Elementary Principles in Statistical Mechanics

Front matter

ELEMENTARY PRINCIPLES

IN

STATISTICAL MECHANICS

front matter

Yale Bicentennial Publications

Yale Bicentennial Publications

With the approval of the President and Fellows of Yale University, a series of volumes has been prepared by a number of the Professors and Instructors, to be issued in connection with the Bicentennial Anniversary, as a partial indication of the character of the studies in which the University teachers are engaged.

Chapter I. General notions. The principle of conservation of extension-in-phase.

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 degrees of freedom, writing for the (generalized) coördinates, for the (generalized) velocities, and for the moment of the forces. We shall call the quantities , the (generalized) forces, and the quantities , defined by the equations where 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 . 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 's and 's. But in expressions like , where the denominator does not determine the question, the kinetic energy is always to be treated in the differentiation as function of the 's and '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 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 () may depend on other variables beside the coördinates . We shall often suppose it to depend in part on coördinates of external bodies, which we shall denote by , , etc. We shall then have for the complete value of the differential of the potential energy where , , 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 () in the most general case is a quadratic function of the 's (or 's) involving also the 's but not the 's ; that the potential energy, when it exists, is function of the 's and 's ; and that the total energy, when it exists, is function of the 's (or s), the 's, and the 's. In expressions like the '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 , either alone or with the coördinates , , etc. of certain external bodies. It is not necessary that they should be derivable from a force-function. The external coördinates , , etc. may vary with the time, but at any given time have fixed values. In this they differ from the internal coördinates , 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 is a function of the 's and 's and in general of 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 dimensions, we may regard the product of differentials in (11) as expressing an element of this extension, and as expressing the density of the systems in that element. We shall call the product an element of extension-in-phase, and 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 's, 's, and '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 () in terms of the 's and 's, and the forces in terms of the 's and 's. The distribution in phase is expressed for the time considered by as function of the 's and 's. To find the value of 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 different ways, viz., by the of a system passing the limit , or the limit , or by the 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 pass into or out of the specified element by passing the limit . It will be convenient, and it is evidently allowable, to suppose so small that the quantities , , etc., which represent the increments of , , etc., in the time 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 passes the limit in the interval are those for which at the commencement of this interval the value of lies between and , as is evident if we consider separately the cases in which is positive and negative. Those systems for which lies between these limits, and the other 's and '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 as a factor, and is evidently negligible, when is sufficiently small, compared with the number which we are seeking to evaluate, and which (with neglect of terms containing ) may be found by substituting for in (10) or for 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 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 to denote summation relative to the suffixes , we get the total decrease in the number of systems within the limits in the time . That is, or where the suffix applied to the differential coefficient indicates that the 's and '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 's and 's, 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 's 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 is a function of , , , 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 due to an increment of (with constant values of the 's and 's), and the rest of the first member represents the increments of due to increments of the 's and 's, expressed by , , etc. But these are precisely the increments which the 's and 's receive in the movement of a system in the time . The whole expression represents the total increment of 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 constant in 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 represent the arbitrary constants of the integral equations of motion, and are suffixed to the differential co 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 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 is constant in time, and therefore the integral which we have called the extension-in-phase, is also constant in time.

Since the system of coördinates employed in the foregoing discussion is entirely arbitrary, the values of the coördinates relating to any configuration and its immediate vicinity do not impose any restriction upon the values relating to other configurations. The fact that the quantity which we have called density-in-phase is constant in time for any given system, implies therefore that its value is independent of the coördinates which are used in its evaluation. For let the density-in-phase as evaluated for the same time and phase by one system of coördinates be , and by another system . A system which at that time has that phase will at another time have another phase. Let the density as calculated for this second time and phase by a third system of coördinates be . Now we may imagine a system of coördinates which at and near the first configuration will coincide with the first system of coördinates, and at and near the second configuration will coincide with the third system of coördinates. This will give . Again we may imagine a system of coördinates which at and near the first configuration will coincide with the second system of coördinates, and at and near the second configuration will coincide with the third system of coördinates. This will give . We have therefore .

It follows, or it may be proved in the same way, that the value of an extension-in-phase is independent of the system of coördinates which is used in its evaluation. This may easily be verified directly. If , are two systems of coördinates, and , the corresponding momenta, we have to prove that when the multiple integrals are taken within limits consisting of the same phases. And this will be evident from the principle on which we change the variables in a multiple integral, if we prove that where the first member of the equation represents a Jacobian or functional determinant. Since all its elements of the form are equal to zero, the determinant reduces to a product of two, and we have to prove that We may transform any element of the first of these determinants as follows. By equations (2) and (3), and in view of the fact that the 's are linear functions of the 's and therefore of the 's, with coefficients involving the 's, so that a differential coefficient of the form is function of the 's alone, we get But since Therefore, The equation to be proved is thus reduced to which is easily proved by the ordinary rule for the multiplication of determinants.

The numerical value of an extension-in-phase will however depend on the units in which we measure energy and time. For a product of the form has the dimensions of energy multiplied by time, as appears from equation (2), by which the momenta are defined. Hence an extension-in-phase has the dimensions of the th power of the product of energy and time. In other words, it has the dimensions of the th power of action, as the term is used in the `principle of Least Action.'

If we distinguish by accents the values of the momenta and coördinates which belong to a time , the unaccented letters relating to the time , the principle of the conservation of extension-in-phase may be written or more briefly the limiting phases being those which belong to the same systems at the times and respectively. But we have identically for such limits. The principle of conservation of extension-in-phase may therefore be expressed in the form This equation is easily proved directly. For we have identically where the double accents distinguish the values of the momenta and coördinates for a time . If we vary , while and remain constant, we have Now since the time is entirely arbitrary, nothing prevents us from making if identical with at the moment considered. Then the determinant will have unity for each of the elements on the principal diagonal, and zero for all the other elements. Since every term of the determinant except the product of the elements on the principal diagonal will have two zero factors, the differential of the determinant will reduce to that of the product of these elements, i. e., to the sum of the differentials of these elements. This gives the equation Now since , the double accents in the second member of this equation may evidently be neglected. This will give, in virtue of such relations as (16), which substituted in (34) will give The determinant in this equation is therefore a constant, the value of which may be determined at the instant when , when it is evidently unity. Equation (33) is therefore demonstrated.

Again, if we write for a system of arbitrary constants of the integral equations of motion, , , etc. will be functions of , and , and we may express an extension-in-phase in the form If we suppose the limits specified by values of , a system initially at the limits will remain at the limits. The principle of conservation of extension-in-phase requires that an extension thus bounded shall have a constant value. This requires that the determinant under the integral sign shall be constant, which may be written This equation, which may be regarded as expressing the principle of conservation of extension-in-phase, may be derived directly from the identity in connection with equation (33).

Since the coördinates and momenta are functions of , and , the determinant in (36) must be a function of the same variables, and since it does not vary with the time, it must be a function of alone. We have therefore

It is the relative numbers of systems which fall within different limits, rather than the absolute numbers, with which we are most concerned. It is indeed only with regard to relative numbers that such discussions as the preceding will apply with literal precision, since the nature of our reasoning implies that the number of systems in the smallest element of space which we consider is very great. This is evidently inconsistent with a finite value of the total number of systems, or of the density-in-phase. Now if the value of is infinite, we cannot speak of any definite number of systems within any finite limits, since all such numbers are infinite. But the ratios of these infinite numbers may be perfectly definite. If we write for the total number of systems, and set may remain finite, when and become infinite. The integral taken within any given limits, will evidently express the ratio of the number of systems falling within those limits to the whole number of systems. This is the same thing as the probability that an unspecified system of the ensemble (i. e. one of which we only know that it belongs to the ensemble) will lie within the given limits. The product expresses the probability that an unspecified system of the ensemble will be found in the element of extension-in-phase . We shall call the coefficient of probability of the phase considered. Its natural logarithm we shall call the index of probability of the phase, and denote it by the letter .

If we substitute and for in equation (19), we get and The condition of statistical equilibrium may be expressed by equating to zero the second member of either of these equations.

The same substitutions in (22) give and That is, the values of and , like those of , are constant in time for moving systems of the ensemble. From this point of view, the principle which otherwise regarded has been called the principle of conservation of density-in-phase or conservation of extension-in-phase, may be called the principle of conservation of the coefficient (or index) of probability of a phase varying according to dynamical laws, or more briefly, the principle of conservation of probability of phase. It is subject to the limitation that the forces must be functions of the coördinates of the system either alone or with the time.

The application of this principle is not limited to cases in which there is a formal and explicit reference to an ensemble of systems. Yet the conception of such an ensemble may serve to give precision to notions of probability. It is in fact customary in the discussion of probabilities to describe anything which is imperfectly known as something taken at random from a great number of things which are completely described. But if we prefer to avoid any reference to an ensemble of systems, we may observe that the probability that the phase of a system falls within certain limits at a certain time, is equal to the probability that at some other time the phase will fall within the limits formed by phases corresponding to the first. For either occurrence necessitates the other. That is, if we write for the coefficient of probability of the phase at the time , and for that of the phase at the time , where the limits in the two cases are formed by corresponding phases. When the integrations cover infinitely small variations of the momenta and coördinates, we may regard and as constant in the integrations and write Now the principle of the conservation of extension-in-phase, which has been proved (viz., in the second demonstration given above) independently of any reference to an ensemble of systems, requires that the values of the multiple integrals in this equation shall be equal. This gives

With reference to an important class of cases this principle may be enunciated as follows.

When the differential equations of motion are exactly known, but the constants of the integral equations imperfectly determined, the coefficient of probability of any phase at any time is equal to the coefficient of probability of the corresponding phase at any other time. By corresponding phases are meant those which are calculated for different times from the same values of the arbitrary constants of the integral equations.

Since the sum of the probabilities of all possible cases is necessarily unity, it is evident that we must have where the integration extends over all phases. This is indeed only a different form of the equation which we may regard as defining .

The values of the coefficient and index of probability of phase, like that of the density-in-phase, are independent of the system of coördinates which is employed to express the distribution in phase of a given ensemble.

In dimensions, the coefficient of probability is the reciprocal of an extension-in-phase, that is, the reciprocal of the th power of the product of time and energy. The index of probability is therefore affected by an additive constant when we change our units of time and energy. If the unit of time is multiplied by and the unit of energy is multiplied by , all indices of probability relating to systems of degrees of freedom will be increased by the addition of

Chapter II. Application of the principle of conservation of extension-in-phase to the theory of errors.

CHAPTER II.

APPLICATION OF THE PRINCIPLE OF CONSERVATION OF EXTENSION-IN-PHASE TO THE THEORY OF ERRORS.

Let us now proceed to combine the principle which has been demonstrated in the preceding chapter and which in its different applications and regarded from different points of view has been variously designated as the conservation of density-in-phase, or of extension-in-phase, or of probability of phase, with those approximate relations which are generally used in the 'theory of errors.'

We suppose that the differential equations of the motion of a system are exactly known, but that the constants of the integral equations are only approximately determined. It is evident that the probability that the momenta and coördinates at the time fall between the limits and , and , etc., may be expressed by the formula where (the index of probability for the phase in question) is a function of the coördinates and momenta and of the time.

Let , , etc. be the values of the coördinates and momenta which give the maximum value to , and let the general value of be developed by Taylor's theorem according to ascending powers and products of the differences , , etc. and let us suppose that we have a sufficient approximation without going beyond terms of the second degree in these differences. We may therefore set where is independent of the differences , , etc., and is a homogeneous quadratic function of these differences. The terms of the first degree vanish in virtue of the maximum condition, which also requires that must have a positive value except when all the differences mentioned vanish. If we set we may write for the probability that the phase lies within the limits considered is evidently the maximum value of the coefficient of probability at the time considered.

In regard to the degree of approximation represented by these formulæ, it is to be observed that we suppose, as is usual in the 'theory of errors' that the determination (explicit or implicit) of the constants of motion is of such precision that the coefficient of probability or is practically zero except for very small values of the differences , , etc. For very small values of these differences the approximation is evidently in general sufficient, for larger values of these differences the value of will be sensibly zero, as it should be, and in this sense the formula will represent the facts.

We shall suppose that the forces to which the system is subject are functions of the coördinates either alone or with the time. The principle of conservation of probability of phase will therefore apply, which requires that at any other time () the maximum value of the coefficient of probability shall be the same as at the time and that the phase which has this greatest probability-coefficient, shall be that which corresponds to the phase , i. e., which is calculated from the same values of the constants of the integral equations of motion.

We may therefore write for the probability that the phase at the time falls within the limits and , and , etc., where represents the same value as in the preceding formula, viz., the constant value of the maximum coefficient of probability, and is a quadratic function of the differences , , etc., the phase being that which at the time corresponds to the phase at the time .

Now we have necessarily when the integration is extended over all possible phases. It will be allowable to set for the limits of all the coördinates and momenta, not because these values represent the actual limits of possible phases, but because the portions of the integrals lying outside of the limits of all possible phases will have sensibly the value zero. With for limits, the equation gives where is the discriminant of , and that of . This discriminant is therefore constant in time, and like an absolute invariant in respect to the system of coördinates which may be employed. In dimensions, like , it is the reciprocal of the 2nth power of the product of energy and time.

Let us see precisely how the functions and are related. The principle of the conservation of the probability-coefficient requires that any values of the coördinates and momenta at the time shall give the function the same value as the corresponding coördinates and momenta at the time give to . Therefore may be derived from by substituting for their values in terms of . Now we have approximately and as in terms of higher degree than the second are to be neglected, these equations may be considered accurate for the purpose of the transformation required. Since by equation (33) the eliminant of these equations has the value unity, the discriminant of will be equal to that of , as has already appeared from the consideration of the principle of conservation of probability of phase, which is, in fact, essentially the same as that expressed by equation (33).

At the time the phases satisfying the equation where is any positive constant, have the probability-coefficient . At the time , the corresponding phases satisfy the equation and have the same probability-coefficient. So also the phases within the limits given by one or the other of these equations are corresponding phases, and have probability-coefficients greater than , while phases without these limits have less probability-coefficients. The probability that the phase at the time falls within the limits is the same as the probability that it falls within the limits at the time , since either event necessitates the other. This probability may be evaluated as follows. We may omit the accents, as we need only consider a single time. Let us denote the extension-in-phase within the limits by , and the probability that the phase falls within these limits by , also the extension-in-phase within the limits by . We have then by definition But since is a homogeneous quadratic function of the differences we have identically That is whence But if varies, equations (58) and (59) give

Since the factor has the constant value in the last multiple integral, we have We may determine the constant of integration by the condition that vanishes with . This gives We may determine the value of the constant by the condition that for . This gives , and

It is worthy of notice that the form of these equations depends only on the number of degrees of freedom of the system, being in other respects independent of its dynamical nature, except that the forces must be functions of the coördinates either alone or with the time.

If we write for the value of which substituted in equation (68) will give , the phases determined by the equation will have the following properties.

The probability that the phase falls within the limits formed by these phases is greater than the probability that it falls within any other limits enclosing an equal extension-in-phase. It is equal to the probability that the phase falls without the same limits.

These properties are analogous to those which in the theory of errors in the determination of a single quantity belong to values expressed by , when is the most probable value, and the 'probable error.'

Chapter III. Application of the principle of conservation of extension-in-phase to the integration of the differential equations of motion.

CHAPTER III.