We derive rigorous results on the link between the principle of maximum entropy production and the principle of maximum Kolmogorov-Sinai entropy using a Markov model of the passive scalar diffusion called the Zero Range Process. We show analytically
that both the entropy production and the Kolmogorov-Sinai entropy seen as functions of f admit a unique maximum denoted fmaxEP and fmaxKS. The behavior of these two maxima is explored as a function of the system disequilibrium and the system resolution N. The main result of this article is that fmaxEP and fmaxKS have the same Taylor expansion at _rst order in the deviation of equilibrium. We find that fmaxEP hardly depends on N whereas fmaxKS depends strongly on N. In particular, for a fixed difference of potential between the reservoirs, fmaxEP (N) tends towards a non-zero value, while fmaxKS (N) tends to 0 when N goes to infinity. For values of N typical of that adopted by Paltridge and climatologists we show that fmaxEP and fmaxKS coincide even far from equilibrium. Finally, we show that one can find an optimal resolution N_ such that fmaxEP and fmaxKS coincide, at least up to a second order parameter proportional to the non-equilibrium uxes imposed to the boundaries.
Many modern techniques employed in physics, such a computation of path integrals, rely on random walks on graphs that can be represented as Markov chains. Traditionally, estimates of running times of such sampling algorithms are computed using the nu
mber of steps in the chain needed to reach the stationary distribution. This quantity is generally defined as mixing time and is often difficult to compute. In this paper, we suggest an alternative estimate based on the Kolmogorov-Sinai entropy, by establishing a link between the maximization of KSE and the minimization of the mixing time. Since KSE are easier to compute in general than mixing time, this link provides a new faster method to approximate the minimum mixing time that could be interesting in computer sciences and statistical physics. Beyond this, our finding will also be of interest to the out-of-equilibrium community, by providing a new rational to select stationary states in out-of-equilibrium physics: it seems reasonable that in a physical system with two simultaneous equiprobable possible dynamics, the final stationary state will be closer to the stationary state corresponding to the fastest dynamics (smallest mixing time).Through the empirical link found in this letter, this state will correspond to a state of maximal Kolmogorov-Sinai entropy. If this is true, this would provide a more satisfying rule for selecting stationary states in complex systems such as climate than the maximization of the entropy production.
We introduce a model of interacting lattices at different resolutions driven by the two-dimensional Ising dynamics with a nearest-neighbor interaction. We study this model both with tools borrowed from equilibrium statistical mechanics as well as non
-equilibrium thermodynamics. Our findings show that this model keeps the signature of the equilibrium phase transition. Moreover the critical temperature of the equilibrium models correspond to the state maximizing the entropy and delimits two out-of-equilibrium regimes, one satisfying the Onsager relations for systems close to equilibrium and one resembling convective turbulent states. Since the model preserves the entropy and energy fluxes in the scale space, it seems a good candidate for parametric studies of out-of-equilibrium turbulent systems.
