Coarsening in granular systems

We review a few representative examples of granular experiments or models where phase separation, accompanied by domain coarsening, is a relevant phenomenon. We first elucidate the intrinsic non-equilibrium, or athermal, nature of granular media. Thereafter, dilute systems, the so-called "granular gases" are discussed: idealized kinetic models, such as the gas of inelastic hard spheres in the cooling regime, are the optimal playground to study the slow growth of correlated structures, e.g. shear patterns, vortices and clusters. In fluidized experiments, liquid-gas or solid-gas separations have been observed. In the case of monolayers of particles, phase coexistence and coarsening appear in several different setups, with mechanical or electrostatic energy input. Phenomenological models describe, even quantitatively, several experimental measures, both for the coarsening dynamics and for the dynamic transition between different granular phases. The origin of the underlying bistability is in general related to negative compressibility from granular hydrodynamics computations, even if the understanding of the mechanism is far from complete. A relevant problem, with important industrial applications, is related to the demixing or segregation of mixtures, for instance in rotating tumblers or on horizontally vibrated plates. Finally, the problem of compaction of highly dense granular materials, which has many important applications, is usually described in terms of coarsening dynamics: there, bubbles of mis-aligned grains evaporate, allowing the coalescence of optimally arranged islands and a progressive reduction of total occupied volume.Comment: 12 pages, 10 figures, to appear in "Dynamics of coarsening" Comptes Rendus Physique special issue, https://sites.google.com/site/ppoliti/crp-special-issu

Nonequilibrium Brownian motion beyond the effective temperature

The condition of thermal equilibrium simplifies the theoretical treatment of fluctuations as found in the celebrated Einstein's relation between mobility and diffusivity for Brownian motion. Several recent theories relax the hypothesis of thermal equilibrium resulting in at least two main scenarios. With well separated timescales, as in aging glassy systems, equilibrium Fluctuation-Dissipation Theorem applies at each scale with its own "effective" temperature. With mixed timescales, as for example in active or granular fluids or in turbulence, temperature is no more well-defined, the dynamical nature of fluctuations fully emerges and a Generalized Fluctuation-Dissipation Theorem (GFDT) applies. Here, we study experimentally the mixed timescale regime by studying fluctuations and linear response in the Brownian motion of a rotating intruder immersed in a vibro-fluidized granular medium. Increasing the packing fraction, the system is moved from a dilute single-timescale regime toward a denser multiple-timescale stage. Einstein's relation holds in the former and is violated in the latter. The violation cannot be explained in terms of effective temperatures, while the GFDT is able to impute it to the emergence of a strong coupling between the intruder and the surrounding fluid. Direct experimental measurements confirm the development of spatial correlations in the system when the density is increased.Comment: 10 pages, 5 figure

Cages and anomalous diffusion in vibrated dense granular media

A vertically shaken granular medium hosts a blade rotating around a fixed vertical axis, which acts as a mesorheological probe. At high densities, independently from the shaking intensity, the blade's dynamics show strong caging effects, marked by transient sub-diffusion and a maximum in the velocity power density spectrum (vpds), at a resonant frequency $\sim 10$ Hz. Interpreting the data through a diffusing harmonic cage model allows us to retrieve the elastic constant of the granular medium and its collective diffusion coefficient. For high frequencies $f$, a tail $\sim 1/f$ in the vpds reveals non-trivial correlations in the intra-cage micro-dynamics. At very long times (larger than $10$ s), a super-diffusive behavior emerges, ballistic in the most extreme cases. Consistently, the distribution of slow velocity inversion times $\tau$ displays a power-law decay, likely due to persistent collective fluctuations of the host medium.Comment: 5 pages + 4 page of supplemental material, 6 figures, to be published on Phys. Rev. Let

Velocity fluctuations in cooling granular gases

We study the formation and the dynamics of correlations in the velocity field for 1D and 2D cooling granular gases with the assumption of negligible density fluctuations (``Homogeneous Velocity-correlated Cooling State'', HVCS). It is shown that the predictions of mean field models fail when velocity fluctuations become important. The study of correlations is done by means of molecular dynamics and introducing an Inelastic Lattice Maxwell Models. This lattice model is able to reproduce all the properties of the Homogeneous Cooling State and several features of the HVCS. Moreover it allows very precise measurements of structure functions and other crucial statistical indicators. The study suggests that both the 1D and the 2D dynamics of the velocity field are compatible with a diffusive dynamics at large scale with a more complex behavior at small scale. In 2D the issue of scale separation, which is of interest in the context of kinetic theories, is addressed.Comment: 24 pages, 16 figures, conference proceedin

A kinetic model for the finite-time thermodynamics of small heat engines

We study a molecular engine constituted by a gas of $N \sim 10^2$ molecules enclosed between a massive piston and a thermostat. The force acting on the piston and the temperature of the thermostat are cyclically changed with a finite period $\tau$. In the adiabatic limit $\tau \to \infty$, even for finite size $N$, the average work and heats reproduce the thermodynamic values, recovering the Carnot result for the efficiency. The system exhibits a stall time $\tau^*$ where net work is zero: for $\tau<\tau^*$ it consumes work instead of producing it, acting as a refrigerator or as a heat sink. At $\tau>\tau^*$ the efficiency at maximum power is close to the Curzorn-Ahlborn limit. The fluctuations of work and heat display approximatively a Gaussian behavior. Based upon kinetic theory, we develop a three-variables Langevin model where the piston's position and velocity are linearly coupled together with the internal energy of the gas. The model reproduces many of the system's features, such as the inversion of the work's sign, the efficiency at maximum power and the approximate shape of fluctuations. A further simplification in the model allows to compute analytically the average work, explaining its non-trivial dependence on $\tau$.Comment: 8 pages, 6 figures, accepted for publication on Physical Review

Clausius relation for active particles: what can we learn from fluctuations?

Many kinds of active particles, such as bacteria or active colloids, move in a thermostatted fluid by means of self-propulsion. Energy injected by such a non-equilibrium force is eventually dissipated as heat in the thermostat. Since thermal fluctuations are much faster and weaker than self-propulsion forces, they are often neglected, blurring the identification of dissipated heat in theoretical models. For the same reason, some freedom - or arbitrariness - appears when defining entropy production. Recently three different recipes to define heat and entropy production have been proposed for the same model where the role of self-propulsion is played by a Gaussian coloured noise. Here we compare and discuss the relation between such proposals and their physical meaning. One of these proposals takes into account the heat exchanged with a non-equilibrium active bath: such an "active heat" satisfies the original Clausius relation and can be experimentally verified.Comment: 10 pages, submitted to Entropy journal for the special issue "Thermodynamics and Statistical Mechanics of Small Systems" (see http://www.mdpi.com/journal/entropy/special_issues/small_systems

Coulomb friction driving Brownian motors

We review a family of models recently introduced to describe Brownian motors under the influence of Coulomb friction, or more general non-linear friction laws. It is known that, if the heat bath is modeled as the usual Langevin equation (linear viscosity plus white noise), additional non-linear friction forces are not sufficient to break detailed balance, i.e. cannot produce a motor effect. We discuss two possibile mechanisms to elude this problem. A first possibility, exploited in several models inspired to recent experiments, is to replace the heat bath's white noise by a ``collisional noise'', that is the effect of random collisions with an external equilibrium gas of particles. A second possibility is enlarging the phase space, e.g. by adding an external potential which couples velocity to position, as in a Klein-Kramers equation. In both cases, non-linear friction becomes sufficient to achieve a non-equilibrium steady state and, in the presence of an even small spatial asymmetry, a motor effect is produced.Comment: 19 pages, 10 figures, Proceedings of the Conference "Small system nonequilibrium fluctuations, dynamics and stochastics, and anomalous behavior", KITPC, Beijing, Chin

Linear and non-linear thermodynamics of a kinetic heat engine with fast transformations

We investigate a kinetic heat engine model constituted by particles enclosed in a box where one side acts as a thermostat and the opposite side is a piston exerting a given pressure. Pressure and temperature are varied in a cyclical protocol of period $\tau$ : their relative excursions, $\delta$ and $\epsilon$ respectively, constitute the thermodynamic forces dragging the system out-of-equilibrium. The analysis of the entropy production of the system allows to define the conjugated fluxes, which are proportional to the extracted work and the consumed heat. In the limit of small $\delta$ and $\epsilon$ the fluxes are linear in the forces through a $\tau$-dependent Onsager matrix whose off-diagonal elements satisfy a reciprocal relation. The dynamics of the piston can be approximated, through a coarse-graining procedure, by a Klein-Kramers equation which - in the linear regime - yields analytic expressions for the Onsager coefficients and the entropy production. A study of the efficiency at maximum power shows that the Curzon-Ahlborn formula is always an upper limit which is approached at increasing values of the thermodynamic forces, i.e. outside of the linear regime. In all our analysis the adiabatic limit $\tau \to \infty$ and the the small force limit $\delta,\epsilon \to 0$ are not directly related.Comment: 10 pages, 9 figure

Langevin equations from experimental data: the case of rotational diffusion in granular media

A model has two main aims: predicting the behavior of a physical system and understanding its nature, that is how it works, at some desired level of abstraction. A promising recent approach to model building consists in deriving a Langevin-type stochastic equation from a time series of empirical data. Even if the protocol is based upon the introduction of drift and diffusion terms in stochastic differential equations, its implementation involves subtle conceptual problems and, most importantly, requires some prior theoretical knowledge about the system. Here we apply this approach to the data obtained in a rotational granular diffusion experiment, showing the power of this method and the theoretical issues behind its limits. A crucial point emerged in the dense liquid regime, where the data reveal a complex multiscale scenario with at least one fast and one slow variable. Identifying the latter is a major problem within the Langevin derivation procedure and led us to introduce innovative ideas for its solution