11*. Mechanical Conditions

The correctness of the conclusion that all the states of distribution considered occur with equal probability we see still more clearly when we remember that a change of condition does not occur as a consequence of chance, but that each alteration of the molecular motions takes place in accordance with fixed and invariable mechanical laws. Each distribution of speed that at any moment exists was the necessary consequence of that which preceded it, and from it in its turn necessarily arises a new state with a distribution that is by no means arbitrary, but completely determined.

Before, therefore, the final state, which seems to the observer one of equilibrium, that does not alter with the time, is arrived at in a mass of gas left to itself, the function which expresses the probability of a given value of the speed continually changes its nature according to the fixed laws of mechanics. But when the final state is attained the form of this function remains always the same; only the arguments-the values of the components-then alter at each encounter, this alteration also being subject to the general laws of mechanics. Each system of simultaneous values of the speeds appears therefore as often as that from which it arises and as that which results from it; in other words, the probability of occurrence of all these systems is the same.

The laws from which this conclusion is the necessary consequence are contained in the theorems which deal with the mechanics of systems of free particles in motion. For our problem, the establishment of the law of distribution of the energy and speed, only those theorems come into consideration which contain and determine these magnitudes alone. These theorems

are

1. The principle of the conservation of energy;

2. The principle of the conservation of the motion of the centroid.

Further hypotheses are not needed for the present; indeed both these theorems depend on a single common basis, if we may assume that the action exerted by one particle on another is equal to the reaction which at each moment it experiences itself from the other in the reverse direction.

In respect to the application of these theorems to our proof,

only the former, the principle of the conservation of energy, requires a few remarks to be made, since it may be applied in different ways according as the molecules are to be looked on as simple massive points or as made up of atoms. In the former case, of the two kinds of energy, kinetic and potential, the sum of which, according to the principle, possesses a value that is constant for all time and under all circumstances, only the former comes into account; for since two molecules act on each other only at the moment of a collision, the potential energy can be neglected if we take into account in the calculation those speeds with which the particles move between two collisions, and not during a collision. We have then to consider simply the sum of the kinetic energies as invariable, or to introduce the theorem that no kinetic energy is lost at a collision.

We shall first of all limit our consideration to this simpler case, and postpone that of composite molecules for later investigation in § 21*.

12*. Determining Equations

By § 10* our problem consists in finding the function F(u, v, w) which has the property that the product

F(u1, v1, w1)F(U2, V2, W2). . . F(ux, vx, WN)

has the same constant value C for all values of the components u, v, w of the molecular velocities that occur together. According to the last discussion the values of these components are not magnitudes that vary arbitrarily and independently, but they are subject to the conditions that they must satisfy the two named theorems of mechanics. If, then, we denote by E the mean value of the kinetic energy of one of the N molecules in question, by m the mass of a molecule, and finally by a, b, c the components of the velocity with which the centroid of the whole system of gaseous particles moves, the two theorems are expressed by the equations

2

2

2

NE=m(u,2 + v12 + w 12 + U q2 + v22 + W22 +

.. ·

+ UN

אט +.

+ WN

For our problem these equations represent the conditions con

necting the variable magnitudes u, v, w with the given constants E, a, b, c. We have then from the functional equation.

C = F(u1, v1, w1) F(u2, v2, w2). . . F(ux, VN, Wx),

coupled with the conditions represented by the above four equations, to determine the function F(u, v, w).

This is done by the known processes of the calculus of variations. If we represent a second system of values which satisfy the equations by the symbols un + dun, vn + dvn, wn + own, where n may represent any integer between 1 and N, then we have also

C=F(u +âu, tôi tà)...F(y+ÒN, NtCN, UN +2N) NE=&m {(u, +èu,)2 + (v, + dv1)2 + (w、 + îw1)2 + . . .

+ (ux + dux)2 + (VN + dvÑ)2 + (Wx+ òwx)2}

On subtracting the one system of equations from the other we obtain five equations from which the constants C, E, a, b, c are absent. In these equations we take the variations du, dv, èw as infinitely small, being justified in this if we choose both systems of values of u, v, w to be such as to differ infinitely little from each other; developing, then, the equations in powers of du, dv, dw, and neglecting their higher powers, we obtain five equations whose terms are all of the first order.

0=F2F3... Fx.F+F1F3... F.F2+...+FF2... FN-1.CFx,

which, on dividing by the product of all the functions F, we may write in the form

In addition to this we obtain from the four other equations the simple conditions

to which the variations are subject. The variations are therefore not perfectly arbitrary magnitudes, but are in such wise dependent on each other that four of them are determined by the remaining 3N-4. The values of these last 3N — 4 are limited only by the condition that they must be infinitely small: for the rest, however, it remains perfectly arbitrary what values we assign to the variations, and what ratios we take between their values. If, therefore, by means of the last four equations we eliminate from the principal equation

four of the 3N variations, we obtain a formula which we can so arrange that its 3N 4 terms contain each a factor du, dv, dw which may have any value whatever. The formula therefore breaks up into 3N - 4 independent equations which do not contain the variations, but only the function F and its arguments.

This elimination is most easily performed by the help of initially undetermined coefficients by which the equations of condition are multiplied before being added to the principal equation: these coefficients, which I will take as 2km, 2kma, - 2kmß, - 2kmy, are then so determined that four variations out of the whole disappear; then, by reason of the 3N 4 other variations being quite arbitrary, their factors are also zero. We thus obtain 3N equations of the form

(" + 2km(w, − y),

1 dFn

=

Fdw

in which for n are to be taken all the integers from 1 to N. In all these equations the four magnitudes k, a, ß, y have each one and

the same value, which is therefore a constant independent of n and of u, v, w.

13*. Another Method

We may arrive at these formulæ in another, perhaps simpler, way from the functional equation

C = F(u, v1, w1)F(uq, v2, w2)... F(UN, UN, WN),

by comparing together two states of the molecular system of which the one immediately follows the other. A change of the state occurs at every collision between two particles: we compare therefore the state of the system before a collision between any two particles with its state immediately after the collision. Of all the particles, then, only those two which collided have changed their motion. We may, then, in the product neglect the factors that have remained unchanged, and thus conclude that

F(u1, v1, w1)F(u2, V2, W2) = F(U1, V1, W1)F(U2, V2, W2),

if u1, v1, w1 and u2, v2, w2 are the components of velocity of the two particles before collision, and U, V, W, and U2, V2, W2 the corresponding values after collision.

We thus arrive at a form of functional equation to which other methods of proof have also led. It first occurs in Maxwell's second proof, and then in the memoirs of Boltzmann,2 Lorentz,3 and others. It occurs in these memoirs as expression for the stability of Maxwell's state of distribution: the equation may also be interpreted in such wise that the number of collisions. depending on the product F(u,, v1, w1) F(u2, v2, w2), in which the components u, v, w are changed into U, V, W, is exactly as great as the number similarly determined by F(U1, V1, W1)F(U2, V2, W2) in which the components u, v, w take the place of the values U, V, W.

The functional equation is subject to the conditions

u12 + v12 + w12 + u22 + v22 + w22

1 Phil. Trans. 1866, p. 157; Scientific Papers, 1890, ii. p. 45.

2 Wiener Sitzungsber. Iviii. 1868, p. 517; lxvi. 1872, p. 275; xcvi. 1887, p. 891; &c.

3 Ibid. xcv. 1887, p. 115.