Transport properties of the Lorentz gas: Fourier's law

We investigate the stationary nonequilibrium (heat transporting) states of the Lorentz gas. This is a gas of classical point particles moving in a region gL containing also fixed (hard sphere) scatterers of radius R. The stationary state considered is obtained by imposing stochastic boundary conditions at the top and bottom of Λ, i.e., a particle hitting one of these walls comes off with a velocity distribution corresponding to temperatures T₁ and T₂ respectively, T₁ <T₂. Letting ρ be the average density of the randomly distributed scatterers we show that in the Boltzmann-Grad limit, ρ→∞, R→ 0 with the mean free path fixed, the stationary distribution of the Lorentz gas converges in the L¹-norm to the stationary distribution of the corresponding linear Boltzmann equation with the same boundary conditions. In particular, the steady state heat flow in the Lorentz gas converges to that of the linear Boltzmann equation, which is known to behave as (T₂-T₁)/L for large L, where L is the distance from the bottom to the top wall: i.e., Fourier's law of heat conduction is valid in the limit. The heat flow converges even in probability. Generalizations of our result for scatterers with a smooth potential as well as the related diffusion problem are discussed.