A gradient flow approach to the Boltzmann equation

Abstract

We show that the spatially homogeneous Boltzmann equation evolves as the gradient flow of the entropy with respect to a suitable geometry on the space of probability measures which takes the collision process into account. This gradient flow structure allows to give a new proof for the convergence of Kac's random walk to the homogeneous Boltzmann equation, exploiting the stability of gradient flows.

Final version as to appear in the journal. Modifications to previous version include generalisation of the main results to a class of soft potentials, discussion of generalised gradient structures for the Boltzmann equation

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