Statistical Mechanics of Vibration-Induced Compaction of Powders

We propose a theory which describes the density relaxation of loosely packed, cohesionless granular material under mechanical tapping. Using the compactivity concept we develope a formalism of statistical mechanics which allows us to calculate the density of a powder as a function of time and compactivity. A simple fluctuation-dissipation relation which relates compactivity to the amplitude and frequency of a tapping is proposed. Experimental data of E.R.Nowak et al. [{\it Powder Technology} 94, 79 (1997) ] show how density of initially deposited in a fluffy state powder evolves under carefully controlled tapping towards a random close packing (RCP) density. Ramping the vibration amplitude repeatedly up and back down again reveals the existence of reversible and irreversible branches in the response. In the framework of our approach the reversible branch (along which the RCP density is obtained) corresponds to the steady state solution of the Fokker-Planck equation whereas the irreversible one is represented by a superposition of "excited states" eigenfunctions. These two regimes of response are analyzed theoretically and a qualitative explanation of the hysteresis curve is offered.Comment: 11 pages, 2 figures, Latex. Revised tex

Dynamic mechanical response of polymer networks

The dynamic-mechanical response of flexible polymer networks is studied in the framework of tube model, in the limit of small affine deformations, using the approach based on Rayleighian dissipation function. The dynamic complex modulus G* is calculated from the analysis of a network strand relaxation to the new equilibrium conformation around the distorted primitive path. Chain equilibration is achieved via a sliding motion of polymer segments along the tube, eliminating the inhomogeneity of the polymer density caused by the deformation. The characteristic relaxation time of this motion separates the low-frequency limit of the complex modulus from the high-frequency one, where the main role is played by chain entanglements, analogous to the rubber plateau in melts. The dependence of storage and loss moduli, G' and G'', on crosslink and entanglement densities gives an interpolation between polymer melts and crosslinked networks. We discuss the experimental implications of the rather short relaxation time and the slow square-root variation of the moduli and the loss factor tan at higher frequencies.Comment: Journal of Chemical Physics (Oct-2000); Lates, 4 EPS figures include

Mechanism for the failure of the Edwards hypothesis in the SK spin glass

The dynamics of the SK model at T=0 starting from random spin configurations is considered. The metastable states reached by such dynamics are atypical of such states as a whole, in that the probability density of site energies, $p(\lambda)$, is small at $\lambda=0$. Since virtually all metastable states have a much larger $p(0)$, this behavior demonstrates a qualitative failure of the Edwards hypothesis. We look for its origins by modelling the changes in the site energies during the dynamics as a Markov process. We show how the small $p(0)$ arises from features of the Markov process that have a clear physical basis in the spin-glass, and hence explain the failure of the Edwards hypothesis.Comment: 5 pages, new title, modified text, additional reference

Dynamical response functions in models of vibrated granular media

In recently introduced schematic lattice gas models for vibrated dry granular media, we study the dynamical response of the system to small perturbations of shaking amplitudes and its relations with the characteristic fluctuations. Strong off equilibrium features appear and a generalized version of the fluctuation dissipation theorem is introduced. The relations with thermal glassy systems and the role of Edwards' compactivity are discussed.Comment: 12 pages, 2 postscript figure

Force correlations and arches formation in granular assemblies

In the context of a simple microscopic schematic scalar model we study the effects of spatial correlations in force transmission in granular assemblies. We show that the parameters of the normalized weights distribution function, $P(v)\sim v^{\alpha}\exp(-v/\phi)$, strongly depend on the spatial extensions, $\xi_V$, of such correlations. We show, then, the connections between measurable macroscopic quantities and microscopic mechanisms enhancing correlations. In particular we evaluate how the exponential cut-off, $\phi(\xi_V)$, and the small forces power law exponent, $\alpha(\xi_V)$, depend on the correlation length, $\xi_V$. If correlations go to infinity, weights are power law distributed.Comment: 6 page

Midgap states and charge inhomogeneities in corrugated graphene

We study the changes induced by the effective gauge field due to ripples on the low energy electronic structure of graphene. We show that zero energy Landau levels will form, associated to the smooth deformation of the graphene layer, when the height corrugation, $h$, and the length of the ripple, $l$, are such that $h^2 / (l a) \gtrsim 1$, where $a$ is the lattice constant. The existence of localized levels gives rise to a large compressibility at zero energy, and to the enhancement of instabilities arising from electron-electron interactions including electronic phase separation. The combined effect of the ripples and an external magnetic field breaks the valley symmetry of graphene leading to the possibility of valley selection

Forces and Conservation Laws for Motion on Our Spheroidal Earth

We explore the forces and conservation laws that govern the motion of a hockey puck that slides without friction on a smooth, rotating, self-gravitating spheroid. The earth\u27s oblate spheroidal shape (apart from small-scale surface features) is determined by balancing the gravitational forces that hold it together against the centrifugal forces that try to tear it apart. The earth achieves this shape when the apparent gravitational force on the puck, defined as the vector sum of the gravitational and centrifugal forces, is perpendicular to the earth\u27s surface at every point on the surface. Thus, the earth\u27s spheroidal deformations neutralize the centrifugal and gravitational forces on the puck, leaving only the Coriolis force to govern its motion. Motion on the spheroid therefore differs profoundly from motion on a rotating sphere, for which the centrifugal force plays a key role. Kinetic energy conservation reflects this difference: On a stably rotating spheroid, the kinetic energy is conserved in the rotating frame, whereas on a rotating sphere, it is conserved in the inertial frame. We derive these results and illustrate them using CorioVis software for visualizing the motion of a puck on the earth\u27s spheroidal surface

Periodic nonlinear sliding modes for two uniformly magnetized spheres

A uniformly magnetized sphere slides without friction along the surface of a second, identical sphere that is held fixed in space, subject to the magnetic force and torque of the fixed sphere and the normal force. The free sphere has two stable equilibrium positions and two unstable equilibrium positions. Two small-amplitude oscillatory modes describe the sliding motion of the free sphere near each stable equilibrium, and an unstable oscillatory mode describes the motion near each unstable equilibrium. The three oscillatory modes remain periodic at finite amplitudes, one bifurcating into mixed modes and circumnavigating the free sphere at large energies. For small energies, the free sphere is confined to one of the two discontiguous domains, each surrounding a stable equilibrium position. At large energies, these domains merge and the free sphere may visit both positions. The critical energy at which these domains merge coincides with the cumulation point of an infinite cascade of mixed-mode bifurcations. These findings exploit the equivalence of the force and torque between two uniformly magnetized spheres and the force and torque between two equivalent point dipoles, and offer clues to the rich nonlinear dynamics of this system