Dario Villamaina

Irreversible dynamics of a massive intruder in dense granular fluids

A Generalized Langevin Equation with exponential memory is proposed for the dynamics of a massive intruder in a dense granular fluid. The model reproduces numerical correlation and response functions, violating the Equilibrium Fluctuation-Dissipation Relations. The source of memory is identified in the coupling of the tracer velocity V with a spontaneous local velocity field U in the surrounding fluid: fluctuations of this field introduce a new time scale with its associated length scale. Such identification allows us to measure the intruders fluctuating entropy production as a function of V and U, obtaining a neat verification of the fluctuation relation.

Fluctuating hydrodynamics and correlation lengths in a driven granular fluid

Static and dynamical structure factors for shear and longitudinal modes of the velocity and density fields are computed for a granular system fluidized by a stochastic bath with friction. Analytical expressions are obtained through fluctuating hydrodynamics and are successfully compared with numerical simulations up to a volume fraction ~ 50%. The hydrodynamic noise is the sum of the external noise due to the bath and the internal one due to collisions. Only the latter is assumed to satisfy the fluctuation-dissipation relation with the average granular temperature. The static velocity structure factors and display a general non-constant behavior with two plateaux at large and small k, representing the granular temperature Tg and the bath temperature Tb > Tg respectively. From this behavior, two different velocity correlation lengths are measured, both increasing as the packing fraction is raised. This growth of spatial order is in agreement with the behavior of dynamical structure factors, the decay of which becomes slower and slower at increasing density.

Nonequilibrium and information: the role of cross correlations.

We discuss the relevance of information contained in cross correlations among different degrees of freedom, which is crucial in nonequilibrium systems. In particular we consider a stochastic system where two degrees of freedom X{1} and X{2}-in contact with two different thermostats-are coupled together. The production of entropy and the violation of equilibrium fluctuation-dissipation theorem (FDT) are both related to the cross correlation between X{1} and X{2}. Information about such cross correlation may be lost when single-variable reduced models for X_{1} are considered. Two different procedures are typically applied: (a) one totally ignores the coupling with X{2}; and (b) one models the effect of X{2} as an average memory effect, obtaining a generalized Langevin equation. In case (a) discrepancies between the system and the model appear both in entropy production and linear response; the latter can be exploited to define effective temperatures, but those are meaningful only when time scales are well separated. In case (b) linear response of the model well reproduces that of the system; however the loss of information is reflected in a loss of entropy production. When only linear forces are present, such a reduction is dramatic and makes the average entropy production vanish, posing problems in interpreting FDT violations.

Thinking outside the box: fluctuations and finite size effects

The isothermal compressibility of an interacting or non-interacting system may be extracted from the fluctuations of the number of particles in a well-chosen control volume. Finite size effects are prevalent and should be accounted for to obtain a meaningful, thermodynamic compressibility. In the traditional computational setup, where a given simulation box is replicated with periodic boundary conditions, we study particle number fluctuations outside the box (i.e. when the control volume exceeds the box itself), which bear relevant thermodynamic information. We also investigate the related problem of extracting the compressibility from the structure factor in the small wave-vector limit (k → 0). The calculation should be restricted to the discrete set of wave-vectors k that are compatible with the periodicity of the system, and we assess the consequences of considering other k values, a widespread error among beginners.

The out of equilibrium response function in sub-diffusive systems

We study the Einstein relation between spontaneous fluctuations and the response to an external perturbation for the comb model and the single file, which are examples of systems with sub-diffusive transport properties. The relevance of nonequilibrium conditions is investigated: when a stationary current (in the form of a drift or an energy flux) is present, the Einstein relation breaks down. In the case of the comb model, a general relation—appearing in the recent literature—between the response function and an unperturbed suitable correlation function allows us to explain the obtained results. This suggests that the relevant ingredient in breaking the Einstein formula, for stationary regimes, is not anomalous diffusion but the presence of currents driving the system out of equilibrium.

Einstein relation in superdiffusive systems

We study the Einstein relation between diffusion and response to an external field in systems showing superdiffusion. In particular, we investigate a continuous time Levy walk where the velocity remains constant for a time τ with distribution Pτ(τ) ~ τ−g. At varying g the diffusion can be standard or anomalous; in spite of this, if in the unperturbed system a current is absent, the Einstein relation holds. In the case where a current is present the scenario is more complicated and the usual Einstein relation fails. This suggests that the main ingredient for the breaking of the Einstein relation is not the anomalous diffusion but the presence of a mean drift (current).

Blast Dynamics in a Dissipative Gas.

The blast caused by an intense explosion has been extensively studied in conservative fluids, where the Taylor-von Neumann-Sedov hydrodynamic solution is a prototypical example of self-similarity driven by conservation laws. In dissipative media, however, energy conservation is violated, yet a distinctive self-similar solution appears. It hinges on the decoupling of random and coherent motion permitted by a broad class of dissipative mechanisms. This enforces a peculiar layered structure in the shock, for which we derive the full hydrodynamic solution, validated by a microscopic approach based on molecular dynamics simulations. We predict and evidence a succession of temporal regimes, as well as a long-time corrugation instability, also self-similar, which disrupts the blast boundary. These generic results may apply from astrophysical systems to granular gases, and invite further cross-fertilization between microscopic and hydrodynamic approaches of shock waves.

Microscopic origin of self-similarity in granular blast waves

The self-similar expansion of a blast wave, well-studied in air, has peculiar counterparts in dense and dissipative media such as granular gases. Recent results have shown that, while the traditional Taylor-von Neumann-Sedov (TvNS) derivation is not applicable to such granular blasts, they can nevertheless be well understood via a combination of microscopic and hydrodynamic insights. In this article, we provide a detailed analysis of these methods associating molecular dynamics simulations and continuum equations, which successfully predict hydrodynamic profiles, scaling properties, and the instability of the self-similar solution. We also present new results for the energy conserving case, including the particle-level analysis of the classic TvNS solution and its breakdown at higher densities.

Large Deviations of Brownian Motors

We review some recent results on the behavior of fluctuations in the framework of molecular motors. We present both theoretical and experimental studies, pointing out some interesting analogies shown by the large deviations of quantities such as work and entropy production in different systems. These common features reveal some underlying symmetry properties governing the nonequilibrium behavior of Brownian motors.

Anomalous Transport and Non-Equilibrium

This chapter the additional ingredient of anomalous diffusion, combined with non equilibrium conditions, is studied. The analysis starts with a random walk on a comb lattice, which can be analytically solvable. A detailed analysis of the “single file model” is then shown and the response analysis is similar to that one in higher dimensions. The chapter ends with the study of a ratchet effect in a fragile glass former. Because of the presence of disorder, under certain conditions, an intruder in a glass former exhibits subdiffusion. Despite of the great differences from the “family” of the non equilibrium steady states, also in this system it is possible to observe a ratchet effect, although characterized by a subvelocity due to the disorder.