Fluctuations of entropy production in the isokinetic ensemble

We discuss the microscopic definition of entropy production rate in a model of a dissipative system: a sheared fluid in which the kinetic energy is kept constant via a Gaussian thermostat. The total phase space contraction rate is the sum of two statistically independent contributions: the first one is due to the work of the conservative forces, is independent of the driving force and does not vanish at zero drive, making the system non-conservative also in equilibrium. The second is due to the work of the dissipative forces, and is responsible for the average entropy production; the distribution of its fluctuations is found to verify the Fluctuation Relation of Gallavotti and Cohen. The distribution of the fluctuations of the total phase space contraction rate also verify the Fluctuation Relation. It is compared with the same quantity calculated in the isoenergetic ensemble: we find that the two ensembles are equivalent, as conjectured by Gallavotti. Finally, we discuss the implication of our results for experiments trying to verify the validity of the FR.