Foundations of Physics

, Volume 45, Issue 4, pp 404–438 | Cite as

Lanford’s Theorem and the Emergence of Irreversibility

  • Jos Uffink
  • Giovanni ValenteEmail author
Original Paper


It has been a longstanding problem to show how the irreversible behaviour of macroscopic systems can be reconciled with the time-reversal invariance of these same systems when considered from a microscopic point of view. A result by Lanford (Dynamical systems, theory and applications, 1975, Asterisque 40:117–137, 1976, Physica 106A:70–76, 1981) shows that, under certain conditions, the famous Boltzmann equation, describing the irreversible behaviour of a dilute gas, can be obtained from the time-reversal invariant Hamiltonian equations of motion for the hard spheres model. Here, we examine how and in what sense Lanford’s theorem succeeds in deriving this remarkable result. Many authors have expressed different views on the question which of the ingredients in Lanford’s theorem is responsible for the emergence of irreversibility. We claim that these interpretations miss the target. In fact, we argue that there is no time-asymmetric ingredient at all.


Statistical mechanics Irreversibility Time-reversal invariance Lanford 



The authors would like to thank Herbert Spohn for extensive comments on the technical and conceptual subtleties of Lanford’s theorem, as well as two anonymous referees for helpful remarks.


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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of MinnesotaMinneapolisUSA
  2. 2.PittsburghUSA

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