Kinetic approach to the Gaussian thermostat in a dilute sheared gas in the thermodynamic limit

A dilute gas of particles with short range interactions is considered in a shearing stationary state. A Gaussian thermostat keeps the total kinetic energy constant. For infinitely many particles it is shown that the thermostat becomes a friction force with constant friction coefficient. For finite number of particles N, the fluctuations around this constant are of order one over the square root of N, and distributed approximately Gaussian with deviations for large fluctuations. These deviations prohibit a derivation of fluctuation-dissipation relations far from equilibrium, based on the Fluctuation Theorem.