(Global and local) fluctuations of phase space contraction in deterministic stationary nonequilibrium

We studied numerically the validity of the fluctuation relation introduced in Evans et al. [Phys. Rev. Lett. 71, 2401-2404 (1993)] and proved under suitable conditions by Gallavotti and Cohen [J. Stat. Phys. 80, 931-970 (1995)] for a two-dimensional system of particles maintained in a steady shear flow by Maxwell demon boundary conditions [Chernov and Lebowitz, J. Stat. Phys. 86, 953-990 (1997)]. The theorem was found to hold if one considers the total phase space contraction sigma occurring at collisions with both walls: sigma=sigma( upward arrow )+sigma( downward arrow ). An attempt to extend it to more local quantities sigma( upward arrow ) and sigma( downward arrow ), corresponding to the collisions with the top or bottom wall only, gave negative results. The time decay of the correlations in sigma( upward arrow, downward arrow ) was very slow compared to that of sigma.