1 92 THE POPULAR SCIENCE MONTHLY 



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 equi- 

 librium. Maxwell's law of final distribution of velocities as determined 

 by Boltzmann's probability coefficient is, therefore, a sufficient condi- 

 tion 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 dis- 

 tribution, H varies with the time and is found to be constantly diminish- 

 ing 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, 128 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." 129 Applying similar 

 reasoning to the material universe, Boltzmann finds that the following 

 assumptions are possible : either the whole universe is in a highly im- 

 probable (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 



12S " 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 cer- 

 tain purpose, but haphazard governs freely the probability that H decreases ia 

 always greater than it increases." Boltzmann, Nature, 1894-5, LI., 414. 



129 Tr. Connect. Acad., III., 229. 



