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analogous approximate average index average value canonical ensemble Chapter coefficient of probability condition conservation of extension-in-phase considered constants of motion coordinates and momenta corresponding degrees of freedom denote density density-in-phase determined distributed in phase element ensemble of systems entropy evidently expressed extension-in-configuration extension-in-velocity external bodies external coordinates finite value forces formula give given limits grand ensemble identical independent index of probability infinite infinitesimal integral equations kinetic energy linear functions microcanonical ensemble modulus multiple integral number of degrees number of systems order of magnitude petit ensemble potential energy principle of conservation probability of phase probability-coefficient quadratic function quantities regarded relating represented respect specific phases statistical equilibrium statistical mechanics system of coordinates temperature theorem thermodynamic equation thermodynamic equilibrium thermodynamics third ensemble tion vanishes variables velocities whole ensemble whole number zero
Page x - Certainly, one is building on an insecure foundation, who rests his work on hypotheses concerning the constitution of matter. Difficulties of this kind have deterred the author from attempting to explain the mysteries of nature, and have forced him to be contented with the more modest aim of deducing some of the more obvious propositions relating to the statistical branch of mechanics.
Page iii - University, a series of volumes has been prepared by a number of the Professors and Instructors, to be issued in connection with the Bicentennial Anniversary, as a partial indication of the character of the studies in which the University teachers are engaged.
Page iii - Sons, of New York. With the approval of the President and Fellows of Yale University, the series has been prepared by a number of the Professors and Instructors, to be issued in connection with the Bicentennial Anniversary, as a partial indication of the studies in which the University teachers are engaged. The list of volumes includes some of a special and technical nature, others of a more general character.
Page x - Difficulties of this kind have deterred the author from attempting to explain the mysteries of nature, and have forced him to be contented with the more modest aim of deducing some of the more obvious propositions relating to the statistical branch of mechanics. Here, there can be no mistake in regard to the agreement of the hypotheses with the facts of nature, for nothing is assumed in that respect. The only error into which one can fall is the want of agreement between the premises and the conclusions,...
Page x - It is well known that while theory would assign to the gas six degrees of freedom per molecule, in our experiments on specific heat we cannot account for more than five. Certainly one is building on an insecure foundation, who rests his work on hypotheses concerning the constitution of matter.
Page 165 - ANALOGIES. IF we wish to find in rational mechanics an a priori foundation for the principles of thermodynamics* we must seek mechanical definitions of temperature and entropy. The quantities thus defined must satisfy (under conditions and with limitations which again must be specified in the language of mechanics) the differential equation de...
Page ix - Moreover, we avoid the gravest difficulties when, giving up the attempt to frame hypotheses concerning the constitution of material bodies, we pursue statistical inquiries as a branch of rational mechanics.
Page vii - We may imagine a great number of systems of the same nature, but differing in the configurations and velocities which they have at a given instant, and differing not merely infinitesimally, but it may be so as to embrace every conceivable combination of configuration and velocities.
Page 166 - But it should be distinctly stated that, if the results obtained whe1. the numbers of degrees of freedom are enormous coincide sensibly with the general laws of thermodynamics, however interesting and significant this coincidence may be, we are still far from having explained the phenomena of nature with respect to these laws. For, as compared with the case of nature, the systems which we have considered are of an ideal simplicity.