p. 299
From these equations it can again be proved that
must always decrease [immer abnehmen muss], unless $\htmlClass{sym sym-u}{u_1^2} = \htmlClass{sym sym-u}{u_1} \htmlClass{sym sym-u}{u_3},\; \htmlClass{sym sym-u}{u_2^2} = \htmlClass{sym sym-u}{u_1} \htmlClass{sym sym-u}{u_3},\; \htmlClass{sym sym-u}{u_3^2} = \htmlClass{sym sym-u}{u_2} \htmlClass{sym sym-u}{u_4} \dots$ (that is, unless all expressions multiplied by the coefficients $\htmlClass{sym sym-B}{B}$ in equation (35)) vanish. Equations (35) have an inconvenient feature: while they can be written compactly with summation formulae, they resist being written out fully and explicitly. For clarity, we therefore begin with the simplest case, then move stepwise toward the general one.
First let $p = 3$; the molecules can have only three kinetic energies, $\htmlClass{sym sym-eps}{\epsilon}$, $2\htmlClass{sym sym-eps}{\epsilon}$, and $3\htmlClass{sym sym-eps}{\epsilon}$. Then the system of equations (35) reduces to the following three equations:
and the expression for $\htmlClass{sym sym-E}{E}$ becomes
Differentiating gives
p. 300
or, after rearranging,
The sum in parentheses vanishes by equations (36). Thus $\htmlClass{sym sym-deriv}{dE/dt}$ is obtained by multiplying the first equation by $\htmlClass{sym sym-log}{\log} \htmlClass{sym sym-u}{u_1}$, the second by $\htmlClass{sym sym-log}{\log} \htmlClass{sym sym-u}{u_2}$, the third by $\htmlClass{sym sym-log}{\log} \htmlClass{sym sym-u}{u_3}$, then adding. This yields
or
Consider the two factors multiplying $\htmlClass{sym sym-B}{B_{11}^{22}}$. When $\htmlClass{sym sym-u}{u_2^2} > \htmlClass{sym sym-u}{u_1}\htmlClass{sym sym-u}{u_3}$, the first factor is positive and the logarithm is negative. When $\htmlClass{sym sym-u}{u_2^2} < \htmlClass{sym sym-u}{u_1}\htmlClass{sym sym-u}{u_3}$, the first factor is negative and the logarithm is positive. In either case their product is always negative. Since $\htmlClass{sym sym-B}{B_{11}^{22}}$ is positive, $\htmlClass{sym sym-deriv}{dE/dt}$ is always negative or zero. Equality holds only when $\htmlClass{sym sym-u}{u_2^2} = \htmlClass{sym sym-u}{u_1}\htmlClass{sym sym-u}{u_3}$.
It is also easy to see that $\htmlClass{sym sym-E}{E}$ cannot run to minus infinity. None of $\htmlClass{sym sym-u}{u_1},\htmlClass{sym sym-u}{u_2},\htmlClass{sym sym-u}{u_3}$ may be negative or imaginary. For positive $u$, the quantity $u\htmlClass{sym sym-log}{\log} u$ cannot be smaller than $-1/e$. Hence $\htmlClass{sym sym-E}{E}$ cannot be smaller than
where $e$ is the base of natural logarithms. Therefore, since its derivative cannot be positive, $\htmlClass{sym sym-E}{E}$ must move steadily toward a minimum where $\htmlClass{sym sym-deriv}{dE/dt} = 0$, namely where $\htmlClass{sym sym-u}{u_2^2} = \htmlClass{sym sym-u}{u_1}\htmlClass{sym sym-u}{u_3}$.
The same proof does not carry over unchanged when $p > 3$. Here I treat only the case $p = 4$.
p. 301
The "?" marks are not typos — they appear in Boltzmann's original 1872 paper. He is working out notation on the fly, and the terms are ambiguous even in the source. We preserve them as historical artifacts.
For $\htmlClass{sym sym-E}{E}$ one finds
If one substitutes here for the derivatives their values from equations (37), he obtains, with a suitable rearrangement of terms,
p. 302
I remark that the change in the order of the summands, which is necessary here, is analogous to our previous transformation of definite integrals. From the above expression one sees at once that $\htmlClass{sym sym-deriv}{dE/dt}$ is again necessarily negative, unless simultaneously we have
which can also be written
Likewise one finds in the general case that $\htmlClass{sym sym-deriv}{dE/dt}$ is necessarily negative so that $\htmlClass{sym sym-E}{E}$ must decrease unless
Since $\htmlClass{sym sym-E}{E}$ cannot have a larger negative value than
it must necessarily approach a minimum value for which equations (38) hold. Thus it continually approaches the distribution of states determined by equations (38).
We now have to prove that equations (38) uniquely determine the distribution of states. If we add together all the equations (35), we obtain
hence
p. 303
In a similar way we find that
where $\htmlClass{sym sym-a}{a}$ and $\htmlClass{sym sym-b}{b}$ are constants. The meaning of these equations is obvious. In particular, $\htmlClass{sym sym-u}{u_1} + \htmlClass{sym sym-sqrt}{\sqrt{2}}\htmlClass{sym sym-u}{u_2} + \htmlClass{sym sym-sqrt}{\sqrt{3}}\htmlClass{sym sym-u}{u_3} + \dots = \htmlClass{sym sym-a}{a}$ is the total number of molecules in unit volume, while $\htmlClass{sym sym-b}{b}$ is their total kinetic energy. Equations (40) and (41) therefore tell us that these two quantities are constant.
Suppose that the two quantities $\htmlClass{sym sym-a}{a}$ and $\htmlClass{sym sym-b}{b}$ are given. Then we set the quotient $\htmlClass{sym sym-u}{u_2}/\htmlClass{sym sym-u}{u_1}$ equal to $\htmlClass{sym sym-y}{y}$. Equations (38) then reduce to
If one substitutes these values into equations (40) and (41), then he finds easily:
Since all the $\htmlClass{sym sym-u}{u}$'s are necessarily positive, we see immediately that $\frac{\htmlClass{sym sym-b}{b}}{\htmlClass{sym sym-eps}{\epsilon}} - \htmlClass{sym sym-a}{a}$ must be positive while $\frac{\htmlClass{sym sym-b}{b}}{\htmlClass{sym sym-eps}{\epsilon}} - p\htmlClass{sym sym-a}{a}$ must be negative. Hence $\htmlClass{sym sym-b}{b}$ must lie between $\htmlClass{sym sym-eps}{\epsilon} \htmlClass{sym sym-a}{a}$ and $p\htmlClass{sym sym-eps}{\epsilon} \htmlClass{sym sym-a}{a}$. Hence, in equation (42) the coefficient of $\htmlClass{sym sym-y}{y}^{p-1}$ is positive, while the term independent of $\htmlClass{sym sym-y}{y}$ must be negative. The polynomial is therefore positive for $\htmlClass{sym sym-y}{y} = \infty$, and negative for $\htmlClass{sym sym-y}{y} = 0$; therefore there is one and only one positive root for $\htmlClass{sym sym-y}{y}$, since the series of coefficients changes sign only once. Negative or imaginary values for $\htmlClass{sym sym-y}{y}$ are of course meaningless. But from $\htmlClass{sym sym-y}{y}$ we can determine uniquely all the $\htmlClass{sym sym-u}{u}$'s and also all the $\htmlClass{sym sym-w}{w}$'s. Hence, whatever may be the initial distribution of states, there is one and only one distribution which it approaches with increasing time. This distribution depends only on the constants $\htmlClass{sym sym-a}{a}$ and $\htmlClass{sym sym-b}{b}$, the total number and total kinetic energy of the molecules (density and temperature of the gas).
p. 304
This theorem was proved first only for the case that the distribution of states is initially uniform. It must also hold, however, when this is not true, provided only that the molecules are distributed in such a way that they tend to become mixed as time progresses, so that the distribution becomes uniform after a very long time. This will always happen with the exception of certain special cases, for example, when the molecules move initially in a straight line and are reflected back in this straight line at the walls. Since we have established this for arbitrary $p$ and $\htmlClass{sym sym-eps}{\epsilon}$, we can immediately go to the case where $\frac{1}{p}$ and $\htmlClass{sym sym-eps}{\epsilon}$ become infinitesimal.
† For very large $p$, the expression (39) will be very large, of order $p$. In this case it is necessary to look for a smaller negative value that $\htmlClass{sym sym-E}{E}$ can never exceed. The quantity denoted here by $\htmlClass{sym sym-E}{E}$ differs by a constant from the one earlier so denoted. If we wish to obtain the quantity denoted by $\htmlClass{sym sym-E1}{E_1}$ in equation (17a), page 113, which again differs only by a constant from the other quantities denoted by this letter, then we must add to our present $\htmlClass{sym sym-E}{E}$, $-\frac{3\htmlClass{sym sym-log}{\log}\htmlClass{sym sym-eps}{\epsilon}}{2}(\htmlClass{sym sym-u}{u_1}+\htmlClass{sym sym-sqrt}{\sqrt{2}}\htmlClass{sym sym-u}{u_2}+\dots)$. Therefore
It is clear now that $\htmlClass{sym sym-E1}{E_1}$ is a real and continuous function of the $\htmlClass{sym sym-u}{u}$'s for all real positive values of it. Furthermore, if we say that a negative quantity is smaller, the greater its numerical value is, then $\htmlClass{sym sym-E}{E}$ is not smaller than the expression (39), hence $\htmlClass{sym sym-E1}{E_1}$ is not smaller than
Hence, $\htmlClass{sym sym-E1}{E_1}$ must have a minimum if the $\htmlClass{sym sym-u}{u}$'s run through all real positive values compatible with equations (40) and (41). One can then easily show that for this minimum none of the $\htmlClass{sym sym-u}{u}$'s can be equal to zero, so that the minimum cannot lie on the boundary of the space formed from the $\htmlClass{sym sym-u}{u}$'s, and consequently it can be found by applying the usual rules of differential calculus. If we add to the total differential of $\htmlClass{sym sym-E1}{E_1}$ that of the two equations (40) and (41), multiplying the former with the undetermined multiplier $\htmlClass{sym sym-lam}{\lambda}$, and the latter by $\htmlClass{sym sym-lam}{\mu}$, then we obtain
At the minimum, the factor of each differential must vanish, whence on elimination of $\htmlClass{sym sym-lam}{\lambda}$ and $\htmlClass{sym sym-lam}{\mu}$ one obtains
or $\htmlClass{sym sym-u}{u_2}/\htmlClass{sym sym-u}{u_1} = \htmlClass{sym sym-u}{u_3}/\htmlClass{sym sym-u}{u_2} = \htmlClass{sym sym-u}{u_4}/\htmlClass{sym sym-u}{u_3} = \dots$, which we recognize to be the same as equations (38). These equations therefore determine the smallest value that $\htmlClass{sym sym-E1}{E_1}$ can have when the $\htmlClass{sym sym-u}{u}$'s take all possible values consistent with equations (40) and (41). However, since the $\htmlClass{sym sym-u}{u}$'s are actually subject to equations (40) and (41) during the entire process, this is the smallest value of $\htmlClass{sym sym-E1}{E_1}$ during the entire process. In order to calculate it, we set again $\htmlClass{sym sym-u}{u_2} = \htmlClass{sym sym-u}{u_1}\htmlClass{sym sym-y}{y},\; \htmlClass{sym sym-u}{u_3} = \htmlClass{sym sym-u}{u_1}\htmlClass{sym sym-y}{y}^2,\dots$ We know that we then find from equations (38), (40) and (41) a unique positive value for $\htmlClass{sym sym-y}{y}$, which must correspond to the actual minimum of $\htmlClass{sym sym-E1}{E_1}$. This minimum value of $\htmlClass{sym sym-E1}{E_1}$ is therefore
$\htmlClass{sym sym-E1}{E_1}$ cannot have a smaller value than this. This value remains finite for infinitesimal $\htmlClass{sym sym-eps}{\epsilon}$ and infinite $p$. Taking account of equations (43), we see that it reduces to $\htmlClass{sym sym-a}{a}\htmlClass{sym sym-log}{\log} \htmlClass{sym sym-C}{C} - \htmlClass{sym sym-b}{b}\htmlClass{sym sym-h}{h}$ or, since $\htmlClass{sym sym-a}{a} = 2\int \htmlClass{sym sym-C}{C}$ and $\htmlClass{sym sym-b}{b} = 2/\htmlClass{sym sym-h}{h}$, one can write for it $\frac{1}{2}\htmlClass{sym sym-a}{a}(\htmlClass{sym sym-log}{\log} \htmlClass{sym sym-C}{C} - 1)$ which is a finite quantity. Hence $\htmlClass{sym sym-E1}{E_1}$ cannot be minus infinity. On the other hand, it may be plus infinity. We still have to show that in that case there cannot be thermal equilibrium. This proof, as well as an explicit discussion of the exceptional case where $\lim_{\tau\to0} \frac{\htmlClass{sym sym-eps}{\epsilon}}{\tau}[\dots]$ comes out to be different according as $\htmlClass{sym sym-eps}{\epsilon}/\tau$ or $\tau/\htmlClass{sym sym-eps}{\epsilon}$ vanishes, will not be discussed further here.
p. 305
We have first:
For infinitesimal $\htmlClass{sym sym-eps}{\epsilon}$ we can again set:
and obtain
which is again the Maxwell distribution. Likewise one can convince himself that the sum which we have here denoted by $\htmlClass{sym sym-E}{E}$ reduces, aside from a constant additive term, to the integral in equation (17a); we therefore obtain by this method all the results that we earlier found by transformations of definite integrals, but it is the advantage of being much simpler and clearer. One only has to accept the abstraction that a molecule may have only a finite number of kinetic energies as a transition stage.
If one sets the time derivatives in equations (35) equal to zero, he obtains the conditions that the distribution of states does not change with time but is stationary. The condition that the distribution be stationary is obtained by setting $\frac{\partial \htmlClass{sym sym-f}{f}(x,t)}{\partial t} = 0$ in equation (16). This gives:
p. 306
A solution of this equation is
which is the Maxwell distribution. From what has been said previously it follows that there are infinitely many other solutions, which are not useful however since $\htmlClass{sym sym-f}{f}(x)$ comes out negative or imaginary for some values of $x$. Hence, it follows very clearly that Maxwell's attempt to prove a priori that his solution is the only one must fail, since it is not the only one but rather it is the only one that gives purely positive probabilities, and therefore it is the only one that is useful.