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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.
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More than 30 years ago Edwards and co-authors proposed a model to describe the statistics of granular packings by an ensemble of equiprobable jammed states. Experimental tests of this model remained scarce so far. We introduce a simple system to analyze statistical properties of jammed granular ensembles to test Edwards theory. Identical spheres packed in a nearly two-dimensional geometrical confinement were studied in experiments and numerical simulations. When tapped, the system evolves towards a ground state, but due to incompatible domain structures it gets trapped. Analytical calculations reproduce relatively well our simulation results, which allows us to test Edwards theory on a coupled system of two subsystems with different properties. We find that the joint system can only be described by the Edwards theory if considered as a single system due to the constraints in the stresses. The results show counterintuitive effects as in the coupled system the change in the order parameter is opposite to what is expected from the change in the compactivity.
We study a simple model of periodic contraction and extension of large intruders in a granular bed to understand the mechanism for swimming in an otherwise solid media. Using an event-driven simulation, we find optimal conditions that idealized swimmers must use to critically fluidize a sand bed so that it is rigid enough to support a load when needed, but fluid enough to permit motion with minimal resistance. Swimmers - or other intruders - that agitate the bed too rapidly produce large voids that prevent traction from being achieved, while swimmers that move too slowly cannot travel before the bed re-solidifies around them i.e., the swimmers locally probe the fundamental time-scale in a granular packing.
A driven granular material, e.g. a vibrated box full of sand, is a stationary system which may be very far from equilibrium. The standard equilibrium statistical mechanics is therefore inadequate to describe fluctuations in such a system. Here we present numerical and analytical results concerning energy and injected power fluctuations. In the first part we explain how the study of the probability density function (pdf) of the fluctuations of total energy is related to the characterization of velocity correlations. Two different regimes are addressed: the gas driven at the boundaries and the homogeneously driven gas. In a granular gas, due to non-Gaussianity of the velocity pdf or lack of homogeneity in hydrodynamics profiles, even in the absence of velocity correlations, the fluctuations of total energy are non-trivial and may lead to erroneous conclusions about the role of correlations. In the second part of the chapter we take into consideration the fluctuations of injected power in driven granular gas models. Recently, real and numerical experiments have been interpreted as evidence that the fluctuations of power injection seem to satisfy the Gallavotti-Cohen Fluctuation Relation. We will discuss an alternative interpretation of such results which invalidates the Gallavotti-Cohen symmetry. Moreover, starting from the Liouville equation and using techniques from large deviation theory, the general validity of a Fluctuation Relation for power injection in driven granular gases is questioned. Finally a functional is defined using the Lebowitz-Spohn approach for Markov processes applied to the linear inelastic Boltzmann equation relevant to describe the motion of a tracer particle. Such a functional results to be different from injected power and to satisfy a Fluctuation Relation.
The terminology granular matter refers to systems with a large number of hard objects (grains) of mesoscopic size ranging from millimeters to meters. Geological examples include desert sand and the rocks of a landslide. But the scope of such systems is much broader, including powders and snow, edible products such a seeds and salt, medical products like pills, and extraterrestrial systems such as the surface regolith of Mars and the rings of Saturn. The importance of a fundamental understanding for granular matter properties can hardly be overestimated. Practical issues of current concern range from disaster mitigation of avalanches and explosions of grain silos to immense economic consequences within the pharmaceutical industry. In addition, they are of academic and conceptual importance as well as examples of systems far from equilibrium. Under many conditions of interest, granular matter flows like a normal fluid. In the latter case such flows are accurately described by the equations of hydrodynamics. Attention is focused here on the possibility for a corresponding hydrodynamic description of granular flows. The tools of nonequilibrium statistical mechanics, developed over the past fifty years for fluids composed of atoms and molecules, are applied here to a system of grains for a fundamental approach to both qualitative questions and practical quantitative predictions. The nonlinear Navier-Stokes equations and expressions for the associated transport coefficients are obtained.
We study the dynamical behavior of a square lattice Ising model with exchange and dipolar interactions by means of Monte Carlo simulations. After a sudden quench to low temperatures we find that the system may undergo a coarsening process where stripe phases with different orientations compete or alternatively it can relax initially to a metastable nematic phase and then decay to the equilibrium stripe phase through nucleation. We measure the distribution of equilibration times for both processes and compute their relative probability of occurrence as a function of temperature and system size. This peculiar relaxation mechanism is due to the strong metastability of the nematic phase, which goes deep in the low temperature stripe phase. We also measure quasi-equilibrium autocorrelations in a wide range of temperatures. They show a distinct decay to a plateau that we identify as due to a finite fraction of frozen spins in the nematic phase. We find indications that the plateau is a finite size effect. Relaxation times as a function of temperature in the metastable region show super-Arrhenius behavior, suggesting a possible glassy behavior of the system at low temperatures.
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