(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 σ occurring at collisions with both walls: ${\mathrm{\sigma =\sigma }}^{\uparrow }{\mathrm{+\sigma }}^{\downarrow }.$ An attempt to extend it to more local quantities ${\sigma }^{\uparrow }$ and ${\sigma }^{\downarrow },$ corresponding to the collisions with the top or bottom wall only, gave negative results. The time decay of the correlations in ${\sigma }^{\mathrm{\uparrow ,\downarrow }}$ was very slow compared to that of σ.