Nonequilibrium ensembles for the three-dimensional Navier-Stokes equations.

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 three-dimensional 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.

Ensembles, turbulence and fluctuation theorem.

The fluctuation theorem is considered intrinsically linked to reversibility and therefore its phenomenological consequence, the fluctuation relation, is sometimes considered not applicable. Nevertheless here is considered the paradigmatic example of irreversible evolution, the 2D Navier-Stokes incompressible flow, to show how universal properties of fluctuations in systems evolving irreversibily could be predicted in a general context. Together with a formulation of the theoretical framework several open questions are formulated and a few more simulations are provided to illustrate the results and to stimulate further checks.

Equivalence of nonequilibrium ensembles in turbulence models.

Understanding under what conditions it is possible to construct equivalent ensembles is key to advancing our ability to connect microscopic and macroscopic properties of nonequilibrium statistical mechanics. In the case of fluid dynamical systems, one 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, cases where the time symmetry of the evolution equations is exactly preserved, as it must be in the corresponding microscopic systems, when available.

Resonances within chaos.

A chaotic system under periodic forcing can develop a periodically visited strange attractor. We discuss simple models in which the phenomenon, quite easy to see in numerical simulations, can be completely studied analytically.

On thermostats: isokinetic or Hamiltonian? Finite or infinite?

The relation between finite isokinetic thermostats and infinite Hamiltonian thermostats is studied and their equivalence in the thermodynamic limit is heuristically discussed.

Entropy, thermostats, and chaotic hypothesis.

The chaotic hypothesis is proposed as a basic for a general theory of nonequilibrium stationary states.

Irreversibility time scale.

Entropy creation rate is introduced for a system interacting with thermostats (i.e., for a system subject to internal conservative forces interacting with "external" thermostats via conservative forces) and a fluctuation theorem for it is proved.

Entropy production and thermodynamics of nonequilibrium stationary states: a point of view.

Entropy might be a not well defined concept if the system can undergo transformations involving stationary nonequilibria. It might be analogous to the heat content (once called "caloric") in transformations that are not isochoric (i.e.

Nonequilibrium stationary states and entropy.

In transformations between nonequilibrium stationary states, entropy might not be a well defined concept. It might be analogous to the "heat content" in transformations in equilibrium which is not well defined either, if they are not isochoric (i.e.

Chaotic dynamics, fluctuations, nonequilibrium ensembles.

The ideas and the conceptual steps leading from the ergodic hypothesis for equilibrium statistical mechanics to the chaotic hypothesis for equilibrium and nonequilibrium statistical mechanics are illustrated. The fluctuation theorem linear law and universal slope prediction for reversible systems is briefly derived. Applications to fluids are briefly alluded to.