energies," Maxwell claims, "is also the only form." At this point the work of Boltzmann becomes of central importance, especially on account of its profound influence on the later works of Gibbs. In Boltzmann's application of probabilities to Maxwell's problem, the starting point or initial stage of any sequence of events is called a "highly improbable one," because its certainty decreases the more the events proceed to some final or "most probable" state. For example, the blowing up of the Maine is to us a moral or mathematical certainty, but it may not be so aeons hence, while its predisposing or exciting causes are even now "highly improbable" in that we know nothing positive about them. When a gas is brought into a new physical state, its initial stage is, in Boltzmann's argument, a highly improbable one from which the system of molecules will continually hasten towards successive states of greater probability until it finally attains the most probable one, or Maxwell's state of equilibrated partition of energy and thermal equilibrium. Maxwell's law of final distribution of velocities as determined by Boltzmann's probability coefficient is, therefore, a sufficient condition for thermal equilibrium, and Boltzmann found that the entropy of any state of gas molecules is proportional to the logarithm of the probability of its occurrence; or as Larmor puts it, the principle that the trend of an isolated system is towards states for which the entropy continually increases is analogous to the principle that the general trend of a system of molecules is through a succession of states whose intrinsic probability of occurrence continually increases. As a measure of the degree of variation of the gas molecules from Maxwell's state, Boltzmann introduces a function H such that, as the distribution of molecular velocities constantly tends toward the most probable distribution, H varies with the time and is found to be constantly diminishing in value. The necessary condition for thermal equilibrium is, therefore, that H should irreversibly attain a minimum value. Thus Boltzmann's "minimum theorem" becomes, like the Clausius doctrine of maximum entropy, a theorem of extreme probability,[1] or to quote the aphorism of Gibbs which Boltzmann chose as a motto for his Gastheorie: "The impossibility of an uncompensated decrease of entropy seems to be reduced to an improbability."[2] Applying similar reasoning to the material universe, Boltzmann finds that the following assumptions are possible: either the whole universe is in a highly improbable (i. e. initial) state, or, as the facts of physical astronomy would seem to indicate, the part of it known to us is in a state of thermal equilibrium, with certain districts, such as the earth we live
- ↑ "It can never be proved from the equations of motions alone, that the minimum function H must always decrease. It can only be deduced from the laws of probability, that if the initial state is not specially arranged for a certain purpose, but haphazard governs freely the probability that H decreases is always greater than it increases." Boltzmann, Nature, 1894-5, LI., 414.
- ↑ Tr. Connect. Acad., III., 229.
