Microscopic Aspects of Entropy and the Statistical Foundations of Nonequilibrium Thermodynamics
The aim of this paper is to show how recent work in nonequilibrium statistical mechanics leads to a mechanical interpretation of the second law of thermodynamics.
First, we recall Boltzmann’s definition of entropy and the difficulties associated with it (extension to dense systems, Loschmidt’s paradox, etc.). On a simple model (McKean—Kac’s model), we show that Boltzmann’s definition is certainly not valid for all possible evolutions but that it is, nevertheless, possible to find out a Liapounoff function which is positive and can only decrease as a result of the time evolution. This leads to a microscopic definition of entropy for this model. The entropy defined in this way reduces to Boltzmann’s entropy close to equilibrium.
We then give a short summary of the work in nonequilibrium statistical mechanics of the Brussels school and emphasise the fact that irreversibility appears as a symmetry breaking of the time reversal invariance which results from the appearance of dynamic operators which are even in the Liouville—von Neumann operator L. This symmetry breaking is a consequence of causality when applied to large systems formed by many interacting degrees of freedom. The introduction of nonunitary (called star unitary) transformation leads to a representation in which the time change of the distribution function is split into two parts, one odd and one even in the Liouville—von Neumann operators. The first corresponds to reversible changes, the second to irreversible ones. The correspondence with the second law is therefore complete. We may now introduce a Liapounoff function which leads to a general microscopic model of entropy. This definition is valid whatever the initial conditions. No additional probabilistic arguments are needed to derive the increase of entropy. On the contrary the probabilistic interpretation when valid is a consequence of dynamics. We also show that there is no Loschmidt paradox associated with our new definition of entropy, and that it reduces to Boltzmann’s definition for systems close to equilibrium.
In the concluding section we stress the fact that our approach leads to a general expression for the entropy production independently of any assumption of closeness to equilibrium. The perspectives opened by this new development which unifies dynamics and thermodynamics are briefly discussed.
Unable to display preview. Download preview PDF.
- 2.I. Prigogine, C. George, F. Henin and L. Rosenfeld. Chemica Scripta, 4 (1973), 5Google Scholar
- 3.I. Prigogine. The Development of the Physicist’s Conception of Nature in the Twentieth Century, ed. by J. Mehra, Reidel, Dordrecht-Holland (1973), p. 697, 561Google Scholar
- 4.I. Prigogine. Proceedings of the International Symposium 100 Years Boltzmann Equation, ed. by E. G. D. Cohen and W. Thirring, Acta Physica Austriaca, Suppl. X, Springer Verlag (1973), p. 401Google Scholar
- 6.M. Planck. Vorlesungen über Thermodynamik, Berlin and Leipzig, Walter de Gruyter (1930), p. 83Google Scholar
- 7.P. Glansdorff and I. Prigogine. Structure, Stability and Fluctuations, Wiley-Interscience (1971)Google Scholar
- 8.L. Boltzmann. Weitere studien über das wärmegleichgewicht unter gasmolekülen. Wien Ber., 66 (1872), 275. See Wissenschaftliche Abhandlungen, Verlag von Johann Ambrosius Barth, Leipzig, 1 (1909)Google Scholar
- 9.L. Boltzmann. Der zweite hauptsatz der mechanische wärmetheorie. In Populäre Schriften, Verlag von Johan Ambrosius Barth, Leipzig (1919), p. 25Google Scholar
- 10.J. O. Hirschfelder, C. F. Curtiss and R. B. Bird. The Molecular Theory of Gases and Liquids, Wiley, New York (1959)Google Scholar
- 12.A. Bellemans and J. Orban. Phys. Lett., 24A (1967), 620Google Scholar
- 13.J. Loschmidt. Wien Ber., 73 (1876), 139Google Scholar
- 14.P. and T. Ehrenfest. Begriffliche grundlagender statistischen auffasung der mechanik. Encycl. Math. Wissenschafter, 4 (1911), 4Google Scholar
- 15.M. Kac. Foundations of kinetic theory. Proc. Third Berkeley Symp. on Math. Stat. and Prob., vol. III (1956), 171Google Scholar
- 16.M. Kac. Probability and Related Topics in Physical Sciences, Interscience, New York (1959)Google Scholar
- 19.F. Herrin and I. Prigogine, P.N.A.S., to appear 1974; F. Henin, Physica, to appear 1974; F. Henin, Ac. Roy. Belg., Bull. Cl. Sc. to appear 1974Google Scholar
- 20.C. George, I. Prigogine and L. Rosenfeld. Koningl. Dansk. Vid. Mat-fys. Medd., 38 (1972), 12Google Scholar
- 21.I. Prigogine, Non Equilibrium Statistical Mechanics, Wiley-Interscience, New York (1962)Google Scholar
- 23.M. Baus. Acad. Roy. Belg. Bull. Cl. Sc., 53 (1967), 1291, 1332, 1352Google Scholar
- 26.I. Prigogine and P. Résibois. Atti del Simposio Lagrangiano, Academia delle Scienze, Torino (1964)Google Scholar
- 29.M. Baus. Thesis, Free University of Brussels (1968)Google Scholar
- 31.G. Stey. Physica, 68 (1973), 273Google Scholar
- 33.I. Prigogine, C. George, A. Grecos. P.N.A.S. to appear 1974Google Scholar
- 35.A. Grecos and I. Prigogine. P.N.A.S., 69 (1972), 1629Google Scholar