Without giving a full proof of the law we will give the main steps which lead to its analytical statement, in so doing following the presentation given by HANS LORENZ on pp. 526-529 of his "Technische Wärmelehre," and will then point out its main features and consequences.
We suppose the gas to contain in a unit of volume molecules each possessing a different velocity and direction. Let there be a system of three co-ordinate axes, . A fraction of the total number of molecules will possess a velocity in the direction, whose values lie between . The number of molecules which at the same time possess velocities in the direction, lying between , will be , since no preference can be given to either the or direction. Similarly and finally the number of molecules whose velocity co-ordinates concurrently lie between , and will be represented by the product where the only thing known about function is that the sum of the fractions extended over all the values of must = unity, so that
Now if we suppose all of the velocities of the molecules to be laid off as vectors from a pole , the three directions will constitute about a perfectly arbitrary system of co-ordinates in which designates a volume element[19] and the velocity of a molecule is given by
Now if we put through the origin another system of co-ordinates of which one axis coincides with any arbitrarily chosen velocity , then in this axis the above-found product will be because the two other co-ordinates (outside of ) of the volume element will equal zero and no preference can be given to any direction. Then it can be shown that the form of the function is given by where , are integration constants which stand in a certain relation to each other, namely,
Further mathematical manipulation eliminates the different velocity directions and gives for the number of molecules possessing absolute velocities between and .
This expression (5) is called MAXWELL'S Law of Distribution; it is identical with that found for the probable distribution of error in a great number of observations and is graphically shown by the following figure, with maximum number of molecules for velocity . The constant is therefore a velocity from which most of the molecules differ but little. The development shows that this self-same distribution exists for every straight line that can be drawn in the volume under consideration.
BOLTZMANN, in his Gas Theorie, has shown that for such a state the "number of complexions" is a maximum, that is, the entropy is then a maximum.
From the preceding expression (5) follows that the kinetic energy of the system is where is the mean square of the velocity. Integration gives
It is known that the measure of temperature is the mean kinetic energy of the individual molecule and not simply the mean square of its velocity, and we possess here therefore a perfectly precise definition of temperature. We see also from (7) that temperature in a particular gas is directly proportional to either or .
MAXWELL further shows without any assumption as to the nature of the molecules, or the forces acting between them, that the derived law of distribution is valid for any gas mixture, but that is it modified when the gas is exposed to the action of external forces.
BOLTZMANN found (Wien. Akad. Sitzber. LXXII B, 1875, p. 443) for monatomic gases that in spite of the effect of external forces (a) the velocity of any molecule is equally likely to have any direction whatever, (b) the velocity distribution in any element of space is exactly like that in a gas of equal density and temperature upon which no external forces act, the effect of the external forces consisting only in varying the density from place to place as in hydrodynamics.
BOLTZMANN says this "normal" state is permanent (stationary) for given external conditions because magnitude does not vary; such a normal state has many configurations, but all agree in having same number of complexions.
Also, "MAXWELL'S Velocity Distribution is not a state which assigns to each molecule a particular place (locus) and a particular velocity, which are reached say by the locus and velocity of each molecule asymptotically approaching said assigned locus and velocity. With a finite number of molecules MAXWELL'S state will never be exactly but only approximately realized. MAXWELL'S velocity is not a singular one which is confronted by an immense number of non-Maxwellian velocity distributions. On the contrary, among the immense number of possible velocity distributions by far the greater number possess the characteristics of the MAXWELL velocity distribution."
MAX PLANCK (Festschrift, p. 113) lucidly dwells on thermal equilibrium, entropy and temperature, as follows:
"The mechanical significance of the temperature idea is most closely connected with the mechanical significance of entropy, for the two are connected by . By answering one of these questions we at the same time settle the other."
In the earlier days interest was naturally centered in the directly measurable magnitude temperature and entropy appeared as a more complicated idea which was to be derived from the former. Nowadays this relation is rather reversed and the prime question is to first explain entropy mechanically and this will then define temperature. The reason for this change of attitude is that in all such explanatory efforts to present Thermodynamics mechanically and give temperature a complete mechanical definition it is necessary to come back to the peculiarities of "thermal equilibrium." But the full significance of this equilibrium conception is only to be reached from the standpoint of irreversibility. For thermal equilibrium can only be defined as the final state toward which all irreversible processes strive. In this way the question as to temperature leads necessarily to the nature of irreversibility and this in turn is solely founded on the existence of the entropy function. This magnitude is therefore the primary, general conception which is significant for all kinds of states and changes of state, while temperature emerges from this with the help of the special condition of thermal equilibrium, in which condition the entropy attains its maximum.
[19]In MAXWELL'S distribution the molecules are assumed to be uniformly scattered throughout the unit volume; it is the velocities only that are variously distributed in the different elementary regions. To realize the haphazard character (necessary in Calculus of Probabilities) of the motions of the molecules, we must bear in mind that each of the molecules in the unit volume has a different velocity and direction; here no direction has preference over another, i.e., one direction of a molecule is as likely as another. Here at first we write expression for the number of molecules whose velocities parallel to the co-ordinate axes are respectively confined between the velocity limits: To find the number of molecules thus limited the procedure given above is essentially as follows: Expressed as a fraction = probability of velocities parallel to axis having values between and expressed as a number = number of molecules having such velocities between the assigned limits; similarly, = probability of velocities parallel to axis having velocities between . As these are two independent sets of velocities, the probability of their concurrence is the product and the number of molecules thus concurring is equal to . Similarly, the number of molecules concurrently possessing velocities parallel to each of the three axes is The problem is simpler in this Maxwellian case than in the more general case of any state of the body in which there is an unequal distribution in space of the molecules.