The superdiffusion behavior, i.e. < x 2 (t) >∼ t 2ν , with ν > 1/2, in general is not completely characherized by a unique exponent. We study some systems exhibiting strong anomalous diffusion, i.e. < |x(t)| q >∼ t qν(q) where ν(2) > 1/2 and qν(q) is not a linear function of q. This feature is different from the weak superdiffusion regime, i.e. ν(q) = const > 1/2, as in random shear flows.The strong anomalous diffusion can be generated by nontrivial chaotic dynamics, e.g. Lagrangian motion in 2d time-dependent incompressible velocity fields, 2d symplectic maps and 1d intermittent maps. Typically the function qν(q) is piecewise linear. This corresponds to two mechanisms: a weak anomalous diffusion for the typical events and a ballistic transport for the rare excursions.In order to have strong anomalous diffusion one needs a violation of the hypothesis of the central limit theorem, this happens only in a very narrow region of the control parameters space.In the presence of the strong anomalous diffusion one does not have a unique exponent and therefore one has the failure of the usual scaling of the probability distribution, i.e. P (x, t) = t −ν F (x/t ν ). This implies that the effective equation at large scale and long time for P (x, t), cannot obey neither the usual Fick equation nor other linear equations involving temporal and/or spatial fractional derivatives.PACS number(s): 05.45.+b, 05.60.+w;

We establish a refined version of the Second Law of Thermodynamics for Langevin stochastic processes describing mesoscopic systems driven by conservative or non-conservative forces and interacting with thermal noise. The refinement is based on the Monge-Kantorovich optimal mass transport. General discussion is illustrated by numerical analysis of a model for micron-size particle manipulated by optical tweezers.

Thermodynamics of small systems has become an important field of statistical physics. Such systems are driven out of equilibrium by a control, and the question is naturally posed how such a control can be optimized. We show that optimization problems in small system thermodynamics are solved by (deterministic) optimal transport, for which very efficient numerical methods have been developed, and of which there are applications in cosmology, fluid mechanics, logistics, and many other fields. We show, in particular, that minimizing expected heat released or work done during a nonequilibrium transition in finite time is solved by the Burgers equation and mass transport by the Burgers velocity field. Our contribution hence considerably extends the range of solvable optimization problems in small system thermodynamics.

We study the problem of optimizing released heat or dissipated work in stochastic thermodynamics. In the overdamped limit these functionals have singular solutions, previously interpreted as protocol jumps. We show that a regularization, penalizing a properly defined acceleration, changes the jumps into boundary layers of finite width. We show that in the limit of vanishing boundary layer width no heat is dissipated in the boundary layer, while work can be done. We further give an alternative interpretation of the fact that the optimal protocols in the overdamped limit are given by optimal deterministic transport (Burgers equation).

The Rayleigh-Taylor instability of two immiscible fluids in the limit of small Atwood numbers is studied by means of a phase-field description. In this method the sharp fluid interface is replaced by a thin, yet finite, transition layer where the interfacial forces vary smoothly. This is achieved by introducing an order parameter (the phase field) whose variation is continuous across the interfacial layers and is uniform in the bulk region. The phase field model obeys a Cahn-Hilliard equation and is two-way coupled to the standard Navier-Stokes equations. Starting from this system of equations we have first performed a linear analysis from which we have analytically rederived the known gravity-capillary dispersion relation in the limit of vanishing mixing energy density and capillary width. We have performed numerical simulations and identified a region of parameters in which the known properties of the linear phase (both stable and unstable) are reproduced in a very accurate way. This has been done both in the case of negligible viscosity and in the case of nonzero viscosity. In the latter situation only upper and lower bounds for the perturbation growth-rate are known. Finally, we have also investigated the weaklynonlinear stage of the perturbation evolution and identified a regime characterized by a constant terminal velocity of bubbles/spikes. The measured value of the terminal velocity is in perfect agreement with available theoretical prediction. The phase-field approach thus appears to be a valuable tecnhique for the dynamical description of the stages where hydrodynamic turbulence and wave-turbulence enter into play.

International audienceWe study the effects of a stable density stratification on the turbulent dynamics of thin fluid layers forced at intermediate scales. By means of a set of high-resolution numerical simulations, performed within the Boussinesq approximation, we investigate how the stratification and confinement affect the mechanisms of kinetic and potential energy transfer. The detailed analysis of the statistics of the energy-dissipation rates and energy-exchange rates and of the spectral fluxes of potential and kinetic energy shows that stratification provides a new channel for the energy transfer towards small scales which reduces the large-scale flux of kinetic energy. We also discuss the role of vortex stretching and enstrophy flux in the transfer of kinetic energy into potential energy at small scales

We present a systematic way to compute the scaling exponents of the structure functions of the Kraichnan model of turbulent advection in a series of powers of ξ, adimensional coupling constant measuring the degree of roughness of the advecting velocity field. We also investigate the relation between standard and renormalization group improved perturbation theory. The aim is to shed light on the relation between renormalization group methods and the statistical conservation laws of the Kraichnan model, also known as zero modes.

We investigate the experimental setup proposed in [New J. Phys., 15, 115006 (2013)] for calorimetric measurements of thermodynamic indicators in an open quantum system. As theoretical model we consider a periodically driven qubit coupled with a large yet finite electron reservoir, the calorimeter. The calorimeter is initially at equilibrium with an infinite phonon bath. As time elapses, the temperature of the calorimeter varies in consequence of energy exchanges with the qubit and the phonon bath. We show how under weak coupling assumptions, the evolution of the qubitcalorimeter system can be described by a generalized quantum jump process including as dynamical variable the temperature of the calorimeter. We study the jump process by numeric and analytic methods. Asymptotically with the duration of the drive, the qubit-calorimeter attains a steady state. In this same limit, we use multiscale perturbation theory to derive a Fokker-Planck equation governing the calorimeter temperature distribution. We inquire the properties of the temperature probability distribution close and at the steady state. In particular, we predict the behavior of measurable statistical indicators versus the qubit-calorimeter coupling constant.