**Authors**
: Giovanni Gallavotti, Valerio Lucarini

**Title:** * Equivalence of
Non-Equilibrium Ensembles and Representation }\centerline{\Large \bf of
Friction in Turbulent Flows: The Lorenz 96 Model*

**Abstract**:*We construct
different equivalent non-equilibrium
statistical ensembles in a simple yet instructive $N$-degrees of freedom
model of atmospheric turbulence, introduced by Lorenz in 1996. The vector
field can be decomposed into an energy-conserving, time-reversible part,
plus a non-time reversible part, including forcing and dissipation. We
construct a modified version of the model where viscosity varies with time,
in such a way that energy is conserved, and the resulting
dynamics is fully time-reversible. For each value of the forcing, the
statistical properties of the irreversible and reversible model are in
excellent agreement, if in the latter the energy is kept constant at a
value equal to the time-average realized with the irreversible model. In
particular, the average contraction rate of the phase space of the
time-reversible model agrees with that of the irreversible model, where
instead it is constant by construction. We also show that the phase space
contraction rate obeys the fluctuation relation, and we relate its finite
time corrections to the characteristic time scales of the system. A local
version of the fluctuation relation is explored and successfully checked.
The equivalence between the two non-equilibrium ensembles extends to
dynamical properties such as the Lyapunov exponents, which are shown to
obey to a good degree of approximation a pairing rule. These results have
relevance in motivating the importance of the chaotic hypothesis. in
explaining that we have the freedom to model non-equilibrium systems using
different but equivalent approaches, and, in particular, that using a model
of a fluid where viscosity is kept constant is just one option, and not
necessarily the only option, for describing accurately its statistical and
dynamical properties. *

**Keywords**: Equivalent Equations, Turbulence, Chaotic
Hypothesis, Fluctuation Theorem, Geophysical Flows, Parametrization,
Lyapunov Exponents

Giovanni Gallavotti

INFN Fisica e Accademia dei Lincei

Universita di Roma Sapienza, 00185 Roma, Italy

giovanni.gallavotti@roma1.infn.it

Valerio Lucarini

Institute of Meteorology, Klimacampus, University of Hamburg,Hamburg,
Germany}

Department of Mathematics and Statistics, University of Reading, Reading,
UK

Walker Institute for Climate Change Research, University of Reading,
Reading, UK