European Journal for Philosophy of Science

, Volume 4, Issue 3, pp 309–335 | Cite as

The approach towards equilibrium in Lanford’s theorem

  • Giovanni ValenteEmail author


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.


Approach to equilibrium; Statistical mechanics; Boltzmann 



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.


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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Philosophy DepartmentUniversity of PittsburghPittsburghUSA

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