# Elementary Principles in Statistical Mechanics/Chapter XI

CHAPTER XI.
MAXIMUM AND MINIMUM PROPERTIES OF VARIOUS DISTRIBUTIONS IN PHASE.
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 $\eta$ for the index which is a function of the energy, and $\eta +\Delta \eta$ for any other which gives the same distribution in energy. It is to be proved that
| $\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}(\eta +\Delta \eta )e^{\eta +\Delta \eta }\,dp_{1}\ldots dq_{n}>\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\eta e^{\eta }\,dp_{1}\ldots dq_{n},$ | (419) |
$\eta$
$\Delta \eta$
| $\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{\eta +\Delta \eta }\,dp_{1}\ldots dq_{n}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{\eta }\,dp_{1}\ldots dq_{n}=1,$ | (420) |
$\epsilon '$
| $\mathop {\int \ldots \int } _{\epsilon =\epsilon '}^{\epsilon =\epsilon '+d\epsilon '}e^{\eta +\Delta \eta }\,dp_{1}\ldots dq_{n}=\mathop {\int \ldots \int } _{\epsilon =\epsilon '}^{\epsilon =\epsilon '+d\epsilon '}e^{\eta }\,dp_{1}\ldots dq_{n}.$ | (421) |
$\eta$
$\eta +\Delta \eta$
​Since $\eta$ 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 $\eta$ under the integral sign in both members, which gives
* $\mathop {\int \ldots \int } _{\epsilon =\epsilon '}^{\epsilon =\epsilon '+d\epsilon '}\eta e^{\eta +\Delta \eta }\,dp_{1}\ldots dq_{n}=\mathop {\int \ldots \int } _{\epsilon =\epsilon '}^{\epsilon =\epsilon '+d\epsilon '}\eta e^{\eta }\,dp_{1}\ldots dq_{n}.$
$\epsilon '$
| $\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\Delta \eta e^{\eta +\Delta \eta }\,dp_{1}\ldots dq_{n}>0.$ | (422) |
| $\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}(\Delta \eta e^{\Delta \eta }+1-e^{\Delta \eta })e^{\eta }\,dp_{1}\ldots dq_{n}>0.$ | (423) |
$\Delta \eta e^{\Delta \eta }+1-e^{\Delta \eta }$
$\Delta \eta$
$\Delta \eta$
$\Delta \eta$
$\Delta \eta$
$\Delta \eta =0$
$\Delta \eta =0$
$\Delta \eta =0$
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 $(\psi -\epsilon )/\Theta$, and for another having the same average energy let the index be $(\psi -\epsilon )/\Theta +\Delta \eta$, where $\Delta \eta$ is an arbitrary function of the phase subject only to the limitation involved in the notion of the index, that ​
| $\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{{\frac {\psi -\epsilon }{\Theta }}+\Delta \eta }\,dp_{1}\ldots dq_{n}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=1,$ | (424) |
| $\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\epsilon e^{{\frac {\psi -\epsilon }{\Theta }}+\Delta \eta }\,dp_{1}\ldots dq_{n}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\epsilon e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}.$ | (425) |
It is to be proved that
| ${\begin{aligned}&\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg{\frac {\psi }{\Theta }}-{\frac {\epsilon }{\Theta }}+\Delta \eta {\bigg )}e^{{\frac {\psi -\epsilon }{\Theta }}+\Delta \eta }\,dp_{1}\ldots dq_{n}>\\&\qquad \qquad \qquad \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg{\frac {\psi }{\Theta }}-{\frac {\epsilon }{\Theta }}{\bigg )}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}.\end{aligned}}$ | (426) |
$\psi /\Theta$
$\epsilon /\Theta$
* $\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\Delta \eta e^{{\frac {\psi -\epsilon }{\Theta }}+\Delta \eta }\,dp_{1}\ldots dq_{n}>0,$
| $\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}(\Delta \eta e^{\Delta \eta }+1-e^{\Delta \eta })e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}>0.$ | (427) |
Theorem III. If $\Theta$ is any positive constant, the average value in an ensemble of the expression $\eta +\epsilon /\Theta$ ($\eta$ denoting as usual the index of probability and $\epsilon$ the energy) is less when the ensemble is distributed canonically with modulus $\Theta$, than for any other distribution whatever.
In accordance with our usual notation let us write $(\psi -\epsilon )/\Theta$ for the index of the canonical distribution. In any other distribution let the index be $(\psi -\epsilon )/\Theta +\Delta \eta$.
​In the canonical ensemble $\eta +\epsilon /\Theta$ has the constant value $\psi /\Theta$; in the other ensemble it has the value $\psi /\Theta +\Delta \eta$. The proposition to be proved may therefore be written
| ${\frac {\psi }{\Theta }}<\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg{\frac {\psi }{\Theta }}+\Delta \eta {\bigg )}e^{{\frac {\psi -\epsilon }{\Theta }}+\Delta \eta }\,dp_{1}\ldots dq_{n}>0,$ | (428) |
| $\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{{\frac {\psi -\epsilon }{\Theta }}+\Delta \eta }\,dp_{1}\ldots dq_{n}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=1.$ | (429) |
$\psi /\Theta$
| $0<\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\Delta \eta e^{{\frac {\psi -\epsilon }{\Theta }}+\Delta \eta }\,dp_{1}\ldots dq_{n},$ | (430) |
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, 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 $F_{1}$, $F_{2}$, 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 $F_{1}$, $F_{2}$, 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 $F_{1}$, $F_{2}$, 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 $F_{1}$, $F_{2}$, etc. have the same average values.
Theorem VI. The average value in an ensemble of systems of $\eta +F$ (where $\eta$ denotes as usual the index of probability and $F$ any function of the phase) is less when the ensemble is so distributed that $\eta +F$ 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 $q_{1}\ldots q_{n},p_{1},\ldots p_{n}$, of which $q_{1}\ldots q_{m},p_{1},\ldots p_{m}$ relate to one part of the system, and $q_{m+1}\ldots q_{n},p_{m+1},\ldots p_{n}$ to the other. If the index of probability for the whole system is denoted by $\eta$, the probability that the phase of an unspecified system lies within any given limits is expressed by the integral
| $\int \ldots \int e^{\eta }\,dp_{1}\ldots dq_{n}$ | (431) |
| $\int \ldots \int e^{\eta }\,dp_{m+1}\ldots dp_{n}\,dq_{m+1}\ldots dq_{n}=e^{\eta _{1}},$ | (432) |
| $\int \ldots \int e^{\eta }\,dp_{1}\ldots dp_{m}\,dq_{1}\ldots dq_{m}=e^{\eta _{2}},$ | (433) |
| $\int \ldots \int e^{\eta _{1}}\,dp_{1}\ldots dp_{m}\,dq_{1}\ldots dq_{m},$ | (434) |
| $\int \ldots \int e^{\eta _{2}}\,dp_{m+1}\ldots dp_{n}\,dq_{m+1}\ldots dq_{n},$ | (435) |
$\eta _{1}$
$\eta _{2}$
The main proposition to be proved may be written
| ${\begin{aligned}\int \ldots \int \eta _{1}e^{\eta _{1}}\,dp_{1}\ldots dq_{m}+&\int \ldots \int \eta _{2}e^{\eta _{2}}\,dp_{m+1}\ldots dq_{n}\leq \\&\qquad \int \ldots \int \eta e^{\eta }\,dp_{1}\ldots dq_{n},\end{aligned}}$ | (436) |
| $\int \ldots \int e^{\eta }\,dp_{1}\ldots dq_{n}=1,$ | (437) |
| $\int \ldots \int e^{\eta _{1}}\,dp_{1}\ldots dq_{m}=1,$ | (438) |
| $\int \ldots \int e^{\eta _{2}}\,dp_{m+1}\ldots dq_{n}=1,$ | (439) |
​It appears from the definitions of $\eta _{1}$ and $\eta _{2}$ that (436) may also be written
| ${\begin{aligned}\int \ldots \int \eta _{1}e^{\eta }\,dp_{1}\ldots dq_{n}+&\int \ldots \int \eta _{2}e^{\eta }\,dp_{1}\ldots dq_{n}\leq \\&\qquad \int \ldots \int \eta e^{\eta }\,dp_{1}\ldots dq_{n},\end{aligned}}$ | (440) |
* $\int \ldots \int (\eta -\eta _{1}-\eta _{2})e^{\eta }\,dp_{1}\ldots dq_{n}\geq 0,$
| $\int \ldots \int e^{\eta _{1}+\eta _{2}}\,dp_{1}\ldots dq_{n}=1,$ | (441) |
| $\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}[(\eta -\eta _{1}-\eta _{2})e^{\eta }+e^{\eta _{1}+\eta _{2}}-e^{\eta }]\,dp_{1}\ldots dq_{n}\geq 0.$ | (442) |
| $u=\eta -\eta _{1}-\eta _{2}.$ | (443) |
| $\int \ldots \int (ue^{u}+1-e^{u})e^{\eta _{1}+\eta _{2}}\,dp_{1}\ldots dq_{n}\geq 0.$ | (444) |
$u=0$
$\eta =\eta _{1}+\eta _{2}$
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 $P_{1}$, $P_{2}$, etc. be the probability-coefficients of the original ensembles, and $P$ that of the ensemble formed by combining them; and let $N_{1}$, $N_{2}$, etc. be the numbers of systems in the original ensembles. It is evident that we shall have
| $P=c_{1}P_{1}+c_{2}P_{2}+\mathrm {etc.} =\Sigma (c_{1}P_{1}),$ | (445) |
| $c_{1}={\frac {N_{1}}{\Sigma N_{1}}},\qquad c_{2}={\frac {N_{2}}{\Sigma N_{1}}},\qquad \mathrm {etc.}$ | (446) |
| $\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}P\log P\,dp_{1}\ldots dq_{n}\leq \Sigma {\Biggc_{1}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}P_{1}\log P_{1}\,dp_{1}\ldots dq_{n}{\Bigg ]}$ | (447) |
| $\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}[\Sigma (c_{1}P_{1}\log P_{1})-P\log P]\,dp_{1}\ldots dq_{n}\geq 0.$ | (448) |
If we set
* $Q_{1}=P_{1}\log P_{1}-P_{1}\log P-P_{1}+P$
$Q_{1}$
$P_{1}=P$
$P_{1}$
$P$
* ${\bigg{\frac {dQ_{1}}{dP_{1}}}{\bigg )}_{P}=\log P_{1}-\log P,$
* ${\bigg{\frac {d^{2}Q_{1}}{dP_{1}^{2}}}{\bigg )}_{P}={\frac {1}{P_{1}}}.$
$Q_{1}$
$dQ_{1}/dP_{1}$
$P_{1}=P$
$Q_{1}$
$P_{1}=P$
$Q_{2}$
| $\Sigma (c_{1}Q_{1})\geq 0.$ | (449) |
* $\Sigma (c_{1}P_{1})=P$
* $\Sigma c_{1}=1,$
| $\Sigma (c_{1}Q_{1})=\Sigma (c_{1}P_{1}\log P_{1})-P\log P.$ | (450) |
* $P_{1}=P,\qquad P_{2}=P,\qquad \mathrm {etc.}$
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 $\eta$ be the constant index of the uniform distribution, and $\eta +\Delta \eta$ the index of some other distribution. Since the number of systems within the given limits is the same in the two distributions we have
| $\int \ldots \int e^{\eta +\Delta \eta }\,dp_{1}\ldots dq_{n}=\int \ldots \int e^{\eta }\,dp_{1}\ldots dq_{n},$ | (451) |
| $\int \ldots \int (\eta +\Delta \eta )e^{\eta +\Delta \eta }\,dp_{1}\ldots dq_{n}>\int \ldots \int \eta e^{\eta }\,dp_{1}\ldots dq_{n},$ | (452) |
$\eta$
| $\int \ldots \int (\eta +\Delta \eta )e^{\Delta \eta }\,dp_{1}\ldots dq_{n}>\int \ldots \int \eta \,dp_{1}\ldots dq_{n}.$ | (453) |
$e^{\eta }$
$(\eta +1)$
* $\int \ldots \int (\eta +1)e^{\Delta \eta }\,dp_{1}\ldots dq_{n}=\int \ldots \int (\eta +1)\,dp_{1}\ldots dq_{n}.$
* $\int \ldots \int (\Delta \eta -1)e^{\Delta \eta }\,dp_{1}\ldots dq_{n}>\int \ldots \int -\,dp_{1}\ldots dq_{n}$
| $\int \ldots \int (\Delta \eta e^{\Delta \eta }-e^{\Delta \eta }+1)\,dp_{1}\ldots dq_{n}>0.$ | (454) |
$\Delta \eta =0$
$\Delta \eta$
* ↑ See pages 37-41.
* ↑ See Theorem I, where this is proved of a similar expression.