Thermodynamic entropy production fluctuation in a two-dimensional shear flow model

We investigate fluctuations in the momentum flux across a surface perpendicular to the velocity gradient in a stationary shear flow maintained by either thermostated deterministic or by stochastic boundary conditions. In the deterministic system the fluctuation relation for the probability of large deviations, which holds for the phase space volume contraction giving the Gibbs ensemble entropy production, never seems to hold for the flux which gives the hydrodynamic entropy production. In the stochastic case the fluctuation relation is found to hold for the total flux, as predicted by various exact results, but not for the flux across part of the surface. The latter appear to satisfy a modified fluctuation relation. Similar results are obtained for the heat flux in a steady state produced by stochastic boundaries at different temperatures.