# Elementary Principles in Statistical Mechanics/Chapter IV

CHAPTER IV.
ON THE DISTRIBUTION IN PHASE CALLED CANONICAL, IN WHICH THE INDEX OF PROBABILITY IS A LINEAR FUNCTION OF THE ENERGY.
Let us now give our attention to the statistical equilibrium of ensembles of conservation systems, especially to those cases and properties which promise to throw light on the phenomena of thermodynamics.
The condition of statistical equilibrium may be expressed in the form
| $\Sigma {\bigg{\frac {dP}{dp_{1}}}{\dot {p}}_{1}+{\frac {dP}{dq_{1}}}{\dot {q}}_{1}{\bigg )}=0,$ | (88) |
$P$
$P$
$p$
$q$
* $P=\operatorname {func.} (\epsilon )$
* $\Sigma {\Bigg{\frac {dP}{dp_{1}}}{\frac {d\epsilon }{dp_{1}}}-{\frac {dP}{dq_{1}}}{\frac {d\epsilon }{dq_{1}}}{\Bigg )}=0.$
There are, however, other conditions to which $P$ is subject, which are not so much conditions of statistical equilibrium, as conditions implicitly involved in the definition of the ​coefficient of probability, whether the case is one of equilibrium or not. These are: that $P$ should be single-valued, and neither negative nor imaginary for any phase, and that expressed by equation (46), viz.,
| $\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}P\,dp_{1}\ldots dq_{n}=1.$ | (89) |
* $P=\epsilon \times \mathrm {constant}$
* $P=\mathrm {constant}$
The distribution represented by
| $\eta =\log P={\frac {\psi -\epsilon }{\Theta }}$ | (90) |
| $P=e^{\frac {\psi -\epsilon }{\Theta }}$ | (91) |
$\Theta$
$\psi$
$\Theta$
$\Theta$
$\epsilon$
$\epsilon$
$\psi$
$\Theta$
| $e^{-{\frac {\psi }{\Theta }}}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n}$ | (92) |
When an ensemble of systems is distributed in phase in the manner described, i. e., when the index of probability is a ​linear function of the energy, we shall say that the ensemble is canonically distributed, and shall call the divisor of the energy ($\Theta$) the modulus of distribution.
The fractional part of an ensemble canonically distributed which lies within any given limits of phase is therefore represented by the multiple integral
| $\int \ldots \int e^{\frac {\psi -\epsilon }{\Theta }}dp_{1}\ldots dq_{n}$ | (93) |
Since the value of a multiple integral of the form (23) (which we have called an extension-in-phase) bounded by any given phases is independent of the system of coördinates by which it is evaluated, the same must be true of the multiple integral in (92), as appears at once if we divide up this integral into parts so small that the exponential factor may be regarded as constant in each. The value of $\psi$ is therefore independent of the system of coördinates employed.
It is evident that $\psi$ might be defined as the energy for which the coefficient of probability of phase has the value unity. Since however this coefficient has the dimensions of the inverse $n$th power of the product of energy and time, the energy represented by $\psi$ is not independent of the units of energy and time. But when these units have been chosen, the definition of $\psi$ will involve the same arbitrary constant as $\epsilon$, so that, while in any given case the numerical values of $\psi$ or $\epsilon$ will be entirely indefinite until the zero of energy has also been fixed for the system considered, the difference $\psi -\epsilon$ will represent a perfectly definite amount of energy, which is entirely independent of the zero of energy which we may choose to adopt.
​It is evident that the canonical distribution is entirely determined by the modulus (considered as a quantity of energy) and the nature of the system considered, since when equation (92) is satisfied the value of the multiple integral (93) is independent of the units and of the coördinates employed, and of the zero chosen for the energy of the system.
In treating of the canonical distribution, we shall always suppose the multiple integral in equation (92) to have a finite value, as otherwise the coefficient of probability vanishes, and the law of distribution becomes illusory. This will exclude certain cases, but not such apparently, as will affect the value of our results with respect to their bearing on thermodynamics. It will exclude, for instance, cases in which the system or parts of it can be distributed in unlimited space (or in a space which has limits, but is still infinite in volume), while the energy remains beneath a finite limit. It also excludes many cases in which the energy can decrease without limit, as when the system contains material points which attract one another inversely as the squares of their distances. Cases of material points attracting each other inversely as the distances would be excluded for some values of $\Theta$, and not for others. The investigation of such points is best left to the particular cases. For the purposes of a general discussion, it is sufficient to call attention to the assumption implicitly involved in the formula (92).
The modulus $\Theta$ has properties analogous to those of temperature in thermodynamics. Let the system $A$ be defined as one of an ensemble of systems of $m$ degrees of freedom distributed in phase with a probability-coefficient
* $e^{\frac {\psi _{A}-\epsilon _{A}}{\Theta }},$
$B$
$n$
* $e^{\frac {\psi _{B}-\epsilon _{B}}{\Theta }}$
$q_{1},\ldots q_{m}$
$p_{1},\ldots p_{m}$
$A$
$q_{m+1},\ldots q_{m+n}$
$p_{m+1},\ldots p_{m+n}$
$B$
$A$
$B$
$C$
$m+n$
$q_{1},\ldots q_{m+n}$
$p_{1},\ldots p_{m+n}$
$C$
* $dp_{1},\ldots dp_{m+n}\,dq_{1},\ldots dq_{m+n}$
$A$
$B$
| $e^{\frac {\psi _{A}+\psi _{B}-\epsilon _{A}-\epsilon _{B}}{\Theta }}dp_{1},\ldots dp_{m+n}\,dq_{1},\ldots dq_{m+n}.$ | (94) |
$C$
| $e^{\frac {\psi _{A}+\psi _{B}-\epsilon _{A}-\epsilon _{B}}{\Theta }},$ | (95) |
$\epsilon _{A}+\epsilon _{B}$
$\psi _{A}$
$\psi _{B}$
This result, however, so far as statistical equilibrium is concerned, is rather nugatory, since conceiving of separate systems as forming a single system does not create any interaction between them, and if the systems combined belong to ensembles in statistical equilibrium, to say that the ensemble formed by such combinations as we have supposed is in statistical equilibrium, is only to repeat the data in different ​words. Let us therefore suppose that in forming the system $C$ we add certain forces acting between $A$ and $B$, and having the force-function $-\epsilon _{AB}$. The energy of the system $C$ is now $\epsilon _{A}+\epsilon _{B}+\epsilon _{AB}$ and an ensemble of such systems distributed with a density proportional to
| $e^{\frac {-(\epsilon _{A}+\epsilon _{B}+\epsilon _{AB}}{\Theta }}$ | (96) |
$C$
$\epsilon _{AB}$
$A$
$B$
$C$
$A$
$B$
$\Theta _{A}$
$\Theta _{B}$
$C$
| $e^{{\frac {\psi _{A}-\epsilon _{A}}{\Theta _{A}}}+{\frac {\psi _{B}-\epsilon _{B}}{\Theta _{B}}}},$ | (97) |
Before proceeding farther in the investigation of the distribution in phase which we have called canonical, it will be interesting to see whether the properties with respect to ​statistical equilibrium which have been described are peculiar to it, or whether other distributions may have analogous properties.
Let $\eta '$ and $\eta ''$ be the indices of probability in two independent ensembles which are each in statistical equilibrium, then $\eta '+\eta ''$ will be the index in the ensemble obtained by combining each system of the first ensemble with each system of the second. This third ensemble will of course be in statistical equilibrium, and the function of phase $\eta '+\eta ''$ will be a constant of motion. Now when infinitesimal forces are added to the compound systems, if $\eta '+\eta ''$ or a function differing infinitesimally from this is still a constant of motion, it must be on account of the nature of the forces added, or if their action is not entirely specified, on account of conditions to which they are subject. Thus, in the case already considered, $\eta '+\eta ''$ is a function of the energy of the compound system, and the infinitesimal forces added are subject to the law of conservation of energy.
Another natural supposition in regard to the added forces is that they should be such as not to affect the moments of momentum of the compound system. To get a case in which moments of momentum of the compound system shall be constants of motion, we may imagine material particles contained in two concentric spherical shells, being prevented from passing the surfaces bounding the shells by repulsions acting always in lines passing through the common centre of the shells. Then, if there are no forces acting between particles in different shells, the mass of particles in each shell will have, besides its energy, the moments of momentum about three axes through the centre as constants of motion.
Now let us imagine an ensemble formed by distributing in phase the system of particles in one shell according to the index of probability
| $A-{\frac {\epsilon }{\Theta }}+{\frac {\omega _{1}}{\Omega _{1}}}+{\frac {\omega _{2}}{\Omega _{2}}}+{\frac {\omega _{3}}{\Omega _{3}}},$ | (98) |
$\epsilon$
$\omega _{1}$
$\omega _{2}$
$\omega _{3}$
| $A'-{\frac {\epsilon '}{\Theta }}+{\frac {\omega _{1}'}{\Omega _{1}}}+{\frac {\omega _{2}'}{\Omega _{2}}}+{\frac {\omega _{3}'}{\Omega _{3}}},$ | (99) |
$\Theta$
$\Omega _{1}$
$\Omega _{2}$
$\Omega _{3}$
| $A+A'-{\frac {\epsilon +\epsilon '}{\Theta }}+{\frac {\omega _{1}+\omega _{1}'}{\Omega _{1}}}+{\frac {\omega _{2}+\omega _{2}'}{\Omega _{2}}}+{\frac {\omega _{3}+\omega _{3}'}{\Omega _{3}}},$ | (100) |
Now if we add in each system of this third ensemble infinitesimal conservative forces of attraction or repulsion between particles in different shells, determined by the same law for all the systems, the functions $\omega _{1}+\omega _{1}'$, $\omega _{2}+\omega _{2}'$, and $\omega _{3}+\omega _{3}'$ will remain constants of motion, and a function differing infinitely little from $\epsilon _{1}+\epsilon '$ will be a constant of motion. It would therefore require only an infinitesimal change in the distribution in phase of the ensemble of compound systems to make it a case of statistical equilibrium. These properties are entirely analogous to those of canonical ensembles.
Again, if the relations between the forces and the coördinates can be expressed by linear equations, there will be certain "normal" types of vibration of which the actual motion may be regarded as composed, and the whole energy may be divided ​into parts relating separately to vibrations of these different types. These partial energies will be constants of motion, and if such a system is distributed according to an index which is any function of the partial energies, the ensemble will be in statistical equilibrium. Let the index be a linear function of the partial energies, say
| $A-{\frac {\epsilon _{1}}{\Theta _{1}}}\ldots -{\frac {\epsilon _{n}}{\Theta _{n}}}.$ | (101) |
| $A'-{\frac {\epsilon _{1}'}{\Theta _{1}'}}\ldots -{\frac {\epsilon _{m}'}{\Theta _{m}'}}.$ | (102) |
Since the two ensembles are both in statistical equilibrium, the ensemble formed by combining each system of the first with each system of the second will also be in statistical equilibrium. Its distribution in phase will be represented by the index
| $A+A'-{\frac {\epsilon _{1}}{\Theta _{1}}}\ldots -{\frac {\epsilon _{n}}{\Theta _{n}}}-{\frac {\epsilon _{1}'}{\Theta _{1}'}}\ldots -{\frac {\epsilon _{m}'}{\Theta _{m}'}},$ | (103) |
Now if we add to these compound systems infinitesimal forces acting between the component systems and subject to the same general law as those already existing, viz., that they are conservative and linear functions of the coördinates, there will still be $n+m$ types of normal vibration, and $n+m$ partial energies which are independent constants of motion. If all the original $n+m$ normal types of vibration have different periods, the new types of normal vibration will differ infinitesimally from the old, and the new partial energies, which are constants of motion, will be nearly the same functions of phase as the old. Therefore the distribution in phase of the ​ensemble of compound systems after the addition of the supposed infinitesimal forces will differ infinitesimally from one which would be in statistical equilibrium.
The case is not so simple when some of the normal types of motion have the same periods. In this case the addition of infinitesimal forces may completely change the normal types of motion. But the sum of the partial energies for all the original types of vibration which have any same period, will be nearly identical (as a function of phase, i. e., of the coördinates and momenta,) with the sum of the partial energies for the normal types of vibration which have the same, or nearly the same, period after the addition of the new forces. If, therefore, the partial energies in the indices of the first two ensembles (101) and (102) which relate to types of vibration having the same periods, have the same divisors, the same will be true of the index (103) of the ensemble of compound systems, and the distribution represented will differ infinitesimally from one which would be in statistical equilibrium after the addition of the new forces.
The same would be true if in the indices of each of the original ensembles we should substitute for the term or terms relating to any period which does not occur in the other ensemble, any function of the total energy related to that period, subject only to the general limitation expressed by equation (89). But in order that the ensemble of compound systems (with the added forces) shall always be approximately in statistical equilibrium, it is necessary that the indices of the original ensembles should be linear functions of those partial energies which relate to vibrations of periods common to the two ensembles, and that the coefficients of such partial energies should be the same in the two indices.
​The properties of canonically distributed ensembles of systems with respect to the equilibrium of the new ensembles which may be formed by combining each system of one ensemble with each system of another, are therefore not peculiar to them in the sense that analogous properties do not belong to some other distributions under special limitations in regard to the systems and forces considered. Yet the canonical distribution evidently constitutes the most simple case of the kind, and that for which the relations described hold with the least restrictions.
Returning to the case of the canonical distribution, we shall find other analogies with thermodynamic systems, if we suppose, as in the preceding chapters, that the potential energy ($\epsilon _{q}$) depends not only upon the coördinates $q_{1}\ldots q_{n}$ which determine the configuration of the system, but also upon certain coördinates $a_{1}$, $a_{2}$, etc. of bodies which we call external meaning by this simply that they are not to be regarded as forming any part of the system, although their positions affect the forces which act on the system. The forces exerted by the system upon these external bodies will be represented by $-d\epsilon _{q}/da_{1}$, $-d\epsilon _{q}/da_{2}$, etc., while $-d\epsilon _{q}/dq_{1},\ldots -d\epsilon _{q}/dq_{n}$ represent all the forces acting upon the bodies of the system, including those which depend upon the position of the external bodies, as well as those which depend only upon the configuration of the system itself. It will be understood that $e_{p}$ depends only upon $q_{1},\ldots q_{n}$, $p_{1},\ldots p_{n}$, in other words, that the kinetic energy of the bodies which we call external forms no part of the kinetic energy of the system. It follows that we may write
| ${\frac {d\epsilon }{da_{1}}}={\frac {d\epsilon _{q}}{da_{1}}}=-A_{1},$ | (104) |
​We always suppose these external coördinates to have the same values for all systems of any ensemble. In the case of a canonical distribution, i. e., when the index of probability of phase is a linear function of the energy, it is evident that the values of the external coördinates will affect the distribution, since they affect the energy. In the equation
| $e^{-{\frac {\psi }{\Theta }}}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n}$ | (105) |
$\psi$
$a_{1}$
$a_{2}$
$\epsilon$
$\Theta$
$\psi$
| ${\begin{aligned}e^{-{\frac {\psi }{\Theta }}}{\Bigg-&{\frac {1}{\Theta }}d\psi +{\frac {\psi }{\Theta ^{2}}}d\Theta {\Bigg )}={\frac {1}{\Theta ^{2}}}d\Theta \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\epsilon e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n}\\&-{\frac {1}{\Theta }}da_{1}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\frac {d\epsilon }{da_{1}}}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n}\\&-{\frac {1}{\Theta }}da_{2}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\frac {d\epsilon }{da_{2}}}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n}-\mathrm {etc.} ,\end{aligned}}$ | (106) |
$\Theta e^{\frac {\psi }{\Theta }}$
* $-{\frac {d\epsilon }{da_{1}}}=A_{1},\quad -{\frac {d\epsilon }{da_{2}}}=A_{2},\mathrm {etc.} ,$
| ${\begin{aligned}-d\psi +{\frac {\psi }{\Theta }}d\Theta &={\frac {1}{\Theta }}d\Theta \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\epsilon e^{\frac {\phi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}\\&+da_{1}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}A_{1}e^{\frac {\phi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}\\&+da_{2}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}A_{2}e^{\frac {\phi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}+\mathrm {etc.} \end{aligned}}$ | (107) |
| ${\overline {u}}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}ue^{\frac {\phi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}.$ | (108) |
| $d\psi ={\frac {\psi }{\Theta }}d\Theta -{\frac {\overline {\epsilon }}{\Theta }}d\Theta -{\overline {A}}_{1}da_{1}-{\overline {A}}_{2}da_{2}-\mathrm {etc.}$ | (109) |
| ${\frac {\psi -\epsilon }{\Theta }}=\eta ,$ | (110) |
| ${\frac {\psi -{\overline {\epsilon }}}{\Theta }}={\overline {\eta }},$ | (111) |
| $d\psi ={\overline {\eta }}d\Theta -{\overline {A}}_{1}da_{1}-{\overline {A}}_{2}da_{2}-\mathrm {etc.}$ | (112) |
| $d\psi -d{\overline {\epsilon }}=\Theta d{\overline {\eta }}+{\overline {\eta }}d\Theta ,$ | (113) |
| $d{\overline {\epsilon }}=-\Theta d{\overline {\eta }}-{\overline {A}}_{1}da_{1}-{\overline {A}}_{2}da_{2}-\mathrm {etc.}$ | (114) |
This equation, if we neglect the sign of averages, is identical in form with the thermodynamic equation
| $d\eta ={\frac {d\epsilon +A_{1}da_{1}+A_{2}da_{2}+\mathrm {etc.} }{T}},$ | (115) |
| $d\epsilon =Td\eta -A_{1}da_{1}-A_{2}da_{2}-\mathrm {etc.} ,$ | (116) |
$\eta$
${\overline {\eta }}$
We may also compare equation (112) with the thermodynamic equation
| $d\psi =-\eta dT-A_{1}da_{1}-A_{2}da_{2}-\mathrm {etc.} ,$ | (117) |
$\psi$
How far, or in what sense, the similarity of these equations constitutes any demonstration of the thermodynamic equations, or accounts for the behavior of material systems, as described in the theorems of thermodynamics, is a question of which we shall postpone the consideration until we have further investigated the properties of an ensemble of systems distributed in phase according to the law which we are considering. The analogies which have been pointed out will at least supply the motive for this investigation, which will naturally commence with the determination of the average values in the ensemble of the most important quantities relating to the systems, and to the distribution of the ensemble with respect to the different values of these quantities.
* ↑ See equations (20), (41), (42), also the paragraph following equation (20). The positions of any external bodies which can affect the systems are here supposed uniform for all the systems and constant in time.
* ↑ See Chapter I, p. 19.
* ↑ It will be observed that similar limitations exist in thermodynamics. In order that a mass of gas can be in thermodynamic equilibrium, it is necessary that it be enclosed. There is no thermodynamic equilibrium of a (finite) mass of gas in an infinite space. Again, that two attracting particles should be able to do an infinite amount of work in passing from one configuration (which is regarded as possible) to another, is a notion which, although perfectly intelligible in a mathematical formula, is quite foreign to our ordinary conceptions of matter.
* ↑ It will be observed that the above condition relating to the forces which act between the different systems is entirely analogous to that which must hold in the corresponding case in thermodynamics. The most simple test of the equality of temperature of two bodies is that they remain in equilibrium when brought into thermal contact. Direct thermal contact implies molecular forces acting between the bodies. Now the test will fail unless the energy of these forces can be neglected in comparison with the other energies of the bodies. Thus, in the case of energetic chemical action between the bodies, or when the number of particles affected by the forces acting between the bodies is not negligible in comparison with the whole number of particles (as when the bodies have the form of exceedingly thin sheets), the contact of bodies of the same temperature may produce considerable thermal disturbance, and thus fail to afford a reliable criterion of the equality of temperature.
* ↑ It would not be possible to omit the term relating to energy in the above indices, since without this term the condition expressed by equation (89) cannot be satisfied. The consideration of the above case of statistical equilibrium may be made the foundation of the theory of the thermodynamic equilibrium of rotating bodies,—a subject which has been treated by Maxwell in his memoir "On Boltzmann's theorem on the average distribution of energy in a system of material points." Cambr. Phil. Trans., vol. XII, p. 547, (1878).
* ↑ It is interesting to compare the above relations with the laws respecting the exchange of energy between bodies by radiation, although the phenomena of radiations lie entirely without the scope of the present treatise, in which the discussion is limited to systems of a finite number of degrees of freedom.
* ↑ The above may perhaps be sufficiently illustrated by the simple case where $n=1$ in each system. If the periods are different in the two systems, they may be distributed according to any functions of the energies: but if the periods are the same they must be distributed canonically with same modulus in order that the compound ensemble with additional forces may be in statistical equilibrium.
* ↑ See especially Chapter I, p. 4.