  Gallavotti proposed an equivalence principle in hydrodynamics, which states
that forced-damped fluids can be equally well represented by means of the
Navier-Stokes equations and by means of time reversible dynamical systems
called GNS. In the GNS systems, the usual viscosity is replaced by a
state-dependent dissipation term which fixes one global quantity. The principle
states that the mean values of properly chosen observables are the same for
both representations of the fluid. In the same paper, the chaotic hypothesis of
Gallavotti and Cohen is applied to hydrodynamics, leading to the conjecture
that entropy fluctuations in the GNS system verify a relation first observed in
nonequilibrium molecular dynamics. We tested these ideas in the case of
two-dimensional fluids. We examined the fluctuations of global quadratic
quantities in the statistically stationary state of a) the Navier-Stokes
equations; b) the GNS equations. Our results are consistent with the validity
of the fluctuation relation, and of the equivalence principle, indicating
possible extensions thereof. Moreover, in these results the difference between
the Gallavotti-Cohen fluctuation theorem and the Evans-Searles identity is
evident.
