Derivation of Ohm’s law in a deterministic mechanical model

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 kBT times the symmetric part {σ} ̃ \~{}{}(B) of the conductivity matrix.