Page:Popular Science Monthly Volume 75.djvu/199

Rh the spheres of one another's influence, Gibbs finds that the processes of statistical mechanics are to all human perception analogous to those of thermodynamics, the familiar formulæ of which appear, as Bumstead puts it, "almost spontaneously, as it seems from the consideration of purely mechanical systems.". The differential equation relating to average values in the ensemble is found to correspond with the fundamental equation of thermodynamics; the modulus of distribution of ensembles turns out to be analogous to the temperature, while the average index of probability in phase is the analogue of the entropy with reversed sign, and being a minus quantity, is found to decrease just as entropy increases. Most of the objections filed against Gibbs's statistical demonstration, turn upon the fact that it is difficult, perhaps impossible, to apply the reversible dynamics of ideal, frictionless systems to the spontaneous irreversible phenomena of nature without making some physical assumptions. "Entropy," Burbury objects, "may, for all that appears, either increase or diminish in a system which is dynamically reversible. This then can not be strictly applied to an irreversible process." Gibbs has met these objections fairly. "Our mathematical fictions," he says, to quote Burbury's paraphrase of his argument, "give us no information whether the distribution of phases is towards uniformity or away from it. Our experience with the real world, however, teaches us that it is towards uniformity." All actual mechanical systems are, as Gibbs pointed out long before, in reality thermodynamic, and it seems odd that the critics who rejected Boltzmann's proof, because it did not agree with the facts of nature, should now, for a logical quibble, take exception to Gibbs's because it does. It has been predicted that future truth in physical science will often be found in the sixth place of decimals, for not everything in nature works out according to specifications. We can, if we choose, regard mathematics as a metaphysical diversion or employ it practically as a means of interpreting the physical facts of nature, empirically ascertained by man. In these matters, says Gibbs elsewhere, "Nature herself takes us by the hand and leads us along by easy steps as a mother teaches her child to walk," and he would have agreed with Langley that man may put questions to nature if he will, but is in no position to dictate her answers to them. Nature seems très femme in this respect, especially in regard to mathematical fictions, that is, ideal or limiting cases devised by the finite mind of man. Like any other human