Authors : Luca Biferale, Massimo Cencini, Massimo Depietro, Giovanni Gallavotti, Valerio Lucarini

Title:: Equivalence of Non-Equilibrium Ensembles in Turbulence Models

Abstract: 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 statistical 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 symmetry 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 \textit{macroscopic} observables, up to an error that vanishes as the system becomes more and more chaotic.

Keywords: Fluids, Irreversibility, Reversibility, Friction, Nonequilibrium Ensembles, Navier-Stokes, Fluctuation Relation

Luca Biferale
Dipartimento di Fisica and INFN, Universit\`a di Roma ``Tor Vergata'', Via Ricerca Scientifica 1, 00133 Roma, Italy

Massimo Cencini
Istituto dei Sistemi Complessi, CNR, via dei Taurini 19, 00185 Rome, Italy and INFN ``Tor Vergata''

Massimo De Pietro
Dipartimento di Fisica and INFN, Universit\`a di Roma ``Tor Vergata'', Via Ricerca Scientifica 1, 00133 Roma, Italy

Giovanni Gallavotti
INFN-Roma1 and Universit\`a "La Sapienza” Roma, Italy

Valerio Lucarini
Department of Mathematics and Statistics, University of Reading, Reading, RG66AX, United Kingdom
Centre for the Mathematics of Planet Earth, University of Reading, Reading, RG66AX, United Kingdom
CEN, University of Hamburg, Hamburg, 20144, Germany