilon }{\Theta }}+\phi }\,d\epsilon .$

The second member of this formula evidently vanishes for the value of $\epsilon$, which gives $V=0$, whether this be finite or negative infinity. Therefore, the second member of (359) vanishes at the lower limit also, and we have

$1-{\overline {e^{-\phi }V{\frac {d\phi }{d\epsilon }}}}-{\overline {e^{-\phi }{\frac {V}{\Theta }}}}+{\overline {e^{-\phi }V{\frac {d\phi }{d\epsilon }}}}=0,$

or

${\overline {e^{-\phi }V}}=\Theta .$

(362)

This equation, which is subject to no restriction in regard to the value of $n$, suggests a connection or analogy between the function of the energy of a system which is represented by $e^{-\phi }V$ and the notion of temperature in thermodynamics. We shall return to this subject in Chapter XIV.

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)

We thus obtain the following equations:

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)

or

$2{\overline {e^{\phi }{\frac {d\phi }{d\epsilon }}}}={\overline {e^{\phi }}}{\overline {\frac {d\phi }{d\epsilon }}}={\frac {\overline {e^{\phi }}}{\Theta }},$

(366)

${\overline {(\epsilon -{\overline {\epsilon }}){\bigg (}{\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}{\bigg )}}}={\overline {\epsilon {\frac {d\phi }{d\epsilon }}}}-{\frac {\overline {\epsilon }}{\Theta }}=-1,$

[8](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)

If $n>4$,

${\overline {{\bigg (}{\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}{\bigg )}^{2}}}={\overline {{\bigg (}{\frac {d\phi }{d\epsilon }}{\bigg )}^{2}}}-{\frac {1}{\Theta ^{2}}}=-{\overline {{\bigg (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}^{2}}}.$

[9](369)

If $n>2h$,

${\overline {e^{-\phi }{\frac {d^{h}V}{d\epsilon ^{h}}}{\frac {d\phi }{d\epsilon }}}}-{\frac {1}{\Theta }}{\overline {e^{-\phi }{\frac {d^{h}V}{d\epsilon ^{h}}}}}={\overline {e^{-\phi }{\frac {d^{h}V}{d\epsilon ^{h}}}{\frac {d\phi }{d\epsilon }}}}-{\overline {e^{-\phi }{\frac {d^{h+1}V}{d\epsilon ^{h+1}}}}}$

or

${\overline {e^{-\phi }{\frac {d^{h+1}V}{d\epsilon ^{h+1}}}}}={\frac {1}{\Theta }}{\overline {e^{-\phi }{\frac {d^{h}V}{d\epsilon ^{h}}}}}$

(370)

whence

${\overline {e^{-\phi }{\frac {d^{h+1}V}{d\epsilon ^{h+1}}}}}={\frac {1}{\Theta ^{h}}}.$

(371)

Giving $h$ the values 1, 2, 3, etc., we have

${\overline {\frac {d\phi }{d\epsilon }}}={\frac {1}{\Theta }},\quad \mathrm {if~} n>2,$

${\overline {\frac {d^{2}\phi }{d\epsilon ^{2}}}}+{\overline {{\bigg (}{\frac {d\phi }{d\epsilon }}{\bigg )}^{2}}}={\frac {1}{\Theta ^{2}}}\quad \mathrm {if~} n>4,$

as already obtained. Also

${\overline {\frac {d^{3}\phi }{d\epsilon ^{3}}}}+3{\overline {{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\frac {d\phi }{d\epsilon }}}}+{\overline {{\bigg (}{\frac {d\phi }{d\epsilon }}{\bigg )}^{3}}}={\frac {1}{\Theta ^{3}}}\quad \mathrm {if~} n>6.$

(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

${\overline {e^{-\phi _{q}}V_{q}}}=\Theta$

requires only the condition already mentioned with respect to $V_{q}$. This equation corresponds to (362), which is subject to no restriction with respect to the value of $n$. We may observe, however, that $V$ will always satisfy a condition similar to that mentioned with respect to $V_{q}$.

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

${\overline {\frac {d\phi _{q}}{d\epsilon _{q}}}}={\frac {1}{\Theta }}.$

If $V_{q}$, $e^{\phi _{q}}$ (or $dV_{q}/d\epsilon _{q}$), $d^{2}V_{q}/d\epsilon _{q}{}^{2}$ all satisfy similar conditions, we shall have an equation similar to (369), which was subject to the condition $n>4$. And if $d^{3}V_{q}/d\epsilon _{q}{}^{3}$ also satisfies a similar condition, we shall have an equation similar to (372), for which the condition was $n>6$. Finally, if $V_{q}$ and $h$ successive differential coefficients satisfy conditions of the kind mentioned, we shall have equations like (370) and (371) for which the condition was $n>2h$.

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.

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1. See page 96.
2. 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.
3. 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

   $1+{\bigg (}{\frac {d^{3}\phi }{d\epsilon ^{3}}}{\bigg )}_{0}{\frac {(\epsilon -\epsilon _{0})^{3}}{3!}}+{\bigg (}{\frac {d^{4}\phi }{d\epsilon ^{4}}}{\bigg )}_{0}{\frac {(\epsilon -\epsilon _{0})^{4}}{4!}}+\mathrm {etc.}$

4. Compare (289), (314).
5. 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.
6. A more general equation, which is not limited to ensembles canonically distributed, is

   ${\overline {\frac {du}{d\epsilon }}}+{\overline {u{\frac {d\eta }{d\epsilon }}}}+{\overline {u{\frac {d\phi }{d\epsilon }}}}={\bigg [}u\,e^{\eta +\phi }{\bigg ]}_{V=0}^{\epsilon =\infty }$

   where $\eta$ denotes, as usual, the index of probability of phase.
7. The term finite applied to the modulus is intended to exclude the value zero as well as infinity.
8. This equation may also be obtained from equations (252) and (321). Compare also equation (349) which was derived by an approximative method.
9. Compare equation (350), obtained by an approximative method.