Abstract

Similar to the nonlinear Boltzmann equation of gas dynamics, the linear Boltzmann equation satisfies an H Theorem. This means that there exist entropy functionals of the solution that do not increase in time. Among other things, this implies time-irreversibility of the solution. Contrary to the nonlinear Boltzmann equation, however, a rigorous derivation of the linear Boltzmann equation (more precisely the equation describing a Lorentz gas) is available. This derivation, due to Gallavotti (1972), is interesting for several reasons: (i) It starts from a reversible microscopic system. More precisely, the derivation makes active use of reversibility. (ii) Inspection of the derivation shows where entropy dissipation comes in. (iii) It is a strange derivation in that we derive a solution formula and subsequently identify an equation for that solution formula.

In the lecture, we prove the H Theorem in the linear case and sketch the derivation of the Lorentz gas equation, pointing out where entropy decay emerges.