Communications in Mathematical Physics

, Volume 203, Issue 3, pp 667–706 | Cite as

Sharp Entropy Dissipation Bounds and Explicit Rate of Trend to Equilibrium for the Spatially Homogeneous Boltzmann Equation

  • G. Toscani
  • C. Villani


We derive a new lower bound for the entropy dissipation associated with the spatially homogeneous Boltzmann equation. This bound is expressed in terms of the relative entropy with respect to the equilibrium, and thus yields a differential inequality which proves convergence towards equilibrium in relative entropy, with an explicit rate. Our result gives a considerable refinement of the analogous estimate by Carlen and Carvalho [9, 10], under very little additional assumptions. Our proof takes advantage of the structure of Boltzmann's collision operator with respect to the tensor product, and its links with Fokker–Planck and Landau equations. Several variants are discussed.


Entropy Tensor Product Boltzmann Equation Additional Assumption Relative Entropy 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • G. Toscani
    • 1
  • C. Villani
    • 2
  1. 1.Department of Mathematics, University of Pavia, via Abbiategrasso 209, 27100 Pavia, Italy.¶E-mail: toscani@dragon.ian.pv.cnr.itIT
  2. 2.École Normale Supérieure, DMI, 45 Rue d'Ulm, 75230 Paris Cedex 05, France.¶E-mail: villani@dmi.ens.frFR

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