such unusual phases of matter as radiation in rarefied gases, where the system has no temperature at all, because its internal motions have not settled down to a definite average. Helmholtz's dynamic proof of the second law assumes the existence of cyclic systems with reversible circular motions, like those of the gyroscope or the governor of a steam engine, in other words it assumes matter to be made of rotational or gyrostatic stresses in the ether. Gibbs's "Elementary Principles of Statical Mechanics" (1903)[1] is based upon no assumptions whatever except that the systems involved are mechanical, obeying the equations of motion of Lagrange and Hamilton. "One is building on insecure foundations," he says, "who rests his work on hypotheses concerning the constitution of matter," and his statistics deal, not with the behavior of gas molecules in isolated systems, but with large averages of vast ensembles of systems of the same kind (solid, liquid or gas), "differing in the configurations and velocities which they have at any given instant, and differing not merely infinitesimally, but it may be so as to embrace every conceivable combination of configuration and velocities." The problem is, given the distribution of these ensembles in phase (i. e., in regard to configuration and velocities) at some one time, to find their distribution at any required time. To solve this problem Gibbs establishes a fundamental equation of statistical mechanics, which gives the rate of change of the systems in regard to distribution in phase. A particular case of this equation gives the condition for statistical equilibrium or permanent distribution in phase. Integration of the equation in the general case gives certain constants relating to the extent, density and probability of distribution of the systems in phase, which Gibbs interprets as the principles of conservation of "extension in phase," of "density in phase," and of "probability in phase." Boltzmann found that when the gas molecules have more than two degrees of freedom, the equations can not be integrated and further progress is impossible. He got around this difficulty by using Jacobi's "method of the last multiplier," which integrates the equations of motion. Gibbs found that the principle of "conservation of extension-inphase," supplies such a Jacobian multiplier, "if we have the skill or good fortune (he says) to perceive that the multiplier will make the first member of the equation an exact differential." Boltzmann's probability coefficient is used as the index of the canonical distribution of ensembles, and when the exponent of this coefficient is zero, the latter becomes unity, producing a distribution in phase called "microcanonical," in which all the systems in the ensemble have the same energy, as in Maxwell's "state." After demonstrating the possibility of irreversible phenomena in the various ensembles, and after a careful study of their behavior when isolated, subjected to external forces or to
- ↑ "Yale Bicentennial Publications," 1903. Translated into German by Ernst Zermelo, Leipzig, 1905.
