Turbulent flows are known to enhance turbulent transport. It has then even been suggested that turbulence is a state of maximum energy dissipation. In this paper, we reexamine critically this suggestion in light of several recent works around the Maximum Entropy Production principle (MEP) that has been used in several out-of-equilibrium systems. We provide a set of four different optimization principles, based on maximization of energy dissipation, entropy production, Kolmogorov-Sinai entropy and minimization of mixing time, and study the connection between these principles using simple out-of-equilibrium models describing mixing of a scalar quantity. We find that there is a chained-relationship between most probable stationary states of the system, and their ability to obey one of the four principles. This provides an empirical justification of the Maximum Entropy Production principle in this class of systems, including some turbulent flows, for special boundary conditions. Otherwise, we claim that the minimization of the mixing time would be a more appropriate principle. We stress that this principle might actually be limited to flows where symmetry or dynamics impose pure mixing of a quantity (like angular momentum, momentum or temperature). The claim that turbulence is a state of maximum energy dissipation, a quantity intimately related to entropy production, is therefore limited to special situations that nevertheless include classical systems such as shear flows, Rayleigh-Bénard convection and von Kármán flows, forced with constant velocity or temperature conditions.

Abstract:The asymmetric simple exclusion process (ASEP) has become a paradigmatic toy-model of a non-equilibrium system, and much effort has been made in the past decades to compute exactly its statistics for given dynamical rules. Here, a different approach is developed; analogously to the equilibrium situation, we consider that the dynamical rules are not exactly known. Allowing for the transition rate to vary, we show that the dynamical rules that maximize the entropy production and those that maximise the rate of variation of the dynamical entropy, known as the Kolmogorov-Sinai entropy coincide with good accuracy. We study the dependence of this agreement on the size of the system and the couplings with the reservoirs, for the original ASEP and a variant with Langmuir kinetics.

We establish a link between the maximization of Kolmogorov Sinai entropy (KSE) and the minimization of the mixing time for general Markov chains. Since the maximisation of KSE is analytical and easier to compute in general than mixing time, this link provides a new faster method to approximate the minimum mixing time dynamics. It could be interesting in computer sciences and statistical physics, for computations that use random walks on graphs that can be represented as Markov chains.