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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, ie, J (B, E)= σ (B)⋅ E+ higher order terms, with the diffusion matrix D (B) at E= 0 given by k B T times the symmetric part σ̃ (B) of the conductivity matrix.