Revealing the Nonequilibrium Nature of a Granular Intruder: The Crucial Role of Non-Gaussian Behavior.

The characterization of the distance from equilibrium is a debated problem in particular in the treatment of experimental signals. If the signal is a one-dimensional time series, such a goal becomes challenging. A paradigmatic example is the angular diffusion of a rotator immersed in a vibro-fluidized granular gas. Here, we experimentally observe that the rotator's angular velocity exhibits significant differences with respect to an equilibrium process. Exploiting the presence of two relevant timescales and non-Gaussian velocity increments, we quantify the breakdown of time-reversal asymmetry, which would vanish in the case of a 1D Gaussian process. We deduce a new model for the massive probe, with two linearly coupled variables, incorporating both Gaussian and Poissonian noise, the latter motivated by the rarefied collisions with the granular bath particles. Our model reproduces the experiment in a range of densities, from dilute to moderately dense, with a meaningful dependence of the parameters on the density. We believe the framework proposed here opens the way to a more consistent and meaningful treatment of out-of-equilibrium and dissipative systems.

Handy fluctuation-dissipation relation to approach generic noisy systems and chaotic dynamics.

We introduce a general formulation of the fluctuation-dissipation relations (FDRs) holding also in far-from-equilibrium stochastic dynamics. A great advantage of this version of the FDR is that it does not require explicit knowledge of the stationary probability density function. Our formula applies to Markov stochastic systems with generic noise distributions: When the noise is additive and Gaussian, the relation reduces to those known in the literature; for multiplicative and non-Gaussian distributions (e.g., Cauchy noise) it provides exact results in agreement with numerical simulations. Our formula allows us to reproduce, in a suitable small-noise limit, the response functions of deterministic, strongly nonlinear dynamical models, even in the presence of chaotic behavior: This could have important practical applications in several contexts, including geophysics and climate. As a case of study, we consider the Lorenz '63 model, which is paradigmatic for the chaotic properties of deterministic dynamical systems.

On the fluctuation-dissipation relation in non-equilibrium and non-Hamiltonian systems.

We review generalized fluctuation-dissipation relations, which are valid under general conditions even in "nonstandard systems," e.g., out of equilibrium and/or without a Hamiltonian structure. The response functions can be expressed in terms of suitable correlation functions computed in the unperturbed dynamics. In these relations, typically, one has nontrivial contributions due to the form of the stationary probability distribution; such terms take into account the interaction among the relevant degrees of freedom in the system. We illustrate the general formalism with some examples in nonstandard cases, including driven granular media, systems with a multiscale structure, active matter, and systems showing anomalous diffusion.

Anomalous mobility of a driven active particle in a steady laminar flow.

We study, via extensive numerical simulations, the force-velocity curve of an active particle advected by a steady laminar flow, in the nonlinear response regime. Our model for an active particle relies on a colored noise term that mimics its persistent motion over a time scale [Formula: see text]. We find that the active particle dynamics shows non-trivial effects, such as negative differential and absolute mobility (NDM and ANM, respectively). We explore the space of the model parameters and compare the observed behaviors with those obtained for a passive particle ([Formula: see text]) advected by the same laminar flow. Our results show that the phenomena of NDM and ANM are quite robust with respect to the details of the considered noise: in particular for finite [Formula: see text] a more complex force-velocity relation can be observed.

Anomalous force-velocity relation of driven inertial tracers in steady laminar flows.

We study the nonlinear response to an external force of an inertial tracer advected by a two-dimensional incompressible laminar flow and subject to thermal noise. In addition to the driving external field F, the main parameters in the system are the noise amplitude [Formula: see text] and the characteristic Stokes time [Formula: see text] of the tracer. The relation velocity vs. force shows interesting effects, such as negative differential mobility (NDM), namely a non-monotonic behavior of the tracer velocity as a function of the applied force, and absolute negative mobility (ANM), i.e. a net motion against the bias. By extensive numerical simulations, we investigate the phase chart in the parameter space of the model, [Formula: see text], identifying the regions where NDM, ANM and more common monotonic behaviors of the force-velocity curve are observed.

Nonlinear Response of Inertial Tracers in Steady Laminar Flows: Differential and Absolute Negative Mobility.

We study the mobility and the diffusion coefficient of an inertial tracer advected by a two-dimensional incompressible laminar flow, in the presence of thermal noise and under the action of an external force. We show, with extensive numerical simulations, that the force-velocity relation for the tracer, in the nonlinear regime, displays complex and rich behaviors, including negative differential and absolute mobility. These effects rely upon a subtle coupling between inertia and applied force that induces the tracer to persist in particular regions of phase space with a velocity opposite to the force. The relevance of this coupling is revisited in the framework of nonequilibrium response theory, applying a generalized Einstein relation to our system. The possibility of experimental observation of these results is also discussed.

Fluctuations in partitioning systems with few degrees of freedom.

We study the behavior of a moving wall in contact with a particle gas and subjected to an external force. We compare the fluctuations of the system observed in the microcanonical and canonical ensembles, by varying the number of particles. Static and dynamic correlations signal significant differences between the two ensembles. Furthermore, velocity-velocity correlations of the moving wall present a complex two-time relaxation that cannot be reproduced by a standard Langevin-like description. Quite remarkably, increasing the number of gas particles in an elongated geometry, we find a typical time scale, related to the interaction between the partitioning wall and the particles, which grows macroscopically.

Discreteness effects in a reacting system of particles with finite interaction radius.

An autocatalytic reacting system with particles interacting at a finite distance is studied. We investigate the effects of the discrete-particle character of the model on properties like reaction rate, quenching phenomenon, and front propagation, focusing on differences with respect to the continuous case. We introduce a renormalized reaction rate depending both on the interaction radius and the particle density, and we relate it to macroscopic observables (e.g., front speed and front thickness) of the system.

Thin front propagation in random shear flows.

Front propagation in time-dependent laminar flows is investigated in the limit of very fast reaction and very thin fronts--i.e., the so-called geometrical optics limit. In particular, we consider fronts stirred by random shear flows, whose time evolution is modeled in terms of Ornstein-Uhlembeck processes. We show that the ratio between the time correlation of the flow and an intrinsic time scale of the reaction dynamics (the wrinkling time tw) is crucial in determining both the front propagation speed and the front spatial patterns. The relevance of time correlation in realistic flows is briefly discussed in light of the bending phenomenon--i.e., the decrease of propagation speed observed at high flow intensities.

Turbulence and coarsening in active and passive binary mixtures.

Phase separation between two fluids in two dimensions is investigated by means of direct numerical simulations of coupled Navier-Stokes and Cahn-Hilliard equations. We study the phase ordering process in the presence of an external stirring acting on the velocity field. For both active and passive mixtures we find that, for a sufficiently strong stirring, coarsening is arrested in a stationary dynamical state characterized by a continuous rupture and formation of finite domains. Coarsening arrest is shown to be independent of the chaotic or regular nature of the flow.

Mixing and reaction efficiency in closed domains.

We present a numerical study of mixing and reaction efficiency in closed domains. In particular, we focus our attention on laminar flows. In the case of inert transport the mixing properties of the flows strongly depend on the details of the Lagrangian transport. We also study the reaction efficiency. Starting with a little spot of product, we compute the time needed to complete the reaction in the container. We find that the reaction efficiency is not strictly related to the mixing properties of the flow. In particular, reaction acts as a "dynamical regulator".

Brownian motion and diffusion: from stochastic processes to chaos and beyond.

One century after Einstein's work, Brownian motion still remains both a fundamental open issue and a continuous source of inspiration for many areas of natural sciences. We first present a discussion about stochastic and deterministic approaches proposed in the literature to model the Brownian motion and more general diffusive behaviors. Then, we focus on the problems concerning the determination of the microscopic nature of diffusion by means of data analysis. Finally, we discuss the general conditions required for the onset of large scale diffusive motion.

Multiple-scale analysis and renormalization for preasymptotic scalar transport.

Preasymptotic transport of a scalar quantity passively advected by a velocity field formed by a large-scale component superimposed on a small-scale fluctuation is investigated both analytically and by means of numerical simulations. Exploiting the multiple-scale expansion one arrives at a Fokker-Planck equation which describes the preasymptotic scalar dynamics. This equation is associated with a Langevin equation involving a multiplicative noise and an effective (compressible) drift. For the general case, no explicit expression for either the affective drift on the effective diffusivity (actually a tensorial field) can be obtained. We discuss an approximation under which an explicit expression for the diffusivity (and thus for the drift) can be obtained. Its expression permits us to highlight the important fact that the diffusivity explicitly depends on the large-scale advecting velocity. Finally, the robustness of the aforementioned approximation is checked numerically by means of direct numerical simulations.

Relaxation of finite perturbations: beyond the fluctuation-response relation.

We study the response of dynamical systems to finite amplitude perturbation. A generalized fluctuation-response relation is derived, which links the average relaxation toward equilibrium to the invariant measure of the system and points out the relevance of the amplitude of the initial perturbation. Numerical computations on systems with many characteristic times show the relevance of the above-mentioned relation in realistic cases.

Fluctuation-response relation in turbulent systems.

We address the problem of measuring time properties of response functions (Green functions) in Gaussian models (Orszag-McLaughin) and strongly non-Gaussian models (shell models for turbulence). We introduce the concept of halving-time statistics to have a statistically stable tool to quantify the time decay of response functions and generalized response functions of high order. We show numerically that in shell models for three-dimensional turbulence response functions are inertial range quantities. This is a strong indication that the invariant measure describing the shell-velocity fluctuations is characterized by short range interactions between neighboring shells.

Front speed enhancement in cellular flows.

The problem of front propagation in a stirred medium is addressed in the case of cellular flows in three different regimes: slow reaction, fast reaction and geometrical optics limit. It is well known that a consequence of stirring is the enhancement of front speed with respect to the nonstirred case. By means of numerical simulations and theoretical arguments we describe the behavior of front speed as a function of the stirring intensity, U. For slow reaction, the front propagates with a speed proportional to U(1/4), conversely for fast reaction the front speed is proportional to U(3/4). In the geometrical optics limit, the front speed asymptotically behaves as U/ln U. (c) 2002 American Institute of Physics.

Front propagation in laminar flows.

The problem of front propagation in flowing media is addressed for laminar velocity fields in two dimensions. Three representative cases are discussed: stationary cellular flow, stationary shear flow, and percolating flow. Production terms of Fisher-Kolmogorov-Petrovskii-Piskunov type and of Arrhenius type are considered under the assumption of no feedback of the concentration on the velocity. Numerical simulations of advection-reaction-diffusion equations have been performed by an algorithm based on discrete-time maps. The results show a generic enhancement of the speed of front propagation by the underlying flow. For small molecular diffusivity, the front speed V(f) depends on the typical flow velocity U as a power law with an exponent depending on the topological properties of the flow, and on the ratio of reactive and advective time scales. For open-streamline flows we find always V(f) approximately U, whereas for cellular flows we observe V(f) approximately U(1/4) for fast advection and V(f) approximately U(3/4) for slow advection.

Inverse statistics of smooth signals: the case of two dimensional turbulence.

The problem of inverse statistics (statistics of distances for which the signal fluctuations are larger than a certain threshold) in differentiable signals with power law spectrum, E(k) approximately k(-alpha), 3< or =alpha<5, is discussed. We show that for these signals, with random phases, exit-distance moments follow a bifractal distribution. We also investigate two dimensional turbulent flows in the direct cascade regime, which display a more complex behavior. We give numerical evidences that the inverse statistics of 2D turbulent flows is described by a multifractal probability distribution; i.e., the statistics of laminar events is not simply captured by the exponent alpha characterizing the spectrum.

Driven granular gases with gravity.

We study fluidized granular gases in a stationary state determined by the balance between external driving and bulk dissipation. The two considered situations are inspired by recent experiments, where gravity plays a major role as a driving mechanism: in the first case, gravity acts only in one direction and the bottom wall is vibrated; in the second case, gravity acts in both directions and no vibrating walls are present. Simulations performed under the molecular chaos assumption show averaged profiles of density, velocity, and granular temperature that are in good agreement with the experiments. Moreover, we measure velocity distributions that show strong non-Gaussian behavior, as experiments pointed out, but also density correlations accounting for clustering, at odds with the experimental results. The hydrodynamics of the first model is discussed and an exact solution is found for the density and granular temperature as functions of the distance from the vibrating wall. The limitations of such a solution, in particular in a broad layer near the wall injecting energy, are discussed.

Chaos or noise: difficulties of a distinction

In experiments, the dynamical behavior of systems is reflected in time series. Due to the finiteness of the observational data set, it is not possible to reconstruct the invariant measure up to an arbitrarily fine resolution and an arbitrarily high embedding dimension. These restrictions limit our ability to distinguish between signals generated by different systems, such as regular, chaotic, or stochastic ones, when analyzed from a time series point of view. We propose to classify the signal behavior, without referring to any specific model, as stochastic or deterministic on a certain scale of the resolution epsilon, according to the dependence of the (epsilon,tau) entropy, h(epsilon, tau), and the finite size Lyapunov exponent lambda(epsilon) on epsilon.