A new theoretical approach to non-equilibrium statistical systems has recently been proposed by the author, a co-author and others. It is based on a variational principle which is associated with the discrepancy of a path through thermodynamical space to one following Liouvillean evolution. In this contribution the approach is extended in such a way that it can be applied to a wide range of practical non-equilibrium statistical systems such as those arising in turbulence but also to a general class of statistical physics models. The new methodology allows for application to autonomous dynamical systems generalizing the previous work which applied only to Hamiltonian systems. Furthermore it provides a general analysis of near equilibrium conditions which allows for a natural analysis of predictability limits in turbulent systems. Finally it describes a method is described for the numerical calculation of far from equilibrium thermodynamical trajectories.
This is a brief survey of recent experimental studies on out-of-equilibrium scaling laws, focusing on two prominent situations where non-trivial universality classes have been identified theoretically: absorbing-state phase transitions and growing interfaces. First the article summarizes main results obtained for electrically-driven turbulent liquid crystal, which exhibited the scaling laws for the directed percolation class at the transition between two turbulent regimes, and those for the Kardar-Parisi-Zhang class in the supercritical phase where one turbulent regime invades the other. Other…
It is shown that correlation function of the mean wind velocity generated by a turbulent thermal convection (Rayleigh number $Ra \sim 10^{11}$) exhibits exponential decay with a very long correlation time, while corresponding largest Lyapunov exponent is certainly positive. These results together with the reconstructed phase portrait indicate presence of chaotic component in the examined mean wind. Telegraph approximation is also used to study relative contribution of the chaotic and stochastic components to the mean wind fluctuations and an equilibrium between these components has been studie…
Two-dimensional turbulent flows, and to some extent, geophysical flows, are systems with a large number of degrees of freedom, which, albeit fluctuating, exhibit some degree of organization: coherent structures emerge spontaneously at large scales. In this short course, we show how the principles of equilibrium statistical mechanics apply to this problem and predict the condensation of energy at large scales and allow for computing the resulting coherent structures. We focus on the structure of the theory using the language of large deviation theory.
An approach that extends equilibrium thermodynamics principles to out-of-equilibrium systems is based on the local equilibrium hypothesis. However, the validity of the a priori assumption of local equilibrium has been questioned due to the lack of sufficient experimental evidence. In this paper, we present experimental results obtained from a pure thermodynamic study of the non-turbulent Rayleigh-Bénard convection at steady-state to verify the validity of the local equilibrium hypothesis. A non-turbulent Rayleigh-Bénard convection at steady-state is an excellent `model thermodynamic system' in…
In this paper we present a detailed description of the statistical and computational techniques that were employed to study a driven far-from-equilibrium steady-state Rayleigh-B{é}nard system in the non-turbulent regime ($Ra\leq 3500$). In our previous work on the Rayleigh-B{é}nard convection system we try to answer two key open problems that are of great interest in contemporary physics: (i) how does an out-of-equilibrium steady-state differ from an equilibrium state and (ii) how do we explain the spontaneous emergence of stable structures and simultaneously interpret the physical notion of t…
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 …
The Kardar-Parisi-Zhang model of non-equilibrium critical behaviour (kinetic surface roughening) with turbulent motion of the environment taken into account is studied by the field theoretic renormalization group approach. The turbulent motion is described by the stochastic Navier-Stokes equation with the random stirring force whose correlation function includes two terms that allow one to account both for a turbulent fluid and for a fluid in thermal equilibrium. The renormalization group analysis performed in the leading order of perturbation theory (one-loop approximation) reveals six possib…
A challenge in fundamental physics and especially in thermodynamics is to understand emergent order in far-from-equilibrium systems. While at equilibrium, temperature plays the role of a key thermodynamic variable whose uniformity in space and time defines the equilibrium state the system is in, this is not the case in a far-from-equilibrium driven system. When energy flows through a finite system at steady-state, temperature takes on a time-independent but spatially varying character. In this study, the convection patterns of a Rayleigh-B{é}nard fluid cell at steady-state is used as a prototy…
The transitional and well-developed regimes of turbulent shear flows exhibit a variety of remarkable scaling laws that are only now beginning to be systematically studied and understood. In the first part of this article, we summarize recent progress in understanding the friction factor of turbulent flows in rough pipes and quasi-two-dimensional soap films, showing how the data obey a two-parameter scaling law known as roughness-induced criticality, and exhibit power-law scaling of friction factor with Reynolds number that depends on the precise form of the nature of the turbulent cascade. The…
The objective of this paper is to unravel any relations that may exist between turbulent shear flows and statistical mechanics, through a detailed numerical investigation in the simplest case where both can be well defined. The shear flow considered for the purpose is the 2D temporal mixing layer, which is a time-dependent flow that is statistically homogeneous in the streamwise direction (x) and evolves from a plane vortex sheet in the direction normal to it (y) in a periodic-in-x domain with period L. The connections to statistical mechanics are explored by revisiting, via extensive computer…
Is there really such a thing as weak turbulence? Here we analyze turbulence of weakly interacting waves using the tools of information theory. It offers a unique perspective for comparing thermal equilibrium and turbulence: the mutual information between modes is shown to be stationary and small in equilibrium but grows linearly with time in weak turbulence. We trace this growth to the concentration of probability on the resonance surfaces, which can go all the way to a singular measure. The surprising conclusion is that no matter how small is the nonlinearity and how close to Gaussian is the …
Oceanic geostrophic turbulence is mostly forced at the surface, yet strong bottom-trapped flows are commonly observed along topographic anomalies. Here we consider the case of a freely evolving, initially surface-intensified velocity field above a topographic bump, and show that the self-organization into a bottom-trapped current can result from its turbulent dynamics. Using equilibrium statistical mechanics, we explain this phenomenon as the most probable outcome of turbulent stirring. We compute explicitly a class of solutions characterized by a linear relation between potential vorticity an…
We suggest that the dynamical spontaneous symmetry breaking reported in a turbulent swirling flow at $Re=40~000$ by Cortet et al., Phys. Rev. Lett., 105, 214501 (2010) can be described through a continuous one parameter family transformation (amounting to a phase shift) of steady states and could be the analogue of the Goldstone mode of the vertical translational symmetry in an ideal system. We investigate a possible mechanism of emergence of such spontaneous symmetry breaking in a toy model of our out-equilibrium system, derived from its equilibrium counterpart. We show that the stationary st…
In the present paper, recent experimental results on large scale coherent steady states observed in experimental von K{á}rm{á}n flows are revisited from a statistical mechanics perspective. The latter is rooted on two levels of description. We first argue that the coherent steady states may be described as the equilibrium states of well-chosen lattice models, that can be used to define global properties of von K{á}rm{á}n flows, such as their temperatures. The equilibrium description is then enlarged, in order to reinterpret a series of results about the stability of those steady states, their …
The statistical properties of physical systems in thermal equilibrium are blatantly different from their far-from-equilibrium counterparts. In the latter, fluctuations often dominate the dynamics and might cluster in ordered patterns in the form of dissipative coherent structures. Here, we study the transition of a cold atomic cloud, driven close to a sharp electronic resonance, from a stable to a turbulent phase. From the atomic density distribution -- measured using a spatially-resolved pump-probe technique -- we have computed the Shannon entropy on two different basis sets. Information comp…
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 reversib…
We experimentally characterize the fluctuations of the non-homogeneous non-isotropic turbulence in an axisymmetric von Kármán flow. We show that these fluctuations satisfy relations analogous to classical Fluctuation-Dissipation Relations (FDRs) in statistical mechanics. We use these relations to measure statistical temperatures of turbulence. The values of these temperatures are found to be dependent on the considered observable as already evidenced in other far from equilibrium systems.
In recent work, Renaud, Venaille, and Bouchet (RVB) revisit the equilibrium statistical mechanics theory of the shallow water equations, within a microcanonical approach, focusing on a more careful treatment of the energy partition between inertial gravity wave and eddy motions in the equilibrium state, and deriving joint probability distributions for the corresponding dynamical degrees of freedom. The authors derive a Liouville theorem that determines the underlying phase space statistical measure, but then, through some physical arguments, actually compute the equilibrium statistics using a …
Effects of strongly anisotropic turbulent mixing on the critical behavior are studied by means of the renormalization group. Two models are considered: the equilibrium model A, which describes purely relaxational dynamics of a nonconserved scalar order parameter, and the Gribov model, which describes the nonequilibrium phase transition between the absorbing and fluctuating states in a reaction-diffusion system. The velocity is modelled by the d-dimensional generalization of the random shear flow introduced by Avellaneda and Majda within the context of passive scalar advection. Existence of new…
An Equation of State for Turbulence in the Gross-Pitaevskii model
Cavendish Laboratory, University of Cambridge（剑桥大学卡文迪许实验室）；Department of Physics, Institute of Science Tokyo（东京科学大学物理系）；Yale University（耶鲁大学）；Yale Quantum Institute, Yale University（耶鲁大学耶鲁量子研究所）
