Edwards statistical mechanics for jammed granular matter

Abstract

In 1989, Sir Sam Edwards made the visionary proposition to treat jammed granular materials using a volume ensemble of equiprobable jammed states in analogy to thermal equilibrium statistical mechanics, despite their inherent athermal features. Since then, the statistical mechanics approach for jammed matter -one of the very few generalizations of Gibbs-Boltzmann statistical mechanics to out of equilibrium matter -has garnered an extraordinary amount of attention by both theorists and experimentalists. Its importance stems from the fact that jammed states of matter are ubiquitous in nature appearing in a broad range of granular and soft materials such as colloids, emulsions, glasses, and biomatter. Indeed, despite being one of the simplest states of matter -primarily governed by the steric interactions between the constitutive particles -a theoretical understanding based on first principles has proved exceedingly challenging. Here, we review a systematic approach to jammed matter based on the Edwards statistical mechanical ensemble. We discuss the construction of microcanonical and canonical ensembles based on the volume function, which replaces the Hamiltonian in jammed systems. The importance of approximation schemes at various levels is emphasized leading to quantitative predictions for ensemble averaged quantities such as packing fractions and contact force distributions. An overview of the phenomenology of jammed states and experiments, simulations, and theoretical models scrutinizing the strong assumptions underlying Edwards' approach is given including recent results suggesting the validity of Edwards ergodic hypothesis for jammed states. A theoretical framework for packings whose constitutive particles range from spherical to non-spherical shapes like dimers, polymers, ellipsoids, spherocylinders or tetrahedra, hard and soft, frictional, frictionless and adhesive, monodisperse and polydisperse particles in any dimensions is discussed providing insight into an unifying phase diagram for all jammed matter. Furthermore, the connection between the Edwards' ensemble of metastable jammed states and metastability in spin-glasses is established. This highlights that the packing problem can be understood as a constraint satisfaction problem for excluded volume and force and torque balance leading to a unifying framework between the Edwards ensemble of equiprobable jammed states and out-of-equilibrium spin-glasses.

Glass and Jamming Transitions: From Exact Results to Finite-Dimensional Descriptions

“…Although relatively few of the many predictions of the exact solution have been tested, (i) and (ii) establish a strong foundation for exploring these predictions. Because (i) and (ii) have already been extensively reviewed (90,114,115), we here only focus on the aspects most related to the exact solution. Two main families of out-of-equilibrium protocols have been developed for generating hard-sphere configurations at jamming.…”

Shape Universality Classes in the Random Sequential Adsorption of Nonspherical Particles

“…Jammed systems are particularly challenging, since they are dominated by the geometry of the particles and are not described by conventional equilibrium statistical mechanics [4]. Exploring the effect of shape variation thus relies on extensive computer simulations [5][6][7][8] or mean-field theories whose solutions require similar computational efforts [9,10].…”

Maximally dense random packings of cubes and cuboids via a novel inverse packing method

“…However, there is no unique density value for the disordered packings of cubes obtained via simulations or theories. 25 The jammed assemblies of soft cubes obtained by Smith et al via the conjugate gradient method exhibit high nematic order 26 and are not amorphous. 27 The packing density of the ideal jamming point of cubes at zero shear stress obtained via the variable-cell method is about 0.734.…”

“…However, the studies of the disordered packings of such packings yield a wide spectrum of packing characteristics. [19][20][21][22][23][24][25][26][27][28][29][30] For example, the density of the disordered packings of cubes varies a lot due to the different particle materials and packing protocols used in experiments or the different particle models (hard vs. soft particles) and packing algorithms (thermal vs. athermal) in simulations. [19][20][21][22][23][24] In addition, the degree of order/disorder in the disordered packing of cubes (often referred as random close packings (RCPs)) generated by different methods varies significantly.…”