​

In the following theorems we suppose, as always, that the systems forming an ensemble are identical in nature and in the values of the external coördinates, which are here regarded as constants.

Theorem I. If an ensemble of systems is so distributed in phase that the index of probability is a function of the energy, the average value of the index is less than for any other distribution in which the distribution in energy is unaltered.

Let us write ${\displaystyle \eta }$ for the index which is a function of the energy, and ${\displaystyle \eta +\Delta \eta }$ for any other which gives the same distribution in energy. It is to be proved that

${\displaystyle \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}(\eta +\Delta \eta )e^{\eta +\Delta \eta }\,dp_{1}\ldots dq_{n}>\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\eta e^{\eta }\,dp_{1}\ldots dq_{n},}$

(419)

where ${\displaystyle \eta }$ is a function of the energy, and ${\displaystyle \Delta \eta }$ a function of the phase, which are subject to the conditions that

${\displaystyle \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{\eta +\Delta \eta }\,dp_{1}\ldots dq_{n}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{\eta }\,dp_{1}\ldots dq_{n}=1,}$

(420)

and that for any value of the energy (${\displaystyle \epsilon '}$)

${\displaystyle \mathop {\int \ldots \int } _{\epsilon =\epsilon '}^{\epsilon =\epsilon '+d\epsilon '}e^{\eta +\Delta \eta }\,dp_{1}\ldots dq_{n}=\mathop {\int \ldots \int } _{\epsilon =\epsilon '}^{\epsilon =\epsilon '+d\epsilon '}e^{\eta }\,dp_{1}\ldots dq_{n}.}$

(421)

Equation (420) expresses the general relations which ${\displaystyle \eta }$ and ${\displaystyle \eta +\Delta \eta }$ must satisfy in order to be indices of any distributions, and (421) expresses the condition that they give the same distribution in energy.

​Since ${\displaystyle \eta }$ is a function of the energy, and may therefore be regarded as a constant within the limits of integration of (421), we may multiply by ${\displaystyle \eta }$ under the integral sign in both members, which gives

${\displaystyle \mathop {\int \ldots \int } _{\epsilon =\epsilon '}^{\epsilon =\epsilon '+d\epsilon '}\eta e^{\eta +\Delta \eta }\,dp_{1}\ldots dq_{n}=\mathop {\int \ldots \int } _{\epsilon =\epsilon '}^{\epsilon =\epsilon '+d\epsilon '}\eta e^{\eta }\,dp_{1}\ldots dq_{n}.}$

Since this is true within the limits indicated, and for every value of ${\displaystyle \epsilon '}$, it will be true if the integrals are taken for all phases. We may therefore cancel the corresponding parts of (419), which gives

${\displaystyle \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\Delta \eta e^{\eta +\Delta \eta }\,dp_{1}\ldots dq_{n}>0.}$

(422)

But by (420) this is equivalent to

${\displaystyle \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}(\Delta \eta e^{\Delta \eta }+1-e^{\Delta \eta })e^{\eta }\,dp_{1}\ldots dq_{n}>0.}$

(423)

Now ${\displaystyle \Delta \eta e^{\Delta \eta }+1-e^{\Delta \eta }}$ is a decreasing function of ${\displaystyle \Delta \eta }$ for negative values of ${\displaystyle \Delta \eta }$, and an increasing function of ${\displaystyle \Delta \eta }$ for positive values of ${\displaystyle \Delta \eta }$. It vanishes for ${\displaystyle \Delta \eta =0}$. The expression is therefore incapable of a negative value, and can have the value 0 only for ${\displaystyle \Delta \eta =0}$. The inequality (423) will hold therefore unless ${\displaystyle \Delta \eta =0}$ for all phases. The theorem is therefore proved.

Theorem II. If an ensemble of systems is canonically distributed in phase, the average index of probability is less than in any other distribution of the ensemble having the same average energy.

For the canonical distribution let the index be ${\displaystyle (\psi -\epsilon )/\Theta }$, and for another having the same average energy let the index be ${\displaystyle (\psi -\epsilon )/\Theta +\Delta \eta }$, where ${\displaystyle \Delta \eta }$ is an arbitrary function of the phase subject only to the limitation involved in the notion of the index, that ​

${\displaystyle \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{{\frac {\psi -\epsilon }{\Theta }}+\Delta \eta }\,dp_{1}\ldots dq_{n}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=1,}$

(424)

and to that relating to the constant average energy, that

${\displaystyle \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\epsilon e^{{\frac {\psi -\epsilon }{\Theta }}+\Delta \eta }\,dp_{1}\ldots dq_{n}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\epsilon e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}.}$

(425)

It is to be proved that

${\displaystyle {\begin{aligned}&\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg (}{\frac {\psi }{\Theta }}-{\frac {\epsilon }{\Theta }}+\Delta \eta {\bigg )}e^{{\frac {\psi -\epsilon }{\Theta }}+\Delta \eta }\,dp_{1}\ldots dq_{n}>\\&\qquad \qquad \qquad \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg (}{\frac {\psi }{\Theta }}-{\frac {\epsilon }{\Theta }}{\bigg )}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}.\end{aligned}}}$

(426)

Now in virtue of the first condition (424) we may cancel the constant term ${\displaystyle \psi /\Theta }$ in the parentheses in (426), and in virtue of the second condition (425) we may cancel the term ${\displaystyle \epsilon /\Theta }$. The proposition to be proved is thus reduced to

${\displaystyle \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\Delta \eta e^{{\frac {\psi -\epsilon }{\Theta }}+\Delta \eta }\,dp_{1}\ldots dq_{n}>0,}$

which may be written, in virtue of the condition (424),

${\displaystyle \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}(\Delta \eta e^{\Delta \eta }+1-e^{\Delta \eta })e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}>0.}$

(427)

In this form its truth is evident for the same reasons which applied to (423).

Theorem III. If ${\displaystyle \Theta }$ is any positive constant, the average value in an ensemble of the expression ${\displaystyle \eta +\epsilon /\Theta }$ (${\displaystyle \eta }$ denoting as usual the index of probability and ${\displaystyle \epsilon }$ the energy) is less when the ensemble is distributed canonically with modulus ${\displaystyle \Theta }$, than for any other distribution whatever.

In accordance with our usual notation let us write ${\displaystyle (\psi -\epsilon )/\Theta }$ for the index of the canonical distribution. In any other distribution let the index be ${\displaystyle (\psi -\epsilon )/\Theta +\Delta \eta }$.

​In the canonical ensemble ${\displaystyle \eta +\epsilon /\Theta }$ has the constant value ${\displaystyle \psi /\Theta }$; in the other ensemble it has the value ${\displaystyle \psi /\Theta +\Delta \eta }$. The proposition to be proved may therefore be written

${\displaystyle {\frac {\psi }{\Theta }}<\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg (}{\frac {\psi }{\Theta }}+\Delta \eta {\bigg )}e^{{\frac {\psi -\epsilon }{\Theta }}+\Delta \eta }\,dp_{1}\ldots dq_{n}>0,}$

(428)

where

${\displaystyle \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{{\frac {\psi -\epsilon }{\Theta }}+\Delta \eta }\,dp_{1}\ldots dq_{n}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=1.}$

(429)

In virtue of this condition, since ${\displaystyle \psi /\Theta }$ is constant, the proposition to be proved reduces to

${\displaystyle 0<\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\Delta \eta e^{{\frac {\psi -\epsilon }{\Theta }}+\Delta \eta }\,dp_{1}\ldots dq_{n},}$

(430)

where the demonstration may be concluded as in the last theorem.

If we should substitute for the energy in the preceding theorems any other function of the phase, the theorems, mutatis mutandis, would still hold. On account of the unique importance of the energy as a function of the phase, the theorems as given are especially worthy of notice. When the case is such that other functions of the phase have important properties relating to statistical equilibrium, as described in Chapter IV,^[1] the three following theorems, which are generalizations of the preceding, may be useful. It will be sufficient to give them without demonstration, as the principles involved are in no respect different.

Theorem IV. If an ensemble of systems is so distributed in phase that the index of probability is any function of ${\displaystyle F_{1}}$, ${\displaystyle F_{2}}$, etc., (these letters denoting functions of the phase,) the average value of the index is less than for any other distribution in phase in which the distribution with respect to the functions ${\displaystyle F_{1}}$, ${\displaystyle F_{2}}$, etc. is unchanged.

​Theorem V. If an ensemble of systems is so distributed in phase that the index of probability is a linear function of ${\displaystyle F_{1}}$, ${\displaystyle F_{2}}$, etc., (these letters denoting functions of the phase,) the average value of the index is less than for any other distribution in which the functions ${\displaystyle F_{1}}$, ${\displaystyle F_{2}}$, etc. have the same average values.

Theorem VI. The average value in an ensemble of systems of ${\displaystyle \eta +F}$ (where ${\displaystyle \eta }$ denotes as usual the index of probability and ${\displaystyle F}$ any function of the phase) is less when the ensemble is so distributed that ${\displaystyle \eta +F}$ is constant than for any other distribution whatever.

Theorem VII. If a system which in its different phases constitutes an ensemble consists of two parts, and we consider the average index of probability for the whole system, and also the average indices for each of the parts taken separately, the sum of the average indices for the parts will be either less than the average index for the whole system, or equal to it, but cannot be greater. The limiting case of equality occurs when the distribution in phase of each part is independent of that of the other, and only in this case.

Let the coördinates and momenta of the whole system be ${\displaystyle q_{1}\ldots q_{n},p_{1},\ldots p_{n}}$, of which ${\displaystyle q_{1}\ldots q_{m},p_{1},\ldots p_{m}}$ relate to one part of the system, and ${\displaystyle q_{m+1}\ldots q_{n},p_{m+1},\ldots p_{n}}$ to the other. If the index of probability for the whole system is denoted by ${\displaystyle \eta }$, the probability that the phase of an unspecified system lies within any given limits is expressed by the integral

${\displaystyle \int \ldots \int e^{\eta }\,dp_{1}\ldots dq_{n}}$

(431)

taken for those limits. If we set

${\displaystyle \int \ldots \int e^{\eta }\,dp_{m+1}\ldots dp_{n}\,dq_{m+1}\ldots dq_{n}=e^{\eta _{1}},}$

(432)

where the integrations cover all phases of the second system, and

${\displaystyle \int \ldots \int e^{\eta }\,dp_{1}\ldots dp_{m}\,dq_{1}\ldots dq_{m}=e^{\eta _{2}},}$

(433)

​where the integrations cover all phases of the first system, the integral (431) will reduce to the form

${\displaystyle \int \ldots \int e^{\eta _{1}}\,dp_{1}\ldots dp_{m}\,dq_{1}\ldots dq_{m},}$

(434)

when the limits can be expressed in terms of the coördinates and momenta of the first part of the system. The same integral will reduce to

${\displaystyle \int \ldots \int e^{\eta _{2}}\,dp_{m+1}\ldots dp_{n}\,dq_{m+1}\ldots dq_{n},}$

(435)

when the limits can be expressed in terms of the coördinates and momenta of the second part of the system. It is evident that ${\displaystyle \eta _{1}}$ and ${\displaystyle \eta _{2}}$ are the indices of probability for the two parts of the system taken separately.

The main proposition to be proved may be written

${\displaystyle {\begin{aligned}\int \ldots \int \eta _{1}e^{\eta _{1}}\,dp_{1}\ldots dq_{m}+&\int \ldots \int \eta _{2}e^{\eta _{2}}\,dp_{m+1}\ldots dq_{n}\leq \\&\qquad \int \ldots \int \eta e^{\eta }\,dp_{1}\ldots dq_{n},\end{aligned}}}$

(436)

where the first integral is to be taken over all phases of the first part of the system, the second integral over all phases of the second part of the system, and the last integral over all phases of the whole system. Now we have

${\displaystyle \int \ldots \int e^{\eta }\,dp_{1}\ldots dq_{n}=1,}$

(437)

${\displaystyle \int \ldots \int e^{\eta _{1}}\,dp_{1}\ldots dq_{m}=1,}$

(438)

and

${\displaystyle \int \ldots \int e^{\eta _{2}}\,dp_{m+1}\ldots dq_{n}=1,}$

(439)

where the limits cover in each case all the phases to which the variables relate. The two last equations, which are in themselves evident, may be derived by partial integration from the first.

​It appears from the definitions of ${\displaystyle \eta _{1}}$ and ${\displaystyle \eta _{2}}$ that (436) may also be written

${\displaystyle {\begin{aligned}\int \ldots \int \eta _{1}e^{\eta }\,dp_{1}\ldots dq_{n}+&\int \ldots \int \eta _{2}e^{\eta }\,dp_{1}\ldots dq_{n}\leq \\&\qquad \int \ldots \int \eta e^{\eta }\,dp_{1}\ldots dq_{n},\end{aligned}}}$

(440)

or

${\displaystyle \int \ldots \int (\eta -\eta _{1}-\eta _{2})e^{\eta }\,dp_{1}\ldots dq_{n}\geq 0,}$

where the integrations cover all phases. Adding the equation

${\displaystyle \int \ldots \int e^{\eta _{1}+\eta _{2}}\,dp_{1}\ldots dq_{n}=1,}$

(441)

which we get by multiplying (438) and (439), and subtracting (437), we have for the proposition to be proved

${\displaystyle \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}[(\eta -\eta _{1}-\eta _{2})e^{\eta }+e^{\eta _{1}+\eta _{2}}-e^{\eta }]\,dp_{1}\ldots dq_{n}\geq 0.}$

(442)

Let

${\displaystyle u=\eta -\eta _{1}-\eta _{2}.}$

(443)

The main proposition to be proved may be written

${\displaystyle \int \ldots \int (ue^{u}+1-e^{u})e^{\eta _{1}+\eta _{2}}\,dp_{1}\ldots dq_{n}\geq 0.}$

(444)

This is evidently true since the quantity in the parenthesis is incapable of a negative value.^[2] Moreover the sign ${\displaystyle =}$ can hold only when the quantity in the parenthesis vanishes for all phases, i. e., when ${\displaystyle u=0}$ for all phases. This makes ${\displaystyle \eta =\eta _{1}+\eta _{2}}$ for all phases, which is the analytical condition which expresses that the distributions in phase of the two parts of the system are independent.

Theorem VIII. If two or more ensembles of systems which are identical in nature, but may be distributed differently in phase, are united to form a single ensemble, so that the probability-coefficient of the resulting ensemble is a linear function ​of the probability-coefficients of the original ensembles, the average index of probability of the resulting ensemble cannot be greater than the same linear function of the average indices of the original ensembles. It can be equal to it only when the original ensembles are similarly distributed in phase.

Let ${\displaystyle P_{1}}$, ${\displaystyle P_{2}}$, etc. be the probability-coefficients of the original ensembles, and ${\displaystyle P}$ that of the ensemble formed by combining them; and let ${\displaystyle N_{1}}$, ${\displaystyle N_{2}}$, etc. be the numbers of systems in the original ensembles. It is evident that we shall have

${\displaystyle P=c_{1}P_{1}+c_{2}P_{2}+\mathrm {etc.} =\Sigma (c_{1}P_{1}),}$

(445)

where

${\displaystyle c_{1}={\frac {N_{1}}{\Sigma N_{1}}},\qquad c_{2}={\frac {N_{2}}{\Sigma N_{1}}},\qquad \mathrm {etc.} }$

(446)

The main proposition to be proved is that

${\displaystyle \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}P\log P\,dp_{1}\ldots dq_{n}\leq \Sigma {\Bigg [}c_{1}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}P_{1}\log P_{1}\,dp_{1}\ldots dq_{n}{\Bigg ]}}$

(447)

or

${\displaystyle \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}[\Sigma (c_{1}P_{1}\log P_{1})-P\log P]\,dp_{1}\ldots dq_{n}\geq 0.}$

(448)

If we set

${\displaystyle Q_{1}=P_{1}\log P_{1}-P_{1}\log P-P_{1}+P}$

${\displaystyle Q_{1}}$ will be positive, except when it vanishes for ${\displaystyle P_{1}=P}$. To prove this, we may regard ${\displaystyle P_{1}}$ and ${\displaystyle P}$ as any positive quantities. Then

${\displaystyle {\bigg (}{\frac {dQ_{1}}{dP_{1}}}{\bigg )}_{P}=\log P_{1}-\log P,}$

${\displaystyle {\bigg (}{\frac {d^{2}Q_{1}}{dP_{1}{}^{2}}}{\bigg )}_{P}={\frac {1}{P_{1}}}.}$

Since ${\displaystyle Q_{1}}$ and ${\displaystyle dQ_{1}/dP_{1}}$ vanish for ${\displaystyle P_{1}=P}$, and the second differential coefficient is always positive, ${\displaystyle Q_{1}}$ must be positive except when ${\displaystyle P_{1}=P}$. Therefore, if ${\displaystyle Q_{2}}$, etc. have similar definitions,

${\displaystyle \Sigma (c_{1}Q_{1})\geq 0.}$

(449)

​But since

${\displaystyle \Sigma (c_{1}P_{1})=P}$

and

${\displaystyle \Sigma c_{1}=1,}$

${\displaystyle \Sigma (c_{1}Q_{1})=\Sigma (c_{1}P_{1}\log P_{1})-P\log P.}$

(450)

This proves (448), and shows that the sign ${\displaystyle =}$ will hold only when

${\displaystyle P_{1}=P,\qquad P_{2}=P,\qquad \mathrm {etc.} }$

for all phases, i. e., only when the distribution in phase of the original ensembles are all identical.

Theorem IX. A uniform distribution of a given number of systems within given limits of phase gives a less average index of probability of phase than any other distribution.

Let ${\displaystyle \eta }$ be the constant index of the uniform distribution, and ${\displaystyle \eta +\Delta \eta }$ the index of some other distribution. Since the number of systems within the given limits is the same in the two distributions we have

${\displaystyle \int \ldots \int e^{\eta +\Delta \eta }\,dp_{1}\ldots dq_{n}=\int \ldots \int e^{\eta }\,dp_{1}\ldots dq_{n},}$

(451)

where the integrations, like those which follow, are to be taken within the given limits. The proposition to be proved may be written

${\displaystyle \int \ldots \int (\eta +\Delta \eta )e^{\eta +\Delta \eta }\,dp_{1}\ldots dq_{n}>\int \ldots \int \eta e^{\eta }\,dp_{1}\ldots dq_{n},}$

(452)

or, since ${\displaystyle \eta }$ is constant,

${\displaystyle \int \ldots \int (\eta +\Delta \eta )e^{\Delta \eta }\,dp_{1}\ldots dq_{n}>\int \ldots \int \eta \,dp_{1}\ldots dq_{n}.}$

(453)

In (451) also we may cancel the constant factor ${\displaystyle e^{\eta }}$, and multiply by the constant factor ${\displaystyle (\eta +1)}$. This gives

${\displaystyle \int \ldots \int (\eta +1)e^{\Delta \eta }\,dp_{1}\ldots dq_{n}=\int \ldots \int (\eta +1)\,dp_{1}\ldots dq_{n}.}$

The subtraction of this equation will not alter the inequality to be proved, which may therefore be written

${\displaystyle \int \ldots \int (\Delta \eta -1)e^{\Delta \eta }\,dp_{1}\ldots dq_{n}>\int \ldots \int -\,dp_{1}\ldots dq_{n}}$

​or

${\displaystyle \int \ldots \int (\Delta \eta e^{\Delta \eta }-e^{\Delta \eta }+1)\,dp_{1}\ldots dq_{n}>0.}$

(454)

Since the parenthesis in this expression represents a positive value, except when it vanishes for ${\displaystyle \Delta \eta =0}$, the integral will be positive unless ${\displaystyle \Delta \eta }$ vanishes everywhere within the limits, which would make the difference of the two distributions vanish. The theorem is therefore proved.