Derivation of Ohm’s law in a deterministic mechanical model

Abstract

We study the Lorentz gas in small external electric and magnetic fields, with the particle kinetic energy held fixed by a Gaussian ‘‘thermostat.’’ Starting from any smooth initial density, a unique stationary, ergodic measure is approached for times t→∞. The steady-state electric current J(B,E) is given by a Kawasaki formula and the entropy production J⋅E/T, with T the ‘‘temperature,’’ is equal to both the asymptotic decay rate of the Gibbs entropy and minus the sum of the Lyapunov exponents. The Einstein and Kubo formulas hold, i.e., J(B,E)=σ(B)⋅E + higher order terms, with the diffusion matrix D(B) at E=0 given by ${\mathit{k}}_{\mathit{B}}$T times the symmetric part σ̃(B) of the conductivity matrix.