Equilibrium distribution of the inherent states and their dynamics in glassy systems and granular media

Abstract

The present paper proposes a Statistical Mechanics approach to the inherent states of glassy systems and granular materials, following the original ideas developed by Edwards for granular materials. Two lattice models, a diluted Spin Glass and a system of hard-spheres under gravity, introduced in the context of glassy systems and granular materials, are evolved using a "tap dynamics" analogous to that of experiments on granular materials. The asymptotic macrostates, reached by the system, are shown to be described by a single thermodynamical parameter, and this parameter to coincide with the temperature, called the "configurational temperature", predicted assuming that the distribution among the inherent states satisfies the principle of maximum entropy.The thermodynamics of macroscopic systems evolving at equilibrium is well described by Statistical Mechanics. However there are many systems, typically found in "frozen states", where they do not evolve at all. These are, for example, supercooled liquids quenched at zero temperature in states, called inherent states [1,2], corresponding to the local minima of the potential energy in the 3N-dimensional configuration space of particle coordinates. Granular materials [3] at rest are another important example of system frozen in mechanically stable microstates. Grains are "frozen" because, due to their large masses [3], the thermal kinetic energy is negligible compared to the gravitational energy; thus the external bath temperature, T bath , can be considered equal to zero (by analogy with supercooled liquids, we call these mechanically stable configurations inherent states).In this paper, following the original ideas by Edwards for granular materials [4] we attempt to develop a unified Statistical Mechanics approach for the inherent states of glassy systems and granular materials along the line of Ref. [5]. The connection between Edwards approach and recent developments on glass theory has received much attention [6][7][8][9][10][11][12].The first step is to introduce a suitable dynamics which allows to explore the configurations of the inherent states. In granular materials the dynamics, from one stable microstate to another, can be induced by sequences of "taps", in which the energy is pumped into the system in pulses. Due to inelastic collisions the kinetic energy is totally dissipated after each tap, and the system is again frozen in one of its inherent states [13]. Similarly, in glass formers at zero temperature the dynamics, from one inherent state to another, can be induced by sequences of taps, where each tap consists in raising the bath temperature and, after a lapse of time τ 0 , quenching it back to zero. By repeating the process cyclically the system explores the space of the inherent states [5,[7][8][9]14,15]. For a tap of infinite length (τ 0 → ∞) the way to explore the inherent states coincides with the one used in [1,2] for a system of Lennard Jones mixture. In the approach of Barrat et al.[6] the system instead evolves in an out of equilibrium quasi-...

Experimental measurement of an effective temperature for jammed granular materials

“…The main assumption of this statistical formulation is that the different jammed configurations are taken to have the same statistical weight. Thus, observables can be obtained as ''flat'' averages of the jammed configurations (12,14,17,19,20,21,41). The validity of this assumption has been extensively debated in the literature (see for instance refs.…”

“…Until recently, the only evidence of the significance of the effective temperature in describing closely packed (jammed) granular media (3) has emerged from computer simulations of granular materials and other soft-matter systems and from analogies with glassy systems (12)(13)(14)(15)(16)(17)(18)(19)(20)(21). The existence of jammed reversible states in granular materials has been also suggested by compaction experiments using tapping, oscillatory compression, or sound propagation as the external perturbation (22)(23)(24)(25)(26).…”

Emergence of Gamma distributions in granular materials and packing models

“…The question whether a statistical mechanics approach can correctly describe granular systems has sparkled an intense debate in the scientific community in recent years [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. Such a statistical mechanics theory of granular matter must be able to connect the observable behavior of large aggregates with the structural and dynamical properties of the grains constituting these systems.…”

Lattice models of disorder with order

“…More generally, "T" shaped particles have also been considered, showing that the model possesses a thermodynamic-like temperature which is numerically measurable through the fluctuation-dissipation relations [14] and which coincides with a configurational pseudo-temperature proposed for granular matter [15]. The latter has also been measured in a system of monodisperse hard spheres on lattice [16]. Dimers placed on lattice with non tilted meshes also exhibit very well defined scaling laws [17] but with stretched exponential (the so called Kohlrausch-Williams-Watts law) rather than logarithmic compaction:…”