Boltzmann’s Entropy and Time’s Arrow

SEP 01, 1993

Given that microscopic physical lows are reversible, why do all macroscopic events have a preferred time direction? Boltzmann’s thoughts on this question have withstood the test of time.

DOI: 10.1063/1.881363

Joel L. Lebowitz

Given the success of Ludwig Boltzmann’s statistical approach in explaining the observed irreversible behavior of macroscopic systems in a manner consistent with their reversible microscopic dynamics, it is quite surprising that there is still so much confusion about the problem of irreversibility. (See figure 1.) I attribute this confusion to the originality of Boltzmann’s ideas: It made them difficult for some of his contemporaries to grasp. The controversies generated by the misunderstandings of Ernst Zermelo and others have been perpetuated by various authors. There is really no excuse for this, considering the clarity of Boltzmann’s later writings. Since next year, 1994, is the 150th anniversary of Boltzmann’s birth, this is a fitting moment to review his ideas on the arrow of time. In Erwin Schrödinger’s words, “Boltzmann’s ideas really give an understanding” of the origin of macroscopic behavior. All claims of inconsistencies that I know of are, in my opinion, wrong; I see no need for alternate explanations. for further reading I highly recommend Boltzmann’s works as well as references 2–7. (See also PHYSICS TODAY, January 1992, page 44, for a marvelous description by Boltzmann of his visit to California in 1906.)

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References

1. L. Boltzmann, Ann. Phys. (Leipzig) 57, 773 (1896); https://doi.org/ANPYA2
translated and reprinted in S. G. Brush, Kinetic Theory 2, Pergamon, Elmsford, N.Y. (1966).
2. R. Feynman, The Character of Physical Law, MIT P., Cambridge, Mass. (1967), ch. 5.
R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, Addison‐Wesley, Reading, Mass. (1963), sections 46‐3, 4, 5.
3. O. Penrose, Foundations of Statistical Mechanics, Pergamon, Elmsford, N. Y. (1970), ch. 5.
4. R. Penrose, The Emperor’s New Mind, Oxford U.P., New York (1990), ch. 7.
5. D. Ruelle, Chance and Chaos, Princeton U.P., Princeton, N.J. (1991), ch. 17, 18.
6. J. L. Lebowitz, Physica A 194, 1 (1993).https://doi.org/PHYADX
7. O. Lanford, Physica A 106, 70 (1981).https://doi.org/PHYADX
8. H. Spohn, Large Scale Dynamics of Interacting Particles, Springer‐Verlag, New York (1991).
A. De Masi, E. Presutti, Mathematical Methods for Hydrodynamic Limits, Lecture Notes in Math. 1501, Springer‐Verlag, New York (1991).
J. L. Lebowitz, E. Presutti, H. Spohn, J. Stat. Phys. 51, 841 (1988).https://doi.org/JSTPBS
9. E. Schrödinger, What Is Life? and Other Scientific Essays, Doubleday Anchor Books, New York (1965), section 6.
10. E. L. Hahn, Phys. Rev. 80, 580 (1950). https://doi.org/PHRVAO
See also S. Zhang, B. H. Meier, R. R. Ernst, Phys. Rev. Lett. 69, 2149 (1992).https://doi.org/PRLTAO
11. D. Levesque, L. Verlet, J. Stat. Phys., to appear.
12. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge U.P., New York (1987).
13. D. Dürr, S. Goldstein, N. Zanghi, J. Stat. Phys. 67, 843 (1992). https://doi.org/JSTPBS
M. Gell‐Mann, J. B. Hartle, in Complexity, Entropy and the Physics of Information, W. H. Zurek, ed., Addison‐Wesley, Reading, Mass. (1990).
14. Y. Aharonov, P. G. Bergmann, J. L. Lebowitz, Phys. Rev. B 134, 1410 (1964).https://doi.org/PRVBAK

More about the authors

Joel L. Lebowitz, Rutgers University, New Brunswick, New Jersey.