# The approach towards equilibrium in Lanford’s theorem

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## Abstract

This paper develops a philosophical investigation of the merits and faults of a theorem by Lanford (1975), Lanford (Asterisque *40*, 117–137, 1976), Lanford (Physica *106A*, 70–76, 1981) for the problem of the approach towards equilibrium in statistical mechanics. Lanford’s result shows that, under precise initial conditions, the Boltzmann equation can be rigorously derived from the Hamiltonian equations of motion for a hard spheres gas in the Boltzmann-Grad limit, thereby proving the existence of a unique solution of the Boltzmann equation, at least for a very short amount of time. We argue that, by establishing a statistical *H*-theorem, it offers a prospect to complete Boltzmann’s combinatorial argument, without running against the objections which plug other typicality-based approaches. However, we submit that, while recovering the irreversible approach towards equilibrium for positive times, it fails to predict a monotonic increase of entropy for negative times, and hence it yields the wrong retrodictions about the past evolution of a gas.

## Keywords

Approach to equilibrium; Statistical mechanics; Boltzmann## Notes

### Acknowledgments

The author is grateful to Jos Uffink for several inspiring discussions on the topic. He also wishes to thank Bob Batterman, Harvey Brown, Roman Frigg, John Norton and Charlotte Werndl for helpful comments.

## References

- Batterman, R.W. In
*The Tyranny of Scales, forthcoming in The Oxford Handbook of Philosophy of Physics.*London: Oxford University Press.Google Scholar - Boltzmann, L. (1872). Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen, Wiener Berichte 66, 275–370; in Boltzmann (1909) vol.I. pp. 316–402.Google Scholar
- Boltzmann, L. (1877). Über die beziehung dem zweiten Haubtsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über das Wärmegleichgewicht.
*Wiener Berichte*,*76*, 373–435.Google Scholar - Butterfield, J. (2011). Less is Different: Emergence and Reduction Reconciled, in.
*Foundations of Physical*,*41*, 1065–1135.CrossRefGoogle Scholar - Cercignani, C. (1972). On the Boltzmann equation for rigid spheres, in.
*Transport Theory and Statistical Physics*,*2*, 211.CrossRefGoogle Scholar - Earman, J. (2006). The past-hypothesis: not even false, in.
*History and Philosophy of Modern Physics*,*37*, 399–430.CrossRefGoogle Scholar - Frigg, R. (2008). A Field Guide to Recent Work on the Foundations of Statistical Mechanics. In Rickles, D. (Ed.)
*The Ashgate Companion to Contemporary Philosophy of Physics*, (pp. 99–196). London. Ashgate, 2008.Google Scholar - Frigg, R. (2011). Why Typicality Does Not Explain the Approach to Equilibrium. In Suarez, M. (Ed.),
*Probabilities, Causes and Propensities in Physics*, (pp. 77–93). Dordrecht: Springer. Synthese Library.CrossRefGoogle Scholar - Goldstein, S. (2001). Boltzmann’s approach to statistical mechanics, in Chance in Physics: Foundations and Perspective In Bricmond, J. et al. (Eds.),
*Lecture Notes in Physics 574, p.39-54*. Berlin: Springer.Google Scholar - Goldstein, S. (2012). Typicality and Notions of Probability in Physics, in Probability in Physics.
*The Frontiers Collection*, 59–71.Google Scholar - Grad, H. (1949). On the kinetic theory of rarefied gases, in.
*Communications Pure and Applied Mathematics*,*2*, 331–407.CrossRefGoogle Scholar - Hemmo, M., & Shenker, O. (2012). Measure over Initial Conditions, in Probability in Physics.
*The Frontiers Collection*, 87–98.Google Scholar - Illner, R., & Shinbrot, M. (1984). The Boltzmann equation: Global existence for a rare gas in a infinite vacuum, in .
*Communications of Mathematics Physiological*,*95*, 217–226.CrossRefGoogle Scholar - Illner, R., & Pulvirenti, M. (1986). Global validity of the Boltzmann equation for a two-dimensional rare gas in a vacuum, in.
*Communications of Mathematics Physiological*,*105*, 189–203.CrossRefGoogle Scholar - Illner, R., & Pulvirenti, M. (1989). Global validity of the Boltzmann equation for a two-dimensional and three-dimensional rare gas in vacuum: Erratum and improved result, in.
*Communications of Mathematics Physiological*,*121*, 143–146.CrossRefGoogle Scholar - Lebowitz, J.L. (1981). Microscopic Dynamics and Macroscopic Laws. In Horton, C.W.Jr., Reichl, L.E., Szebehely, V.G. (Eds.)
*Long-Time Prediction in Dynamics, presentations from the workshop held in March 1981 in Lakeway, TX*, (pp. 220–233). New York: John Wiley.Google Scholar - Loschmidt, J. (1876). Über die Zustand des Wärmegleichgewichtes eines Systems von Körpern mit Rücksicht auf die Schwerkraft in
*Wiener Berichte*, 73: 128, 366.Google Scholar - Lanford, O.E. (1975). Time evolution of large classical systems In Moser, J. (Ed.)
*Dynamical Systems, Theory and Applications*, (pp. 1–111). Berlin: Springer. Lecture Notes in Theoretical Physics Vol.38.CrossRefGoogle Scholar - Lanford, O.E. (1976). On the derivation of the Boltzmann equation, in.
*Asterisque*,*40*, 117–137.Google Scholar - Lanford, O.E. (1981). The hard sphere gas in the Boltzmann-Grad limit, in.
*Physica*,*106A*, 70–76.CrossRefGoogle Scholar - King, F.G. (1975). Ph.D, Dissertation, Department of Mathematics, University of California at Berkeley.Google Scholar
- Norton, J. (2012). Approximation and Idealization: Why the Difference Matter, in.
*Philosophy of Science*,*79*, 207–232.CrossRefGoogle Scholar - Pitowsky, I. (2012). Typicality and the Role of the Lebesgue Measure in Statistical Mechanics, in Probability in Physics.
*The Frontiers Collection*, 41–58.Google Scholar - Price, H. (1996). In
*Times arrow and Archimedes point*. New York: Oxford University Press.Google Scholar - Spohn, H. (1991). In
*Large Scale Dynamics of Interacting Particles*. Heidelberg: Springer Verlag.CrossRefGoogle Scholar - Spohn, H. (1997). Loschmidt’s reversibility argument and the H-theorem. In Fleischhacker, W., & Schlonfeld, T. (Eds.),
*Pioneering Ideas for the Physical and Chemical Sciences, Loschmidt’s Contributions and Modern Developments in Structural Organic Chemistry, Atomistics and Statistical Mechanics*, (pp. 153–158). New York: Plenum Press.Google Scholar - Uffink, J. (2007). Compendium of the foundations of classical statistical physics. In Butterfield, J., & Earman, J. (Eds.),
*Handbook for the Philosophy of Physics*. Amsterdam: Elsevier.Google Scholar - Zermelo, E. (1896). Über einen Satz der Dynamik und die mechanische Wärmetheorie.
*Annalen der Physik*,*57*, 485–494.CrossRefGoogle Scholar