Elementary Principles in Statistical Mechanics/Chapter XIII

by Josiah Willard Gibbs

CHAPTER XIII.

EFFECT OF VARIOUS PROCESSES ON AN ENSEMBLE OF SYSTEMS.

In the last chapter and in Chapter I we have considered the changes which take place in the course of time in an ensemble of isolated systems. Let us now proceed to consider the changes which will take place in an ensemble of systems under external influences. These external influences will be of two kinds, the variation of the coördinates which we have called external, and the action of other ensembles of systems. The essential difference of the two kinds of influence consists in this, that the bodies to which the external coördinates relate are not distributed in phase, while in the case of interaction of the systems of two ensembles, we have to regard the fact that both are distributed in phase. To find the effect produced on the ensemble with which we are principally concerned, we have therefore to consider single values of what we have called external coördinates, but an infinity of values of the internal coördinates of any other ensemble with which there is interaction.

Or,—to regard the subject from another point of view,—the action between an unspecified system of an ensemble and the bodies represented by the external coördinates, is the action between a system imperfectly determined with respect to phase and one which is perfectly determined; while the interaction between two unspecified systems belonging to different ensembles is the action between two systems both of which are imperfectly determined with respect to phase.^[1]

We shall suppose the ensembles which we consider to be distributed in phase in the manner described in Chapter I, and represented by the notations of that chapter, especially by the index of probability of phase ($\eta$). There are therefore $2n$ independent variations in the phases which constitute the ensembles considered. This excludes ensembles like the microcanonical, in which, as energy is constant, there are only $2n-1$ independent variations of phase. This seems necessary for the purposes of a general discussion. For although we may imagine a microcanonical ensemble to have a permanent existence when isolated from external influences, the effect of such influences would generally be to destroy the uniformity of energy in the ensemble. Moreover, since the microcanonical ensemble may be regarded as a limiting case of such ensembles as are described in Chapter I, (and that in more than one way, as shown in Chapter X,) the exclusion is rather formal than real, since any properties which belong to the microcanonical ensemble could easily be derived from those of the ensembles of Chapter I, which in a certain sense may be regarded as representing the general case.

Let us first consider the effect of variation of the external coördinates. We have already had occasion to regard these quantities as variable in the differentiation of certain equations relating to ensembles distributed according to certain laws called canonical or microcanonical. That variation of the external coördinates was, however, only carrying the attention of the mind from an ensemble with certain values of the external coördinates, and distributed in phase according to some general law depending upon those values, to another ensemble with different values of the external coördinates, and with the distribution changed to conform to these new values.

What we have now to consider is the effect which would actually result in the course of time in an ensemble of systems in which the external coördinates should be varied in any arbitrary manner. Let us suppose, in the first place, that these coördinates are varied abruptly at a given instant, being constant both before and after that instant. By the definition of the external coördinates it appears that this variation does not affect the phase of any system of the ensemble at the time when it takes place. Therefore it does not affect the index of probability of phase ($\eta$) of any system, or the average value of the index (${\overline {\eta }}$) at that time. And if these quantities are constant in time before the variation of the external coördinates, and after that variation, their constancy is time is not interrupted by that variation. In fact, in the demonstration of the conservation of probability of phase in Chapter I, the variation of the external coördinates was not excluded.

But a variation of the external coördinates will in general disturb a previously existing state of statistical equilibrium. For, although it does not affect (at the first instant) the distribution-in-phase, it does affect the condition necessary for equilibrium. This condition, as we have seen in Chapter IV, is that the index of probability of phase shall be a function of phase which is constant in time for moving systems. Now a change in the external coördinates, by changing the forces which act on the systems, will change the nature of the functions of phase which are constant in time. Therefore, the distribution in phase which was one of statistical equilibrium for the old values of the external coördinates, will not be such for the new values.

Now we have seen, in the last chapter, that when the distribution-in-phase is not one of statistical equilibrium, an ensemble of systems may, and in general will, after a longer or shorter time, come to a state which may be regarded, if very small differences of phase are neglected, as one of statistical equilibrium, and in which consequently the average value of the index (${\overline {\eta }}$) is less than at first. It is evident, therefore, that a variation of the external coördinates, by disturbing a state of statistical equilibrium, may indirectly cause a diminution, (in a certain sense at least,) of the value of ${\overline {\eta }}$.

But if the change in the external coördinates is very small, the change in the distribution necessary for equilibrium will in general be correspondingly small. Hence, the original distribution in phase, since it differs little from one which would be in statistical equilibrium with the new values of the external coördinates, may be supposed to have a value of ${\overline {\eta }}$ which differs by a small quantity of the second order from the minimum value which characterizes the state of statistical equilibrium. And the diminution in the average index resulting in the course of time from the very small change in the external coördinates, cannot exceed this small quantity of the second order.

Hence also, if the change in the external coördinates of an ensemble initially in statistical equilibrium consists in successive very small changes separated by very long intervals of time in which the disturbance of statistical equilibrium becomes sensibly effaced, the final diminution in the average index of probability will in general be negligible, although the total change in the external coördinates is large. The result will be the same if the change in the external coördinates takes place continuously but sufficiently slowly.

Even in cases in which there is no tendency toward the restoration of statistical equilibrium in the lapse of time, a variation of external coördinates which would cause, if it took place in a short time, a great disturbance of a previous state of equilibrium, may, if sufficiently distributed in time, produce no sensible disturbance of the statistical equilibrium.

Thus, in the case of three degrees of freedom, let the systems be heavy points suspended by elastic massless cords, and let the ensemble be distributed in phase with a density proportioned to some function of the energy, and therefore in statistical equilibrium. For a change in the external coördinates, we may take a horizontal motion of the point of suspension. If this is moved a given distance, the resulting disturbance of the statistical equilibrium may evidently be diminished indefinitely by diminishing the velocity of the point of suspension. This will be true if the law of elasticity of the string is such that the period of vibration is independent of the energy, in which case there is no tendency in the course of time toward a state of statistical equilibrium, as well as in the more general case, in which there is a tendency toward statistical equilibrium.

That something of this kind will be true in general, the following considerations will tend to show.

We define a path as the series of phases through which a system passes in the course of time when the external coördinates have fixed values. When the external coördinates are varied, paths are changed. The path of a phase is the path to which that phase belongs. With reference to any ensemble of systems we shall denote by ${\overline {D}}|_{p}$ the average value of the density-in-phase in a path. This implies that we have a measure for comparing different portions of the path. We shall suppose the time required to traverse any portion of a path to be its measure for the purpose of determining this average.

With this understanding, let us suppose that a certain ensemble is in statistical equilibrium. In every element of extension-in-phase, therefore, the density-in-phase $D$ is equal to its path-average ${\overline {D}}|_{p}$. Let a sudden small change be made in the external coördinates. The statistical equilibrium will be disturbed and we shall no longer have $D={\overline {D}}|_{p}$ everywhere. This is not because $D$ is changed, but because ${\overline {D}}|_{p}$ is changed, the paths being changed. It is evident that if $D>{\overline {D}}|_{p}$ in a part of a path, we shall have $D<{\overline {D}}|_{p}$ in other parts of the same path.

Now, if we should imagine a further change in the external coördinates of the same kind, we should expect it to produce an effect of the same kind. But the manner in which the second effect will be superposed on the first will be different, according as it occurs immediately after the first change or after an interval of time. If it occurs immediately after the first change, then in any element of phase in which the first change produced a positive value of $D-{\overline {D}}|_{p}$ the second change will add a positive value to the first positive value, and where $D-{\overline {D}}|_{p}$ was negative, the second change will add a negative value to the first negative value.

But if we wait a sufficient time before making the second change in the external coördinates, so that systems have passed from elements of phase in which $D-{\overline {D}}|_{p}$ was originally positive to elements in which it was originally negative, and vice versa, (the systems carrying with them the values of $D-{\overline {D}}|_{p}$) the positive values of $D-{\overline {D}}|_{p}$ caused by the second change will be in part superposed on negative values due to the first change, and vice versa.

The disturbance of statistical equilibrium, therefore, produced by a given change in the values of the external coördinates may be very much diminished by dividing the change into two parts separated by a sufficient interval of time, and a sufficient interval of time for this purpose is one in which the phases of the individual systems are entirely unlike the first, so that any individual system is differently affected by the change, although the whole ensemble is affected in nearly the same way. Since there is no limit to the diminution of the disturbance of equilibrium by division of the change in the external coördinates, we may suppose as a general rule that by diminishing the velocity of the changes in the external coördinates, a given change may be made to produce a very small disturbance of statistical equilibrium.

If we write ${\overline {\eta }}'$ for the value of the average index of probability before the variation of the external coördinates, and ${\overline {\eta }}''$ for the value after this variation, we shall have in any case

${\overline {\eta }}''\leq {\overline {\eta }}'$

as the simple result of the variation of the external coördinates. This may be compared with the thermodynamic theorem that the entropy of a body cannot be diminished by mechanical (as distinguished from thermal) action.^[2]

If we have (approximate) statistical equilibrium between the times $t'$ and $t''$ (corresponding to ${\overline {\eta }}'$ and ${\overline {\eta }}''$), we shall have approximately

${\overline {\eta }}'={\overline {\eta }}''$

which may be compared with the thermodynamic theorem that the entropy of a body is not (sensibly) affected by mechanical action, during which the body is at each instant (sensibly) in a state of thermodynamic equilibrium.

Approximate statistical equilibrium may usually be attained by a sufficiently slow variation of the external coördinates, just as approximate thermodynamic equilibrium may usually be attained by sufficient slowness in the mechanical operations to which the body is subject.

We now pass to the consideration of the effect on an ensemble of systems which is produced by the action of other ensembles with which it is brought into dynamical connection. In a previous chapter^[3] we have imagined a dynamical connection arbitrarily created between the systems of two ensembles. We shall now regard the action between the systems of the two ensembles as a result of the variation of the external coördinates, which causes such variations of the internal coördinates as to bring the systems of the two ensembles within the range of each other's action.

Initially, we suppose that we have two separate ensembles of systems, $E_{1}$ and $E_{2}$. The numbers of degrees of freedom of the systems in the two ensembles will be denoted by $n_{1}$ and $n_{2}$ respectively, and the probability-coefficients by $e^{\eta _{1}}$ and $e^{\eta _{2}}$. Now we may regard any system of the first ensemble combined with any system of the second as forming a single system of $n_{1}+n_{2}$ degrees of freedom. Let us consider the ensemble ($E_{12}$) obtained by thus combining each system of the first ensemble with each of the second.

At the initial moment, which may be specified by a single accent, the probability-coefficient of any phase of the combined systems is evidently the product of the probability-coefficients of the phases of which it is made up. This may be expressed by the equation,

$e^{\eta _{12}'}=e^{\eta _{1}'}e^{\eta _{2}'},$

(455)

or

$\eta _{12}'=\eta _{1}'+\eta _{2}',$

(456)

which gives

${\overline {\eta }}_{12}'={\overline {\eta }}_{1}'+{\overline {\eta }}_{2}'.$

(457)

The forces tending to vary the internal coördinates of the combined systems, together with those exerted by either system upon the bodies represented by the coördinates called external, may be derived from a single force-function, which, taken negatively, we shall call the potential energy of the combined systems and denote by $\epsilon _{12}$. But we suppose that initially none of the systems of the two ensembles $E_{1}$ and $E_{2}$ come within range of each other's action, so that the potential energy of the combined system falls into two parts relating separately to the systems which are combined. The same is obviously true of the kinetic energy of the combined compound system, and therefore of its total energy. This may be expressed by the equation

$\epsilon _{12}'=\epsilon _{1}'+\epsilon _{2}',$

(458)

which gives

${\overline {\epsilon }}_{12}'={\overline {\epsilon }}_{1}'+{\overline {\epsilon }}_{2}'.$

(459)

Let us now suppose that in the course of time, owing to the motion of the bodies represented by the coördinates called external, the forces acting on the systems and consequently their positions are so altered, that the systems of the ensembles $E_{1}$ and $E_{2}$ are brought within range of each other's action, and after such mutual influence has lasted for a time, by a further change in the external coördinates, perhaps a return to their original values, the systems of the two original ensembles are brought again out of range of each other's action. Finally, then, at a time specified by double accents, we shall have as at first

${\overline {\epsilon }}_{12}''={\overline {\epsilon }}_{1}''+{\overline {\epsilon }}_{2}''.$

(460)

But for the indices of probability we must write^[4]

${\overline {\eta }}_{1}''+{\overline {\eta }}_{2}''\leqq {\overline {\eta }}_{12}''.$

(461)

The considerations adduced in the last chapter show that it is safe to write

${\overline {\eta }}_{12}''\leqq {\overline {\eta }}_{12}'.$

(462)

We have therefore

${\overline {\eta }}_{1}''+{\overline {\eta }}_{2}''\leqq {\overline {\eta }}_{1}'+{\overline {\eta }}_{2}',$

(463)

which may be compared with the thermodynamic theorem that the thermal contact of two bodies may increase but cannot diminish the sum of their entropies.

Let us especially consider the case in which the two original ensembles were both canonically distributed in phase with the respective moduli $\Theta _{1}$ and $\Theta _{2}$. We have then, by Theorem III of Chapter XI,

${\overline {\eta }}_{1}'+{\frac {{\overline {\epsilon }}_{1}'}{\Theta _{1}}}\leq {\overline {\eta }}_{1}''+{\frac {{\overline {\epsilon }}_{1}''}{\Theta _{1}}}$

(464)

${\overline {\eta }}_{2}'+{\frac {{\overline {\epsilon }}_{2}'}{\Theta _{2}}}\leq {\overline {\eta }}_{2}''+{\frac {{\overline {\epsilon }}_{2}''}{\Theta _{2}}}$

(465)

Whence with (463) we have

${\frac {{\overline {\epsilon }}_{1}'}{\Theta _{1}}}+{\frac {{\overline {\epsilon }}_{2}'}{\Theta _{2}}}\leq {\frac {\epsilon _{1}''}{\Theta _{1}}}+{\frac {{\overline {\epsilon }}_{2}''}{\Theta _{2}}}$

(466)

or

${\frac {{\overline {\epsilon }}_{1}''-{\overline {\epsilon }}_{1}'}{\Theta _{1}}}+{\frac {{\overline {\epsilon }}_{2}''-{\overline {\epsilon }}_{2}'}{\Theta _{2}}}\geq 0.$

(467)

If we write ${\overline {W}}$ for the average work done by the combined systems on the external bodies, we have by the principle of the conservation of energy

${\overline {W}}={\overline {\epsilon }}_{12}'-{\overline {\epsilon }}_{12}''={\overline {\epsilon }}_{1}'-{\overline {\epsilon }}_{1}''+{\overline {\epsilon }}_{2}'-{\overline {\epsilon }}_{2}''.$

(468)

Now if ${\overline {W}}$ is negligible, we have

${\overline {\epsilon }}_{1}''-{\overline {\epsilon }}_{1}'=-({\overline {\epsilon }}_{2}''-{\overline {\epsilon }}_{2}')$

(469)

and (467) shows that the ensemble which has the greater modulus must lose energy. This result may be compared to the thermodynamic principle, that when two bodies of different temperatures are brought together, that which has the higher temperature will lose energy.

Let us next suppose that the ensemble $E_{2}$ is originally canonically distributed with the modulus $\Theta _{2}$, but leave the distribution of the other arbitrary. We have, to determine the result of a similar process,

${\overline {\eta }}_{1}''+{\overline {\eta }}_{2}''\leq {\overline {\eta }}_{1}'+{\overline {\eta }}_{2}'$

${\overline {\eta }}_{2}'+{\frac {{\overline {\epsilon }}_{2}'}{\Theta _{2}}}\leq {\overline {\eta }}_{2}''+{\frac {{\overline {\epsilon }}_{2}''}{\Theta _{2}}}$

Hence

${\overline {\eta }}_{1}''+{\frac {{\overline {\epsilon }}_{2}'}{\Theta _{2}}}\leq {\overline {\eta }}_{1}'+{\frac {{\overline {\epsilon }}_{2}''}{\Theta _{2}}}$

(470)

which may be written

${\overline {\eta }}_{1}'-{\overline {\eta }}_{1}''\geq {\frac {{\overline {\epsilon }}_{2}'-{\overline {\epsilon }}_{2}''}{\Theta _{2}}}$

(471)

This may be compared with the thennodynamic principle that when a body (which need not be in thermal equilibrium) is brought into thermal contact with another of a given temperature, the increase of entropy of the first cannot be less (algebraically) than the loss of heat by the second divided by its temperature. Where ${\overline {W}}$ is negligible, we may write

${\overline {\eta }}_{1}''+{\frac {{\overline {\epsilon }}_{1}''}{\Theta _{2}}}\leq {\overline {\eta }}_{1}'+{\frac {{\overline {\epsilon }}_{1}'}{\Theta _{2}}}$

(472)

Now, by Theorem III of Chapter XI, the quantity

${\overline {\eta }}_{1}+{\frac {{\overline {\epsilon }}_{1}}{\Theta _{2}}}$

(473)

has a minimum value when the ensemble to which ${\overline {\eta }}_{1}$ and ${\overline {\epsilon }}_{1}$ relate is distributed canonically with the modulus $\Theta _{2}$. If the ensemble had originally this distribution, the sign $<$ in (472) would be impossible. In fact, in this case, it would be easy to show that the preceding formulae on which (472) is founded would all have the sign $=$. But when the two ensembles are not both originally distributed canonically with the same modulus, the formulae indicate that the quantity (473) may be diminished by bringing the ensemble to which $\epsilon _{1}$ and $\eta _{1}$ relate into connection with another which is canonically distributed with modulus $\Theta _{2}$, and therefore, that by repeated operations of this kind the ensemble of which the original distribution was entirely arbitrary might be brought approximately into a state of canonical distribution with the modulus $\Theta _{2}$. We may compare this with the thermodynamic principle that a body of which the original thermal state may be entirely arbitrary, may be brought approximately into a state of thermal equilibrium with any given temperature by repeated connections with other bodies of that temperature.

Let us now suppose that we have a certain number of ensembles, $E_{0}$, $E_{1}$, $E_{2}$, etc., distributed canonically with the respective moduli $\Theta _{0}$, $\Theta _{1}$, $\Theta _{2}$, etc. By variation of the external coördinates of the ensemble $E_{0}$, let it be brought into connection with $E_{1}$, and then let the connection be broken. Let it then be brought into connection with $E_{2}$, and then let that connection be broken. Let this process be continued with respect to the remaining ensembles. We do not make the assumption, as in some cases before, that the work connected with the variation of the external coördinates is a negligible quantity. On the contrary, we wish especially to consider the case in which it is large. In the final state of the ensemble $E_{0}$, let us suppose that the external coördinates have been brought back to their original values, and that the average energy (${\overline {\epsilon }}_{0}$) is the same as at first.

In our usual notations, using one and two accents to distinguish original and final values, we get by repeated applications of the principle expressed in (463)

${\overline {\eta }}_{0}'+{\overline {\eta }}_{1}'+{\overline {\eta }}_{2}'+\mathrm {etc.} \geq {\overline {\eta }}_{0}''+{\overline {\eta }}_{1}''+{\overline {\eta }}_{2}''+\mathrm {etc.}$

(474)

But by Theorem III of Chapter XI,

${\overline {\eta }}_{0}''+{\frac {{\overline {\epsilon }}_{0}''}{\Theta _{0}}}\geq {\overline {\eta }}_{0}'+{\frac {{\overline {\epsilon }}_{0}'}{\Theta _{0}}},$

(475)

${\overline {\eta }}_{1}''+{\frac {{\overline {\epsilon }}_{1}''}{\Theta _{1}}}\geq {\overline {\eta }}_{1}'+{\frac {{\overline {\epsilon }}_{1}'}{\Theta _{1}}},$

(476)

${\overline {\eta }}_{2}''+{\frac {{\overline {\epsilon }}_{2}''}{\Theta _{2}}}\geq {\overline {\eta }}_{2}'+{\frac {{\overline {\epsilon }}_{2}'}{\Theta _{2}}},$

(477)

$\mathrm {etc.}$

Hence

${\frac {{\overline {\epsilon }}_{0}''}{\Theta _{0}}}+{\frac {{\overline {\epsilon }}_{1}''}{\Theta _{1}}}+{\frac {{\overline {\epsilon }}_{2}''}{\Theta _{2}}}+\mathrm {etc.} \geq {\frac {{\overline {\epsilon }}_{0}'}{\Theta _{0}}}+{\frac {{\overline {\epsilon }}_{1}'}{\Theta _{1}}}+{\frac {{\overline {\epsilon }}_{2}'}{\Theta _{2}}}+\mathrm {etc.}$

(478)

or, since

${\overline {\epsilon }}_{0}'={\overline {\epsilon }}_{0}'',$

$0\geq {\frac {{\overline {\epsilon }}_{1}'-{\overline {\epsilon }}_{1}''}{\Theta _{1}}}+{\frac {{\overline {\epsilon }}_{2}'-{\overline {\epsilon }}_{2}''}{\Theta _{2}}}+\mathrm {etc.}$

(479)

If we write ${\overline {W}}$ for the average work done on the bodies represented by the external coördinates, we have

${\overline {\epsilon }}_{1}'-{\overline {\epsilon }}_{1}''+{\overline {\epsilon }}_{2}'-{\overline {\epsilon }}_{2}''+\mathrm {etc.} ={\overline {W}}.$

(480)

If $E_{0}$, $E_{1}$, and $E_{2}$ are the only ensembles, we have

$W<{\frac {\Theta _{1}-\Theta _{2}}{\Theta _{1}}}({\overline {\epsilon }}_{1}'-{\overline {\epsilon }}_{1}'').$

(481)

It will be observed that the relations expressed in the last three formulae between ${\overline {W}}$, ${\overline {\epsilon }}_{1}'-{\overline {\epsilon }}_{1}''$, ${\overline {\epsilon }}_{2}'-{\overline {\epsilon }}_{2}''$, etc., and $\Theta _{1}$, $\Theta _{2}$, etc. are precisely those which hold in a Carnot's cycle for the work obtained, the energy lost by the several bodies which serve as heaters or coolers, and their initial temperatures.

It will not escape the reader's notice, that while from one point of view the operations which are here described are quite beyond our powers of actual performance, on account of the impossibility of handling the immense number of systems which are involved, yet from another point of view the operations described are the most simple and accurate means of representing what actually takes place in our simplest experiments in thermodynamics. The states of the bodies which we handle are certainly not known to us exactly. What we know about a body can generally be described most accurately and most simply by saying that it is one taken at random from a great number (ensemble) of bodies which are completely described. If we bring it into connection with another body concerning which we have a similar limited knowledge, the state of the two bodies is properly described as that of a pair of bodies taken from a great number (ensemble) of pairs which are formed by combining each body of the first ensemble with each of the second.

Again, when we bring one body into thermal contact with another, for example, in a Carnot's cycle, when we bring a mass of fluid into thermal contact with some other body from which we wish it to receive heat, we may do it by moving the vessel containing the fluid. This motion is mathematically expressed by the variation of the coördinates which determine the position of the vessel. We allow ourselves for the purposes of a theoretical discussion to suppose that the walls of this vessel are incapable of absorbing heat from the fluid. Yet while we exclude the kind of action which we call thermal between the fluid and the containing vessel, we allow the kind which we call work in the narrower sense, which takes place when the volume of the fluid is changed by the motion of a piston. This agrees with what we have supposed in regard to the external coördinates, which we may vary in any arbitrary manner, and are in this entirely unlike the coördinates of the second ensemble with which we bring the first into connection.

When heat passes in any thermodynamic experiment between the fluid principally considered and some other body, it is actually absorbed and given out by the walls of the vessel, which will retain a varying quantity. This is, however, a disturbing circumstance, which we suppose in some way made negligible, and actually neglect in a theoretical discussion. In our case, we suppose the walls incapable of absorbing energy, except through the motion of the external coördinates, but that they allow the systems which they contain to act directly on one another. Properties of this kind are mathematically expressed by supposing that in the vicinity of a certain surface, the position of which is determined by certain (external) coördinates, particles belonging to the system in question experience a repulsion from the surface increasing so rapidly with nearness to the surface that an infinite expenditure of energy would be required to carry them through it. It is evident that two systems might be separated by a surface or surfaces exerting the proper forces, and yet approach each other closely enough to exert mechanical action on each other.

---

1. In the development of the subject, we shall find that this distinction corresponds to the distinction in thermodynamics between mechanical and thermal action.
2. The correspondences to which the reader's attention is called are between $-\eta$ and entropy, and between $\Theta$ and temperature.
3. See Chapter IV, page 37.
4. See Chapter XI, Theorem VII.