Equivalence of nonequilibrium ensembles in turbulence models

Understanding under which conditions it is possible to construct equivalent ensembles is key to advancing our ability to connect microscopic and macroscopic properties of non-equilibrium statisti- cal mechanics. In the case of fluid dynamical systems, a first issue is to test whether different models for viscosity lead to the same macroscopic properties of the fluid systems in different regimes. Such models include, besides the standard choice of constant viscosity, also cases where the time symme- try of the evolution equations is exactly preserved, as it must be in the corresponding microscopic systems, when available. Here a time-reversible dynamics is obtained by imposing the conservation of global observables. We test the equivalence of reversible and irreversible ensembles for the case of a multiscale shell model of turbulence. We verify that the equivalence is obeyed for the mean-values of macroscopic observables, up to an error that vanishes as the system becomes more and more chaotic.