Removing Heat and Conceptual Loops from the Definition of Entropy

Year 2010, Volume: 13 Issue: 2, 67 - 76, 01.06.2010

Enzo Zanchini , Gian Beretta

Abstract

A rigorous and general logical scheme is presented, which provides an operative non-statistical definition of entropy valid also in the nonequilibrium domain and free of the usual conceptual loops and unnecessary assumptions that restrict the traditional definition of entropy to the equilibrium domain. The scheme is based on carefully worded operative definitions for all the fundamental concepts employed, including those of system, state of a system, isolated system, separable system, systems uncorrelated form each other, environment of a system, process and reversible process. The treatment considers also systems with movable internal walls and/or semipermeable walls, with chemical reactions and and/or external force fields, and with small numbers of particles. The definition of entropy involves neither the concept of heat nor that of quasistatic process; it applies to both equilibrium and nonequilibrium states. Simple and rigorous proofs of the additivity of entropy and of the principle of entropy nondecrease complete the logical framework.

Keywords

definition of entropy; Second Law; foundations of thermodynamics

References

• Bennett, C.H., 2008, “The Second Law and Quantum Physics”, pp. 66-79, in Meeting the Entropy Challenge, AIP Conf. Proc. Series, Vol. 1033.
• Beretta, G.P., Gyftopoulos, E.P., Park, J.L., and Hatsopou- los, G.N., 1984, “Quantum Thermodynamics: a New Equa- tion of Motion for a Single Constituent of Matter”, Il Nuovo Cimento B, Vol. 82, pp. 169-191.
• Carath´eodory, C., 1909, “Untersuchungen ueber die Grund- lagen der Thermodynamik”, Math. Ann., Vol. 67, pp. 355- 386.
• Fermi, E., 1937, Thermodynamics, Prentice-Hall.
• Gheorghiu-Svirschevski, S. , 2001a, “Nonlinear Quantum Evolution with Maximal Entropy Production”, Phys. Rev. A, Vol. 63, 022105, pp. 1-15.
• Gheorghiu-Svirschevski, S. , 2001b, “Addendum. Nonlin- ear Quantum Evolution with Maximal Entropy Production”, Phys. Rev. A, Vol. 63, 054102, pp. 1-2.
• Giovannetti, V., Lloyd, S., and Maccone, L., 2003, “Quan- tum Limits to Dynamical Evolution”, Phys. Rev. A, Vol. 67, 052109, pp. 1-8.
• Grmela, M. and ¨Ottinger, H.C., 1997, “Dynamics and Ther- modynamics of Complex Fluids. I. Development of a Gen- eral Formalism”, Phys. Rev. E, Vol. 56, pp.6620-6632.
• Goldstein, S., Lebowitz, J.L., Tumulka, R., and Zangh´ı, N., 2006, “Canonical Typicality”, Phys. Rev. Lett., Vol. 96, 050403, pp. 1-3.
• Gyftopoulos, E.P. and Beretta, G.P., 2005, Thermodynamics. Foundations and Applications, Dover, Mineola (first edition, Macmillan, 1991).
• Hatsopoulos, G.N. and Beretta, G.P., 2008, “Where is the Entropy Challenge?”, in Meeting the Entropy Challenge, AIP Conf. Proc. Series, Vol. 1033, pp. 34-54.
• Hatsopoulos, G.N. and Keenan, J.H., 1965, Principles of General Thermodynamics, Wiley.
• Jou, D., Casas V´azquez, J. and Lebon, G., 2001, Extended Irreversible Thermodynamics, Springer.
• Kim, Y., Yu, R., Kulik, S.P., Shih, Y., and Scully, M.O., 2000, “Delayed Choice Quantum Eraser”, Phys. Rev. Lett., Vol. 84, pp. 1-5.
• Kjelstrup, S. and Bedeaux, D., 2008, Non-equilibrium Ther- modynamics of Heterogeneous Systems, World Scientific Pu. Lloyd, S., 1989, “Use of Mutual Information to Decrease Entropy: Implications for the Second Law of Thermodynam- ics”, Phys. Rev. A, Vol. 39, pp. 5378-5386.
• Lloyd, S., 1997, “Quantum-Mechanical Maxwell’s demon”, Phys. Rev. A, Vol. 56, pp. 3374-3382.
• Lloyd, S., 2008, “The Once and Future Second Law of Ther- modynamics”, pp. 143-157, in Meeting the Entropy Chal- lenge, AIP Conf. Proc. Series, Vol. 1033.
• Maccone, L., 2009, “Quantum Solution to the Arrow-of- Time Dilemma”, Phys. Rev. Lett., Vol. 103, 080401, pp. 1-4. Martyushev, L.M. and Seleznev, V.D., 2006, “Maximum En- tropy Production Principle in Physics, Chemistry and Biol- ogy”, Physics Reports, Vol. 426, pp. 1-45.
• M¨uller, I. and Ruggeri, T., 1998, Rational Extended Thermo- dynamics, Springer.
• Onsager, L., 1931, “Reciprocal Relations in Irreversible Pro- cesses”, Phys. Rev., Vol. 38, pp. 405-426 and Vol. 38, pp. 2265-2279. ¨
• Ottinger, H.C. and Grmela, M., 1997, “Dynamics and Ther- modynamics of Complex Fluids. II. Illustrations of a General Formalism”, Phys. Rev. E, Vol. 56, pp.6633-6655.
• Planck, M., 1927, Treatise on Thermodynamics, translated by A. Oggs from the 7th German edition, Longmans, Green, and Co., London (the first german edition appeared in 1897). Poincar´e, H., 1908, Thermodynamique, Gautier-Villars, Paris.
• Prigogine, I., 1961, Introduction to Thermodynamics of Irre- versible Processes, Interscience, New York.
• Scully, M.O. and Dr¨uhl, K., 1982 “Quantum Eraser: a Pro- posed Photon Correlation Experiment Concerning Observa- tion and Delayed Choice in Quantum Mechanics”, Phys. Rev. A, Vol. 25, pp. 2208-2213.
• Scully, M.O., 2001, “Extracting Work from a Single Thermal Bath via Quantum Negentropy”, Phys. Rev. Lett., Vol. 87, 220601, pp. 1-4.
• Scully, M.O., 2002, “Quantum Afterburner: Improving the Efficiency of an Ideal Heat Engine”, Phys. Rev. Lett., Vol. 88, 050602, pp. 1-4.
• Vilar, J.M.G. and Rubi, M., 2001, “Thermodynamics “Be- yond” Local Equilibrium”, Proc. Natl. Acad. Sci. (U.S.), Vol. 98, 1108111084.
• Zanchini, E., 1986, “On the Definition of Extensive Property Energy by the First Postulate of Thermodynamics”, Found. Phys., Vol. 16, pp. 923-935.
• Zanchini, E., 1988, “Thermodynamics: Energy of Closed and Open Systems”, Il Nuovo Cimento B, Vol. 101, pp. 453- 465.
• Zanchini, E., 1992, “Thermodynamics: Energy of Nonsim- ple Systems and Second Postulate”, Il Nuovo Cimento B, Vol. 107A, pp. 123-139.