Elementary Principles in Statistical Mechanics

ELEMENTARY PRINCIPLES IN STATISTICAL MECHANICS

CHAPTER I.

GENERAL NOTIONS. THE PRINCIPLE OF CONSERVATION OF EXTENSION-IN-PHASE.

WE shall use Hamilton’s form of the equations of motion for a system of n degrees of freedom, writing q1, … qn for the (generalized) coördinates, for the (generalized) velocities, and

for the moment of the forces. We shall call the quantities F1, … Fn the (generalized) forces, and the quantities p1 … pn, defined by the equations

where ∊p denotes the kinetic energy of the system, the (generalized) momenta. The kinetic energy is here regarded as a function of the velocities and coördinates. We shall usually regard it as a function of the momenta and coördinates,* and on this account we denote it by ∊p. This will not prevent us from occasionally using formulae like (2), where it is sufficiently evident the kinetic energy is regarded as function of the and q’s. But in expressions like d∊p/dq1, where the denominator does not determine the question, the kinetic energy is always to be treated in the differentiation as function of the p’s and q’s.

We have then

These equations will hold for any forces whatever. If the forces are conservative, in other words, if the expression (1) is an exact differential, we may set

where ∊q is a function of the coördinates which we shall call the potential energy of the system. If we write ∊ for the total energy, we shall have

and equations (3) may be written

The potential energy (∊q) may depend on other variables beside the coördinates q1… qn. We shall often suppose it to depend in part on coördinates of external bodies, which we shall denote by a1, a2, etc. We shall then have for the complete value of the differential of the potential energy *

where A1, A2, etc., represent forces (in the generalized sense) exerted by the system on external bodies. For the total energy (∊) we shall have

It will be observed that the kinetic energy (∊p) in the most general case is a quadratic function of the p’s (or ’s) involving also the q’s but not the a’s ; that the potential energy, when it exists, is function of the q’s and a’s ; and that the total energy, when it exists, is function of the p’s (or ’s), the q’s, and the a’s. In expressions like d∊/dq1 the p’s, and not the ’s, are to be taken as independent variables, as has already been stated with respect to the kinetic energy.

Let us imagine a great number of independent systems, identical in nature, but differing in phase, that is, in their condition with respect to configuration and velocity. The forces are supposed to be determined for every system by the same law, being functions of the coördinates of the system q1, … qn, either alone or with the coördinates a1, a2, etc. of certain external bodies. It is not necessary that they should be derivable from a force-function. The external coördinates a1, a2, etc. may vary with the time, but at any given time have fixed values. In this they differ from the internal coördinates q1, … qn, which at the same time have different values in the different systems considered.

Let us especially consider the number of systems which at a given instant fall within specified limits of phase, viz., those for which

the accented letters denoting constants. We shall suppose the differences , etc. to be infinitesimal, and that the systems are distributed in phase in some continuous manner,* so that the number having phases within the limits specified may be represented by

or more briefly by

where D is a function of the p’s and q’s and in general of t also, for as time goes on, and the individual systems change their phases, the distribution of the ensemble in phase will in general vary. In special cases, the distribution in phase will remain unchanged. These are cases of statistical equilibrium.

If we regard all possible phases as forming a sort of extension of 2n dimensions, we may regard the product of differentials in (11) as expressing an element of this extension, and D as expressing the density of the systems in that element. We shall call the product

an element of extension-in-phase, and D the density-in-phase of the systems.

It is evident that the changes which take place in the density of the systems in any given element of extension-in-phase will depend on the dynamical nature of the systems and their distribution in phase at the time considered.

In the case of conservative systems, with which we shall be principally concerned, their dynamical nature is completely determined by the function which expresses the energy (∊) in terms of the p’s, q’s, and a’s (a function supposed identical for all the systems) ; in the more general case which we are considering, the dynamical nature of the systems is determined by the functions which express the kinetic energy (∊p) in terms of the p’s and q’s, and the forces in terms of the p’s and q’s. The distribution in phase is expressed for the time considered by D as function of the p’s and q’s. To find the value of dD/dt for the specified element of extension-in-phase, we observe that the number of systems within the limits can only be varied by systems passing the limits, which may take place in 4n different ways, viz., by the p1 of a system passing the limit or the limit or by the q1 of a system passing the limit , or the limit etc. Let us consider these cases separately.

In the first place, let us consider the number of systems which in the time dt pass into or out of the specified element by p1 passing the limit . It will be convenient, and it is evidently allowable, to suppose dt so small that the quantities etc., which represent the increments of p1, q1, etc., in the time dt shall be infinitely small in comparison with the infinitesimal differences etc., which determine the magnitude of the element of extension-in-phase. The systems for which p1 passes the limit in the interval dt are those for which at the commencement of this interval the value of p1 lies between and dt, as is evident if we consider separately the cases in which is positive and negative. Those systems for which p1 lies between these limits, and the other p’s and q’s between the limits specified in (9), will therefore pass into or out of the element considered according as is positive or negative, unless indeed they also pass some other limit specified in (9) during the same interval of time. But the number which pass any two of these limits will be represented by an expression containing the square of dt as a factor, and is evidently negligible, when dt is sufficiently small, compared with the number which we are seeking to evaluate, and which (with neglect of terms containing dt²) may be found by substituting dt for in (10) or for dp1 in (11).

The expression

will therefore represent, according as it is positive or negative, the increase or decrease of the number of systems within the given limits which is due to systems passing the limit . A similar expression, in which however D and will have slightly different values (being determined for instead of ), will represent the decrease or increase of the number of systems due to the passing of the limit . The difference of the two expressions, or

will represent algebraically the decrease of the number of systems within the limits due to systems passing the limits and .

The decrease in the number of systems within the limits due to systems passing the limits and may be found in the same way. This will give

for the decrease due to passing the four limits But since the equations of motion (3) give

the expression reduces to

If we prefix Σ to denote summation relative to the suffixes 1 … n, we get the total decrease in the number of systems within the limits in the time dt. That is,

or

where the suffix applied to the differential coefficient indicates that the p’s and q’s are to be regarded as constant in the differentiation. The condition of statistical equilibrium is therefore

If at any instant this condition is fulfilled for all values of the p’s and q’s (dD/dt)p,q vanishes, and therefore the condition will continue to hold, and the distribution in phase will be permanent, so long as the external coördinates remain constant. But the statistical equilibrium would in general be disturbed by a change in the values of the external coördinates, which would alter the values of the as determined by equations (3), and thus disturb the relation expressed in the last equation.

If we write equation (19) in the form

it will be seen to express a theorem of remarkable simplicity. Since D is a function of t, p1, … pn, q1,… qn, its complete differential will consist of parts due to the variations of all these quantities. Now the first term of the equation represents the increment of D due to an increment of t (with constant values of the p’s and q’s), and the rest of the first member represents the increments of D due to increments of the p’s and q’s, expressed by , etc. But these are precisely the increments which the p’s and q’s receive in the movement of a system in the time dt. The whole expression represents the total increment of D for the varying phase of a moving system. We have therefore the theorem: —

In an ensemble of mechanical systems identical in nature and subject to forces determined by identical laws, but distributed in phase in any continuous manner, the density-in-phase is Constantin time for the varying phases of a moving system; provided, that the forces of a system are functions of its coördinates, either alone or with the time.*

This may be called the principle of conservation of density-in-phase. It may also be written

where a, … h represent the arbitrary constants of the integral equations of motion, and are suffixed to the differential coefficient to indicate that they are to be regarded as constant in the differentiation.

We may give to this principle a slightly different expression. Let us call the value of the integral

taken within any limits the extension-in-phase within those limits.

When the phases bounding an extension-in-phase vary in the course of time according to the dynamical laws of a system subject to forces which are functions of the coördinates either alone or with the time, the value of the extension-in-phase thus bounded remains constant. In this form the principle may be called the principle of conservation of extension-in-phase. In some respects this may be regarded as the most simple statement of the principle, since it contains no explicit reference to an ensemble of systems.

Since any extension-in-phase may be divided into infinitesimal portions, it is only necessary to prove the principle for an infinitely small extension. The number of systems of an ensemble which fall within the extension will be represented by the integral

If the extension is infinitely small, we may regard D as constant in the extension and write

for the number of systems. The value of this expression must be constant in time, since no systems are supposed to be created or destroyed, and none can pass the limits, because the motion of the limits is identical with that of the systems. But we have seen that D is constant in time, and therefore the