(Global and Local) Fluctuations of Phase Space Contraction in Deterministic Stationary Non-equilibrium

We studied numerically the validity of the fluctuation theorem, introduced by Evans,Cohen and Morris and proved by Gallavotti and Cohen, for a 2-dimensional system of particles maintained in a steady shear flow by Maxwell daemon boundary conditions (see Chernov and Lebowitz). The theorem was found to hold if one considers the total phase space contraction $\sigma$ occuring at collisions with both walls: $\sigma ={\sigma }^{\text{su}}+{\sigma }^{\text{giu}}$. An attempt to extend it to more local quantities ${\sigma }^{\text{su}}$ and ${\sigma }^{\text{giu}}$, corresponding to the collisions with the top or bottom wall only, gave negative results. The time decay of the correlations in ${\sigma }^{\text{su},\text{giu}}$ was very slow compared to that of $\sigma$.