The Statistical Physics of Athermal Materials

At the core of equilibrium statistical mechanics lies the notion of statistical ensembles: a collection of microstates, each occurring with a given a priori probability that depends only on a few macroscopic parameters such as temperature, pressure, volume, and energy. In this review article, we discuss recent advances in establishing statistical ensembles for athermal materials. The broad class of granular and particulate materials is immune from the effects of thermal fluctuations because the constituents are macroscopic. In addition, interactions between grains are frictional and dissipative, which invalidates the fundamental postulates of equilibrium statistical mechanics. However, granular materials exhibit distributions of microscopic quantities that are reproducible and often depend on only a few macroscopic parameters. We explore the history of statistical ensemble ideas in the context of granular materials, clarify the nature of such ensembles and their foundational principles, highlight advances in testing key ideas, and discuss applications of ensembles to analyze the collective behavior of granular materials

The Inherent Structure Landscape Connection Between Liquids, Granular materials and the Jamming Phase Diagram

We provide a comprehensive picture of the jamming phase diagram by connecting the athermal, granular ensemble of jammed states and the equilibrium fluid through the inherent structure paradigm for a system hard discs confined to a narrow channel. The J-line is shown to be divided into packings that are thermodynamically accessible from the equilibrium fluid and inaccessible packings. The J-point is found to occur at the transition between these two sets of packings and is located at the maximum the inherent structure distribution. A general thermodynamic argument suggests that the density of the states at the configurational entropy maximum represents a lower bound on the J-point density in hard sphere systems. Finally, we find that the granular and fluid systems only occupy the same set of inherent structures, under the same thermodynamic conditions, at two points, corresponding to zero and infinite pressures, where they sample the J-point states and the most dense packing respectively.Comment: 5 pages, 3 Figure

Experimental and computational studies of jamming

Jamming is a common feature of out of equilibrium systems showing slow relaxation dynamics. Here we review our efforts in understanding jamming in granular materials using experiments and computer simulations. We first obtain an estimation of an effective temperature for a slowly sheared granular material very close to jamming. The measurement of the effective temperature is realized in the laboratory by slowly shearing a closely-packed ensemble of spherical beads confined by an external pressure in a Couette geometry. All the probe particles, independent of their characteristic features, equilibrate at the same temperature, given by the packing density of the system. This suggests that the effective temperature is a state variable for the nearly jammed system. Then we investigate numerically whether the effective temperature can be obtained from a flat average over the jammed configuration at a given energy in the granular packing, as postulated by the thermodynamic approach to grains.Comment: 20 pages, 9 figure

Stabilization of nonlinear velocity profiles in athermal systems undergoing planar shear flow

We perform molecular dynamics simulations of model granular systems undergoing boundary-driven planar shear flow in two spatial dimensions with the goal of developing a more complete understanding of how dense particulate systems respond to applied shear. In particular, we are interested in determining when these systems will possess linear velocity profiles and when they will develop highly localized velocity profiles in response to shear. In previous work on similar systems we showed that nonlinear velocity profiles form when the speed of the shearing boundary exceeds the speed of shear waves in the material. However, we find that nonlinear velocity profiles in these systems are unstable at very long times. The degree of nonlinearity slowly decreases in time; the velocity profiles become linear when the granular temperature and density profiles are uniform across the system at long times. We measure the time $t_l$ required for the velocity profiles to become linear and find that $t_l$ increases as a power-law with the speed of the shearing boundary and increases rapidly as the packing fraction approaches random close packing. We also performed simulations in which differences in the granular temperature across the system were maintained by vertically vibrating one of the boundaries during shear flow. We find that nonlinear velocity profiles form and are stable at long times if the difference in the granular temperature across the system exceeds a threshold value that is comparable to the glass transition temperature in an equilibrium system at the same average density. Finally, the sheared and vibrated systems form stable shear bands, or highly localized velocity profiles, when the applied shear stress is lowered below the yield stress of the static part of the system.Comment: 11 pages, 14 figure

What is the temperature of a granular medium?

In this paper we discuss whether thermodynamical concepts and in particular the notion of temperature could be relevant for the dynamics of granular systems. We briefly review how a temperature-like quantity can be defined and measured in granular media in very different regimes, namely the glassy-like, the liquid-like and the granular gas. The common denominator will be given by the Fluctuation-Dissipation Theorem, whose validity is explored by means of both numerical and experimental techniques. It turns out that, although a definition of a temperature is possible in all cases, its interpretation is far from being obvious. We discuss the possible perspectives both from the theoretical and, more importantly, from the experimental point of view

Dynamic heterogeneity in amorphous materials

Amorphous solids are mechanically rigid while possessing a disordered structure similar to that of dense liquids. Recent research indicates that dynamical heterogeneity, spatio-temporal fluctuations in local dynamical behavior, might help understanding the statistical mechanics of glassy states.Comment: 7 pages; 5 figures -- "Trends" article published by Physics at http://physics.aps.org/articles/v4/4

Evolution of displacements and strains in sheared amorphous solids

The local deformation of two-dimensional Lennard-Jones glasses under imposed shear strain is studied via computer simulations. Both the mean squared displacement and mean squared strain rise linearly with the length of the strain interval $\Delta \gamma$ over which they are measured. However, the increase in displacement does not represent single-particle diffusion. There are long-range spatial correlations in displacement associated with slip lines with an amplitude of order the particle size. Strong dependence on system size is also observed. The probability distributions of displacement and strain are very different. For small $\Delta \gamma$ the distribution of displacement has a plateau followed by an exponential tail. The distribution becomes Gaussian as $\Delta \gamma$ increases to about .03. The strain distributions consist of sharp central peaks associated with elastic regions, and long exponential tails associated with plastic regions. The latter persist to the largest $\Delta \gamma$ studied.Comment: Submitted to J. Phys. Cond. Mat. special volume for PITP Conference on Mechanical Behavior of Glassy Materials. 16 Pages, 8 figure

Rheology of Granular Materials: Dynamics in a Stress Landscape

We present a framework for analyzing the rheology of dense driven granular materials, based on a recent proposal of a stress-based ensemble. In this ensemble fluctuations in a granular system near jamming are controlled by a temperature-like parameter, the angoricity, which is conjugate to the stress of the system. In this paper, we develop a model for slowly driven granular materials based on the stress ensemble and the idea of a landscape in stress space. The idea of an activated process driven by the angoricity has been shown by Behringer et al (2008) to describe the logarithmic strengthening of granular materials. Just as in the Soft Glassy Rheology (SGR) picture, our model represents the evolution of a small patch of granular material (a mesoscopic region) in a stress-based trap landscape. The angoricity plays the role of the fluctuation temperature in SGR. We determine (a) the constitutive equation, (b) the yield stress, and (c) the distribution of stress dissipated during granular shearing experiments, and compare these predictions to experiments of Hartley & Behringer (2003).Comment: 17 pages, 4 figure