Elementary Principles in Statistical Mechanics/Chapter X

by Josiah Willard Gibbs

CHAPTER X.

ON A DISTRIBUTION IN PHASE CALLED MICROCANONICAL IN WHICH ALL THE SYSTEMS HAVE THE SAME ENERGY.

An important case of statistical equilibrium is that in which all systems of the ensemble have the same energy. We may arrive at the notion of a distribution which will satisfy the necessary conditions by the following process. We may suppose that an ensemble is distributed with a uniform density-in-phase between two limiting values of the energy, $\epsilon '$ and $\epsilon ''$, and with density zero outside of those limits. Such an ensemble is evidently in statistical equilibrium according to the criterion in Chapter IV, since the density-in-phase may be regarded as a function of the energy. By diminishing the difference of $\epsilon '$ and $\epsilon ''$, we may diminish the differences of energy in the ensemble. The limit of this process gives us a permanent distribution in which the energy is constant.

We should arrive at the same result, if we should make the density any function of the energy between the limits $\epsilon '$ and $\epsilon ''$, and zero outside of those limits. Thus, the limiting distribution obtained from the part of a canonical ensemble between two limits of energy, when the difference of the limiting energies is indefinitely diminished, is independent of the modulus, being determined entirely by the energy, and is identical with the limiting distribution obtained from a uniform density between limits of energy approaching the same value.

We shall call the limiting distribution at which we arrive by this process microcanonical.

We shall find however, in certain cases, that for certain values of the energy, viz., for those for which $e^{\phi }$ is infinite, this process fails to define a limiting distribution in any such distinct sense as for other values of the energy. The difficulty is not in the process, but in the nature of the case, being entirely analogous to that which we meet when we try to find a canonical distribution in cases when $\psi$ becomes infinite. We have not regarded such cases as affording true examples of the canonical distribution, and we shall not regard the cases in which $e^{\phi }$ is infinite as affording true examples of the microcanonical distribution. We shall in fact find as we go on that in such cases our most important formulae become illusory.

The use of formulae relating to a canonical ensemble which contain $e^{\phi }\,d\epsilon$ instead of $dp_{1}\ldots dq_{n}$, as in the preceding chapters, amounts to the consideration of the ensemble as divided into an infinity of microcanonical elements.

From a certain point of view, the microcanonical distribution may seem more simple than the canonical, and it has perhaps been more studied, and been regarded as more closely related to the fundamental notions of thermodynamics. To this last point we shall return in a subsequent chapter. It is sufficient here to remark that analytically the canonical distribution is much more manageable than the microcanonical.

We may sometimes avoid difficulties which the microcanonical distribution presents by regarding it as the result of the following process, which involves conceptions less simple but more amenable to analytical treatment. We may suppose an ensemble distributed with a density proportional to

$e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}},$

where $\omega$ and $\epsilon '$ are constants, and then diminish indefinitely the value of the constant $\omega$. Here the density is nowhere zero until we come to the limit, but at the limit it is zero for all energies except $\epsilon '$. We thus avoid the analytical complication of discontinuities in the value of the density, which require the use of integrals with inconvenient limits.

In a microcanonical ensemble of systems the energy ($\epsilon$) is constant, but the kinetic energy ($\epsilon _{p}$) and the potential energy ($\epsilon _{q}$) vary in the different systems, subject of course to the condition

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

(373)

Our first inquiries will relate to the division of energy into these two parts, and to the average values of functions of $\epsilon _{p}$ and $\epsilon _{q}$.

We shall use the notation ${\overline {u}}|_{\epsilon }$ to denote an average value in a microcanonical ensemble of energy $\epsilon$. An average value in a canonical ensemble of modulus $\Theta$, which has hitherto been denoted by ${\overline {u}}$, we shall in this chapter denote by ${\overline {u}}|_{\Theta }$, to distinguish more clearly the two kinds of averages.

The extension-in-phase within any limits which can be given in terms of $\epsilon _{p}$ and $\epsilon _{q}$ may be expressed in the notations of the preceding chapter by the double integral

$\iint dV_{p}\,dV_{q}$

taken within those limits. If an ensemble of systems is distributed within those limits with a uniform density-in-phase, the average value in the ensemble of any function ($u$) of the kinetic and potential energies will be expressed by the quotient of integrals

${\frac {\iint u\,dV_{p}\,dV_{q}}{\iint dV_{p}\,dV_{q}}}$

Since $dV_{p}=e^{\phi _{p}}\,d\epsilon _{p}$, and $d\epsilon _{p}=d\epsilon$ when $\epsilon _{q}$ is constant, the expression may be written

${\frac {\iint u\,e^{\phi _{p}}\,d\epsilon \,dV_{q}}{\iint e^{\phi _{p}}\,d\epsilon \,dV_{q}}}$

To get the average value of $u$ in an ensemble distributed microcanonically with the energy $\epsilon$, we must make the integrations cover the extension-in-phase between the energies $\epsilon$ and $\epsilon +d\epsilon$. This gives

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

But by (299) the value of the integral in the denominator is $e^{\phi }$. We have therefore

${\overline {u}}|_{\epsilon }=e^{-\phi }\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }u\,e^{\phi _{p}}\,dV_{q}$

(374)

where $e^{\phi _{p}}$ and $V_{q}$ are connected by equation (373), and $u$, if given as function of $\epsilon _{p}$, or of $\epsilon _{p}$ and $\epsilon _{q}$, becomes in virtue of the same equation a function of $\epsilon _{q}$ alone.

We shall assume that $e^{\phi }$ has a finite value. If $n>1$, it is evident from equation (305) that $e^{\phi }$ is an increasing function of $\epsilon$, and therefore cannot be infinite for one value of $\epsilon$ without being infinite for all greater values of $\epsilon$, which would make $-\psi$ infinite.^[1] When $n>1$, therefore, if we assume that $e^{\phi }$ is finite, we only exclude such cases as we found necessary to exclude in the study of the canonical distribution. But when $n=1$, cases may occur in which the canonical distribution is perfectly applicable, but in which the formulae for the microcanonical distribution become illusory, for particular values of $\epsilon$, on account of the infinite value of $e^{\phi }$. Such failing cases of the microcanonical distribution for particular values of the energy will not prevent us from regarding the canonical ensemble as consisting of an infinity of microcanonical ensembles.^[2]

From the last equation, with (298), we get

${\overline {e^{-\phi _{p}}V_{p}}}|_{\epsilon }=e^{-\phi }\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }V_{p}\,dV_{q}=e^{-\phi }V.$

(375)

But by equations (288) and (289)

$e^{-\phi _{p}}V_{p}={\frac {2}{n}}\epsilon _{p}.$

(376)

Therefore

$e^{-\phi }V={\overline {e^{-\phi _{p}}V_{p}}}|_{\epsilon }={\frac {2}{n}}{\overline {\epsilon _{p}}}|_{\epsilon }.$

(377)

Again, with the aid of equation (301), we get

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

(378)

if $n>2$. Therefore, by (289),

${\frac {d\phi }{d\epsilon }}={\overline {\frac {d\phi _{p}}{d\epsilon _{p}}}}{\bigg |}_{\epsilon }={\bigg (}{\frac {n}{2}}-1{\bigg )}{\overline {\epsilon _{p}^{-1}}}|_{\epsilon },\quad \mathrm {if} \quad n>2.$

(379)

These results are interesting on account of the relations of the functions $e^{-\phi }V$ and ${\frac {d\phi }{d\epsilon }}$ to the notion of temperature in thermodynamics,—a subject to which we shall return hereafter. They are particular cases of a general relation easily deduced from equations (306), (374), (288) and (289). We have

${\frac {d^{h}V}{d\epsilon ^{h}}}=\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }{\frac {d^{h}V}{d\epsilon _{p}^{h}}}\,dV_{q},\quad \mathrm {if} \quad h<{\tfrac {1}{2}}n+1.$

The equation may be written

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

We have therefore

$e^{-\phi }{\frac {d^{h}V}{d\epsilon ^{h}}}={\overline {e^{-\phi _{p}}{\frac {d^{h}V_{p}}{d\epsilon _{p}^{h}}}}}{\bigg |}_{\epsilon }={\frac {\Gamma ({\tfrac {1}{2}}n)}{\Gamma ({\tfrac {1}{2}}n-h+1)}}{\overline {\epsilon _{p}^{1-h}}}|_{\epsilon },$

(380)

if $h<{\tfrac {1}{2}}n+1$. For example, when $n$ is even, we may make $h={\tfrac {1}{2}}n$, which gives, with (307),

$(2\pi )^{\tfrac {n}{2}}e^{-\phi }(V_{q})_{\epsilon _{q}=\epsilon }=\Gamma ({\tfrac {1}{2}}n){\overline {{\epsilon _{p}}^{1-{\tfrac {n}{2}}}}}{\Big |}_{\epsilon }.$

(381)

Since any canonical ensemble of systems may be regarded as composed of microcanonical ensembles, if any quantities $u$ and $v$ have the same average values in every microcanonical ensemble, they will have the same values in every canonical ensemble. To bring equation (380) formally under this rule, we may observe that the first member being a function of $\epsilon$ is a constant value in a microcanonical ensemble, and therefore identical with its average value. We get thus the general equation

${\overline {e^{-\phi }{\frac {d^{h}V}{d\epsilon ^{h}}}}}{\bigg |}_{\Theta }={\overline {e^{-\phi _{p}}{\frac {d^{h}V_{p}}{d\epsilon _{p}^{h}}}}}{\bigg |}_{\Theta }={\frac {\Gamma ({\tfrac {1}{2}}n)}{\Gamma ({\tfrac {1}{2}}n-h+1)}}{\overline {\epsilon _{p}^{1-h}}}|_{\Theta }=\Theta ^{1-h}$

(382)

if $h<{\frac {1}{2}}n+1$.^[3] The equations

$\Theta ={\overline {e^{-\phi }V}}|_{\Theta }={\overline {e^{-\phi _{p}}V_{p}}}|_{\Theta }={\frac {2}{n}}{\overline {\epsilon _{p}}}|_{\Theta },$

(383)

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

(384)

may be regarded as particular cases of the general equation. The last equation is subject to the condition that $n>2$.

The last two equations give for a canonical ensemble, if $n>2$,

${\bigg (}1-{\frac {2}{n}}{\bigg )}{\overline {\epsilon _{p}}}|_{\Theta }\,{\overline {\epsilon _{p}^{-1}}}|_{\Theta }=1.$

(385)

The corresponding equations for a microcanonical ensemble give, if $n>2$,

${\bigg (}1-{\frac {2}{n}}{\bigg )}{\overline {\epsilon _{p}}}|_{\epsilon }\,{\overline {\epsilon _{p}^{-1}}}|_{\epsilon }={\frac {d\phi }{d\log V}},$

(386)

which shows that $d\phi \,d\log V$ approaches the value unity when $n$ is very great.

If a system consists of two parts, having separate energies, we may obtain equations similar in form to the preceding, which relate to the system as thus divided.^[4] We shall distinguish quantities relating to the parts by letters with suffixes, the same letters without suffixes relating to the whole system. The extension-in-phase of the whole system within any given limits of the energies may be represented by the double integral

$\iint dV_{1}\,dV_{2}$

taken within those limits, as appears at once from the definitions of Chapter VIII. In an ensemble distributed with uniform density within those limits, and zero density outside, the average value of any function of $\epsilon _{1}$ and $\epsilon _{2}$ is given by the quotient

${\frac {\iint u\,dV_{1}\,dV_{2}}{\iint dV_{1}\,dV_{2}}}$

which may also be written^[5]

${\frac {\iint u\,e^{\phi _{1}}\,d\epsilon \,dV_{2}}{\iint e^{\phi _{1}}\,d\epsilon \,dV_{2}}}$

If we make the limits of integration $\epsilon$ and $\epsilon +d\epsilon$, we get the average value of $u$ in an ensemble in which the whole system is microcanonically distributed in phase, viz.,

${\overline {u}}|_{\epsilon }=e^{-\phi }\int \limits _{V_{2}=0}^{\epsilon _{2}=\epsilon }u\,e^{\phi _{1}}\,dV_{2},$

(387)

where $\phi _{1}$ and $V_{2}$ are connected by the equation

$\epsilon _{1}+\epsilon _{2}=\mathrm {constant} =\epsilon ,$

(388)

and $u$, if given as function of $\epsilon _{1}$, or of $\epsilon _{1}$ and $\epsilon _{2}$, becomes in virtue of the same equation a function of $\epsilon _{2}$ alone.^[6] Thus

${\overline {e^{-\phi _{1}}V_{1}}}|_{\epsilon }=e^{-\phi }\int \limits _{V_{2}=0}^{\epsilon _{2}=\epsilon }V_{1}\,dV_{2},$

(389)

$e^{-\phi }V={\overline {e^{-\phi _{1}}V_{1}}}|_{\epsilon }={\overline {e^{-\phi _{2}}V_{2}}}|_{\epsilon }.$

(390)

This requires a similar relation for canonical averages

$\Theta ={\overline {e^{-\phi }V}}|_{\Theta }={\overline {e^{-\phi _{1}}V_{1}}}|_{\Theta }={\overline {e^{-\phi _{2}}V_{2}}}|_{\Theta }.$

(391)

Again

${\overline {\frac {d\phi _{1}}{d\epsilon _{1}}}}{\bigg |}_{\epsilon }=e^{-\phi }\int \limits _{V_{2}=0}^{\epsilon _{2}=\epsilon }{\frac {d\phi _{1}}{d\epsilon _{1}}}e^{\phi _{1}}\,dV_{2}.$

(392)

But if $n_{1}>2$, $e^{\phi _{1}}$ vanishes for $V_{1}=0$,^[7] and

${\frac {d}{d\epsilon }}e^{\phi }={\frac {d}{d\epsilon }}\int \limits _{V_{2}=0}^{\epsilon _{2}=\epsilon }e^{\phi _{1}}\,dV_{2}=\int \limits _{V_{2}=0}^{\epsilon _{2}=\epsilon }{\frac {d\phi _{1}}{d\epsilon _{1}}}\,e^{\phi _{1}}\,dV_{2}.$

(393)

Hence, if $n_{1}>2$, and $n_{2}>2$,

${\frac {d\phi }{d\epsilon }}={\overline {\frac {d\phi _{1}}{d\epsilon _{1}}}}{\bigg |}_{\epsilon }={\overline {\frac {d\phi _{2}}{d\epsilon _{2}}}}{\bigg |}_{\epsilon },$

(394)

and

${\frac {1}{\Theta }}={\overline {\frac {d\phi }{d\epsilon }}}{\bigg |}_{\Theta }={\overline {\frac {d\phi _{1}}{d\epsilon _{1}}}}{\bigg |}_{\Theta }={\overline {\frac {d\phi _{2}}{d\epsilon _{2}}}}{\bigg |}_{\Theta }.$

(395)

We have compared certain functions of the energy of the whole system with average values of similar functions of the kinetic energy of the whole system, and with average values of similar functions of the whole energy of a part of the system. We may also compare the same functions with average values of the kinetic energy of a part of the system.

We shall express the total, kinetic, and potential energies of the whole system by $\epsilon$, $\epsilon _{p}$, and $\epsilon _{q}$, and the kinetic energies of the parts by $\epsilon _{1p}$, and $\epsilon _{2p}$. These kinetic energies are necessarily separate: we need not make any supposition concerning potential energies. The extension-in-phase within any limits which can be expressed in terms of $\epsilon _{q}$, $\epsilon _{1p}$, $\epsilon _{2p}$ may be represented in the notations of Chapter VIII by the triple integral

$\iiint dV_{1p}\,dV_{2p}\,dV_{q}$

taken within those limits. And if an ensemble of systems is distributed with a uniform density within those limits, the average value of any function of $\epsilon _{q}$, $\epsilon _{1p}$, $\epsilon _{2p}$ will be expressed by the quotient

${\frac {\iiint u\,dV_{1p}\,dV_{2p}\,dV_{q}}{\iiint dV_{1p}\,dV_{2p}\,dV_{q}}}$

or

${\frac {\iiint u\,e^{\phi _{1p}}\,d\epsilon \,dV_{2p}\,dV_{q}}{\iiint e^{\phi _{1p}}\,d\epsilon \,dV_{2p}\,dV_{q}}}$

To get the average value of $u$ for a microcanonical distribution, we must make the limits $\epsilon$ and $\epsilon +d\epsilon$. The denominator in this case becomes $\epsilon ^{\phi }\,d\epsilon$, and we have

${\overline {u}}|_{\epsilon }=e^{-\phi }\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }\int \limits _{\epsilon _{2p}=0}^{\epsilon _{2p}=\epsilon -\epsilon _{q}}u\,e^{\phi _{1p}}\,dV_{2p}\,dV_{q},$

(396)

where $\phi _{1p}$, $V_{2p}$, and $V_{q}$ are connected by the equation

$\epsilon _{1p}+\epsilon _{2p}+\epsilon _{q}=\mathrm {constant} =\epsilon .$

Accordingly

${\overline {e^{-\phi _{1p}}V_{1p}}}|_{\epsilon }=e^{-\phi }\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }\int \limits _{\epsilon _{2p}=0}^{\epsilon _{2p}=\epsilon -\epsilon _{q}}V_{1p}\,dV_{2p}\,dV_{q}=e^{-\phi }V,$

(397)

and we may write

$e^{-\phi }V={\overline {e^{-\phi _{1p}}V_{1p}}}|_{\epsilon }={\overline {e^{-\phi _{2p}}V_{2p}}}|_{\epsilon }={\frac {2}{n_{1}}}{\overline {\epsilon _{1p}}}|_{\epsilon }={\frac {2}{n_{2}}}{\overline {\epsilon _{2p}}}|_{\epsilon },$

(398)

and

$\Theta ={\overline {e^{-\phi _{p}}V_{p}}}|_{\Theta }={\overline {e^{-\phi _{1p}}V_{1p}}}|_{\Theta }={\overline {e^{-\phi _{2p}}V_{2p}}}|_{\Theta }={\frac {2}{n_{1}}}{\overline {\epsilon _{1p}}}|_{\Theta }={\frac {2}{n_{2}}}{\overline {\epsilon _{2p}}}|_{\Theta }.$

(399)

Again, if $n_{1}>2$,

${\begin{aligned}{\overline {\frac {d\phi _{1p}}{d\epsilon _{1p}}}}{\bigg |}=e^{-\phi }\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }&\int \limits _{\epsilon _{2p}=0}^{\epsilon _{2p}=\epsilon -\epsilon _{q}}{\frac {d\phi _{1p}}{d\epsilon _{1p}}}e^{\phi _{1p}}\,dV_{2p}\,dV_{q}\\&=e^{-\phi }\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }{\frac {d\epsilon ^{\phi _{p}}}{d\epsilon _{p}}}\,dV_{q}=e^{-\phi }{\frac {d\epsilon ^{\phi }}{d\epsilon }}={\frac {d\phi }{d\epsilon }}.\end{aligned}}$

(400)

Hence, if $n_{1}>2$, and $n_{2}>2$,

${\frac {d\phi }{d\epsilon }}={\overline {\frac {d\phi _{1p}}{d\epsilon _{1p}}}}{\bigg |}_{\epsilon }={\overline {\frac {d\phi _{2p}}{d\epsilon _{2p}}}}{\bigg |}_{\epsilon }=({\tfrac {1}{2}}n_{1}-1){\overline {{\epsilon _{1p}}^{-1}}}|_{\epsilon }=({\tfrac {1}{2}}n_{2}-1){\overline {{\epsilon _{2p}}^{-1}}}|_{\epsilon },$

(401)

${\frac {1}{\Theta }}={\overline {\frac {d\phi _{p}}{d\epsilon _{p}}}}{\bigg |}_{\Theta }={\overline {\frac {d\phi _{1p}}{d\epsilon _{1p}}}}{\bigg |}_{\Theta }={\overline {\frac {d\phi _{2p}}{d\epsilon _{2p}}}}{\bigg |}_{\Theta }=({\tfrac {1}{2}}n_{1}-1){\overline {{\epsilon _{1p}}^{-1}}}|_{\Theta }=({\tfrac {1}{2}}n_{2}-1){\overline {{\epsilon _{2p}}^{-1}}}|_{\Theta }.$

(402)

We cannot apply the methods employed in the preceding pages to the microcanonical averages of the (generalized) forces $A_{1}$, $A_{2}$, etc., exerted by a system on external bodies, since these quantities are not functions of the energies, either kinetic or potential, of the whole or any part of the system. We may however use the method described on page 116.

Let us imagine an ensemble of systems distributed in phase according to the index of probability

$c-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}},$

where $\epsilon '$ is any constant which is a possible value of the energy, except only the least value which is consistent with the values of the external coördinates, and $c$ and $\omega$ are other constants. We have therefore

$\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{c-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}}\,dp_{1}\ldots dq_{n}=1,$

(403)

or

$e^{-c}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}}\,dp_{1}\ldots dq_{n},$

(404)

or again

$e^{-c}=\int \limits _{V=0}^{\epsilon =\infty }e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}+\phi }\,d\epsilon .$

(405)

From (404) we have

${\begin{aligned}{\frac {de^{-c}}{da_{1}}}&=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}2{\frac {\epsilon -\epsilon '}{\omega ^{2}}}A_{1}\,e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}}\,dp_{1}\ldots dq_{n}\\&=\int \limits _{V=0}^{\epsilon =\infty }2{\frac {\epsilon -\epsilon '}{\omega ^{2}}}{\overline {A_{1}}}|_{\epsilon }\,e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}+\phi }\,d\epsilon ,\end{aligned}}$

(406)

where ${\overline {A_{1}}}|_{\epsilon }$ denotes the average value of $A_{1}$ in those systems of the ensemble which have any same energy $\epsilon$. (This is the same thing as the average value of $A_{1}$ in a microcanonical ensemble of energy $\epsilon$.) The validity of the transformation is evident, if we consider separately the part of each integral which lies between two infinitesimally differing limits of energy. Integrating by parts, we get

${\begin{aligned}{\frac {de^{-c}}{da_{1}}}&=-{\Big [}{\overline {A_{1}}}|_{\epsilon }\,e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}+\phi }{\Big ]}_{V=0}^{\epsilon =\infty }\\&\qquad {}+\int \limits _{V=0}^{\epsilon =\infty }{\bigg (}{\frac {d{\overline {A_{1}}}|_{\epsilon }}{d\epsilon }}+{\overline {A_{1}}}|_{\epsilon }\,{\frac {d\phi }{d\epsilon }}{\bigg )}e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}+\phi }\,d\epsilon .\end{aligned}}$

(407)

Differentiating (405), we get

${\frac {de^{-c}}{da_{1}}}=\int \limits _{V=0}^{\epsilon =\infty }{\frac {d\phi }{da_{1}}}\,e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}+\phi }\,d\epsilon -{\bigg (}e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}+\phi }\,{\frac {d\epsilon _{a}}{da_{1}}}{\bigg )}_{V=0}$

(408)

where $\epsilon _{a}$ denotes the least value of $\epsilon$ consistent with the external coördinates. The last term in this equation represents the part of $de^{-c}/da_{1}$ which is due to the variation of the lower limit of the integral. It is evident that the expression in the brackets will vanish at the upper limit. At the lower limit, at which $\epsilon _{p}=0$, and $\epsilon _{q}$ has the least value consistent with the external coördinates, the average sign on ${\overline {A_{1}}}|_{\epsilon }$ is superfluous, as there is but one value of $A_{1}$ which is represented by $-d\epsilon _{a}/da_{1}$. Exceptions may indeed occur for particular values of the external coördinates, at which $d\epsilon _{a}/da_{1}$ receive a finite increment, and the formula becomes illusory. Such particular values we may for the moment leave out of account. The last term of (408) is therefore equal to the first term of the second member of (407). (We may observe that both vanish when $n>2$ on account of the factor $e^{\phi }$.)

We have therefore from these equations

$\int \limits _{V=0}^{\epsilon =\infty }{\bigg (}{\frac {d{\overline {A_{1}}}|_{\epsilon }}{d\epsilon }}+{\overline {A_{1}}}|_{\epsilon }\,{\frac {d\phi }{d\epsilon }}{\bigg )}e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}+\phi }\,d\epsilon =\int \limits _{V=0}^{\epsilon =\infty }{\frac {d\phi }{da_{1}}}\,e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}+\phi }\,d\epsilon ,$

or

$\int \limits _{V=0}^{\epsilon =\infty }{\bigg (}{\frac {d{\overline {A_{1}}}|_{\epsilon }}{d\epsilon }}+{\overline {A_{1}}}|_{\epsilon }\,{\frac {d\phi }{d\epsilon }}-{\frac {d\phi }{da_{1}}}{\bigg )}e^{c-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}+\phi }\,d\epsilon =0.$

(409)

That is: the average value in the ensemble of the quantity represented by the principal parenthesis is zero. This must be true for any value of $\omega$. If we diminish $\omega$, the average value of the parenthesis at the limit when $\omega$ vanishes becomes identical with the value for $\epsilon =\epsilon '$. But this may be any value of the energy, except the least possible. We have therefore

${\frac {d{\overline {A_{1}}}|_{\epsilon }}{d\epsilon }}+{\overline {A_{1}}}|_{\epsilon }\,{\frac {d\phi }{d\epsilon }}-{\frac {d\phi }{da_{1}}}=0,$

(410)

unless it be for the least value of the energy consistent with the external coördinates, or for particular values of the external coördinates. But the value of any term of this equation as determined for particular values of the energy and of the external coördinates is not distinguishable from its value as determined for values of the energy and external coördinates indefinitely near those particular values. The equation therefore holds without limitation. Multiplying by $e^{\phi }$, we get

$e^{\phi }{\frac {d{\overline {A_{1}}}|_{\epsilon }}{d\epsilon }}+{\overline {A_{1}}}|_{\epsilon }\,e^{\phi }\,{\frac {d\phi }{d\epsilon }}=e^{\phi }\,{\frac {d\phi }{da_{1}}}={\frac {de^{\phi }}{da_{1}}}={\frac {d^{2}V}{da_{1}\,d\epsilon }}.$

(411)

The integral of this equation is

${\overline {A_{1}}}|_{\epsilon }e^{\phi }={\frac {dV}{da_{1}}}+F_{1},,$

(412)

where $F_{1}$ is a function of the external coördinates. We have an equation of this form for each of the external coördinates. This gives, with (266), for the complete value of the differential of $V$

$dV=e^{\phi }\,d\epsilon +(e^{\phi }{\overline {A_{1}}}|_{\epsilon }-F_{1})da_{1}+(e^{\phi }{\overline {A_{2}}}|_{\epsilon }-F_{2})da_{2}+\mathrm {etc.} ,$

(413)

or

$dV=e^{\phi }(d\epsilon +{\overline {A_{1}}}|_{\epsilon }\,da_{1}+{\overline {A_{2}}}|_{\epsilon }\,da_{2}+\mathrm {etc.} )-F_{1}\,da_{1}-F_{2}\,da_{2}-\mathrm {etc.}$

(414)

To determine the values of the functions $F_{1}$, $F_{2}$, etc., let us suppose $a_{1}$, $a_{2}$, etc. to vary arbitrarily, while $\epsilon$ varies so as always to have the least value consistent with the values of the external coördinates. This will make $V=0$, and $dV=0$. If $n<2$, we shall have also $e^{\phi }=0$, which will give

$F_{1}=0,\quad F_{2}=0,\quad \mathrm {etc.}$

(415)

The result is the same for any value of $n$. For in the variations considered the kinetic energy will be constantly zero, and the potential energy will have the least value consistent with the external coördinates. The condition of the least possible potential energy may limit the ensemble at each instant to a single configuration, or it may not do so; but in any case the values of $A_{1}$, $A_{2}$, etc. will be the same at each instant for all the systems of the ensemble,^[8] and the equation

$d\epsilon +A_{1}da_{1}+A_{2}da_{2}+\mathrm {etc.} =0$

will hold for the variations considered. Hence the functions $F_{1}$, $F_{2}$, etc. vanish in any case, and we have the equation

$dV=e^{\phi }\,d\epsilon +e^{\phi }{\overline {A_{1}}}|_{\epsilon }\,da_{1}+e^{\phi }{\overline {A_{2}}}|_{\epsilon }\,da_{2}+\mathrm {etc.} ,$

(416)

or

$d\log V={\frac {d\epsilon +{\overline {A_{1}}}|_{\epsilon }\,da_{1}+{\overline {A_{2}}}|_{\epsilon }\,da_{2}+\mathrm {etc.} }{e^{-\phi }V}},$

(417)

or again

$d\epsilon =e^{-\phi }V\,d\log V-{\overline {A_{1}}}|_{\epsilon }\,da_{1}-{\overline {A_{2}}}|_{\epsilon }\,da_{2}-\mathrm {etc.}$

(418)

It will be observed that the two last equations have the form of the fundamental differential equations of thermodynamics,$e^{-\phi }V$ corresponding to temperature and $\log V$ to entropy. We have already observed properties of $e^{-\phi }V$ suggestive of an analogy with temperature.^[9] The significance of these facts will be discussed in another chapter.

The two last equations might be written more simply

$dV={\frac {d\epsilon +{\overline {A_{1}}}|_{\epsilon }\,da_{1}+{\overline {A_{2}}}|_{\epsilon }\,da_{2}+\mathrm {etc.} }{e^{-\phi }}},$

$d\epsilon =e^{-\phi }\,dV-{\overline {A_{1}}}|_{\epsilon }\,da_{1}-{\overline {A_{2}}}|_{\epsilon }\,da_{2}-\mathrm {etc.} ,$

and still have the form analogous to the thermodynamic equations, but $e^{-\phi }$ has nothing like the analogies with temperature which we have observed in $e^{-\phi }V$.

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1. See equation (322).
2. An example of the failing case of the microcanonical distribution is afforded by a material point, under the influence of gravity, and constrained to remain in a vertical circle. The failing case occurs when the energy is just sufficient to carry the material point to the highest point of the circle. It will be observed that the difficulty is inherent in the nature of the case, and is quite independent of the mathematical formulae. The nature of the difficulty is at once apparent if we try to distribute a finite number of material points with this particular value of the energy as nearly as possible in statistical equilibrium, or if we ask: What is the probability that a point taken at random from an ensemble in statistical equilibrium with this value of the energy will be found in any specified part of the circle?
3. See equation (292).
4. If this condition is rigorously fulfilled, the parts will have no influence on each other, and the ensemble formed by distributing the whole microcanonically is too arbitrary a conception to have a real interest. The principal interest of the equations which we shall obtain will be in cases in which the condition is approximately fulfilled. But for the purposes of a theoretical discussion, it is of course convenient to make such a condition absolute. Compare Chapter IV, pp. 35 ff., where a similar condition is considered in connection with canonical ensembles.
5. Where the analytical transformations are identical in form with those on the preceding pages, it does not appear necessary to give all the steps with the same detail.
6. In the applications of the equation (387), we cannot obtain all the results corresponding to those which we have obtained from equation (374), because $\phi _{p}$ is a known function of $\epsilon _{p}$, while $\phi _{1}$ must be treated as an arbitrary function of $\epsilon _{1}$, or nearly so.
7. See Chapter VIII, equations (306) and (316).
8. This statement, as mentioned before, may have exceptions for particular values of the external coördinates. This will not invalidate the reasoning, which has to do with varying values of the external coördinates.
9. See Chapter IX, page 111; also this chapter, page 119.