Finding the optimal random packing of non-spherical particles is an open problem with great significance in a broad range of scientific and engineering fields. So far, this search has been performed only empirically on a case-by-case basis, in particular, for shapes like dimers, spherocylinders and ellipsoids of revolution. Here we present a mean-field formalism to estimate the packing density of axisymmetric non-spherical particles. We derive an analytic continuation from the sphere that provides a phase diagram predicting that, for the same coordination number, the density of monodisperse random packings follows the sequence of increasing packing fractions: spheres ooblate ellipsoids oprolate ellipsoids odimers ospherocylinders. We find the maximal packing densities of 73.1% for spherocylinders and 70.7% for dimers, in good agreement with the largest densities found in simulations. Moreover, we find a packing density of 73.6% for lens-shaped particles, representing the densest random packing of the axisymmetric objects studied so far.

The anomalous (i.e. non-Gaussian) dynamics of particles subject to a deterministic acceleration and a series of 'random kicks' is studied. Based on an extension of the concept of continuous time random walks to position-velocity space, a new fractional equation of the Kramers-FokkerPlanck type is derived. The associated collision operator necessarily involves a fractional substantial derivative, representing important nonlocal couplings in time and space. For the force-free case, a closed solution is found and discussed.

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.

We consider joint probability distributions for the class of coupled Langevin equations introduced by Fogedby [H. C. Fogedby, Phys. Rev. E 50, 1657 (1994)]. We generalize well-known results for the single-time probability distributions to the case of N -time joint probability distributions. It is shown that these probability distribution functions can be obtained by an integral transform from distributions of a Markovian process. The integral kernel obeys a partial differential equation with fractional time derivatives reflecting the non-Markovian character of the process.