**Authors**
: Giorgos Margazoglu, Luca Biferale, Massimo
Cencini, Giovanni Gallavotti, Valerio Lucarini

**Title::** *
Non-equilibrium Ensembles for the 3D Navier-Stokes Equations
*

**Abstract**:*
At the molecular level fluid motions are, by first principles,
described by time reversible laws. On the other hand, the coarse
grained macroscopic evolution is suitably described by the Navier
Stokes equations, which are inherently irreversible, due to the
dissipation term. Here, a reversible version of 3D Navier Stokes
is studied, by introducing a fluctuating viscosity constructed in
such a way that enstrophy is conserved, along the lines of the
paradigm of microcanonical versus canonical treatment in
equilibrium statistical mechanics. Through systematic simulations
we provide evidences of the conditions that must be satisfied in
order to have a statistical equivalence between the two
non-equilibrium ensembles. Moreover, we numerically study the
empirical distribution of the fluctuating viscosity by changing
the Reynolds number and the number of modes used in the
discretization of the evolution equation.
*

**Keywords**:
Irreversible fluid equations, Reversible fluid equations, Equivalence
Irreversible-Reversible

Giovanni Gallavotti

INFN-Roma1 and Universit\`a "La Sapienzaâ€ť Roma, Italy