Marginal scaling scenario and analytic results for a glassy compaction model

A diffusion-deposition model for glassy dynamics in compacting granular systems is treated by time scaling and by a method that provides the exact asymptotic (long time) behavior. The results include Vogel-Fulcher dependence of rates on density, inverse logarithmic time decay of densities, exponential distribution of decay times and broadening of noise spectrum. These are all in broad agreement with experiments. The main characteristics result from a marginal rescaling in time of the control parameter (density); this is argued to be generic for glassy systems.Comment: 4 pages, 4 figure

Fluctuation-dissipation relation and the Edwards entropy for a glassy granular compaction model

We analytically study a one dimensional compaction model in the glassy regime. Both correlation and response functions are calculated exactly in the evolving dense and low tapping strength limit, where the density relaxes in a $1/\ln t$ fashion. The response and correlation functions turn out to be connected through a non-equilibrium generalisation of the fluctuation-dissipation theorem. The initial response in the average density to an increase in the tapping strength is shown to be negative, while on longer timescales it is shown to be positive. On short time scales the fluctuation-dissipation theorem governs the relation between correlation and response, and we show that such a relationship also exists for the slow degrees of freedom, albeit with a different temperature. The model is further studied within the statistical theory proposed by Edwards and co-workers, and the Edwards entropy is calculated in the large system limit. The fluctuations described by this approach turn out to match the fluctuations as calculated through the dynamical consideration. We believe this to be the first time these ideas have been analytically confirmed in a non-mean-field model.Comment: 4 pages, 3 figure

Cross-link governed dynamics of biopolymer networks

Cytoskeletal networks of biopolymers are cross-linked by a variety of proteins. Experiments have shown that dynamic cross-linking with physiological linker proteins leads to complex stress relaxation and enables network flow at long times. We present a model for the mechanical properties of transient networks. By a combination of simulations and analytical techniques we show that a single microscopic timescale for cross-linker unbinding leads to a broad spectrum of macroscopic relaxation times, resulting in a weak power-law dependence of the shear modulus on frequency. By performing rheological experiments, we demonstrate that our model quantitatively describes the frequency behavior of actin network cross-linked with $\alpha$-Actinin-$4$ over four decades in frequency.Comment: 4 page

Continuum approach to wide shear zones in quasi-static granular matter

Slow and dense granular flows often exhibit narrow shear bands, making them ill-suited for a continuum description. However, smooth granular flows have been shown to occur in specific geometries such as linear shear in the absence of gravity, slow inclined plane flows and, recently, flows in split-bottom Couette geometries. The wide shear regions in these systems should be amenable to a continuum description, and the theoretical challenge lies in finding constitutive relations between the internal stresses and the flow field. We propose a set of testable constitutive assumptions, including rate-independence, and investigate the additional restrictions on the constitutive relations imposed by the flow geometries. The wide shear layers in the highly symmetric linear shear and inclined plane flows are consistent with the simple constitutive assumption that, in analogy with solid friction, the effective-friction coefficient (ratio between shear and normal stresses) is a constant. However, this standard picture of granular flows is shown to be inconsistent with flows in the less symmetric split-bottom geometry - here the effective friction coefficient must vary throughout the shear zone, or else the shear zone localizes. We suggest that a subtle dependence of the effective-friction coefficient on the orientation of the sliding layers with respect to the bulk force is crucial for the understanding of slow granular flows.Comment: 11 pages, 7 figure

Granular Flows in Split-Bottom Geometries

There is a simple and general experimental protocol to generate slow granular flows that exhibit wide shear zones, qualitatively different from the narrow shear bands that are usually observed in granular materials . The essence is to drive the granular medium not from the sidewalls, but to split the bottom of the container that supports the grains in two parts and slide these parts past each other. Here we review the main features of granular flows in such split-bottom geometries.Comment: 8 pages, 10 figures, accepted for Soft Matte

Exact joint density-current probability function for the asymmetric exclusion process

We study the asymmetric exclusion process with open boundaries and derive the exact form of the joint probability function for the occupation number and the current through the system. We further consider the thermodynamic limit, showing that the resulting distribution is non-Gaussian and that the density fluctuations have a discontinuity at the continuous phase transition, while the current fluctuations are continuous. The derivations are performed by using the standard operator algebraic approach, and by the introduction of new operators satisfying a modified version of the original algebra.Comment: 4 pages, 3 figure

Exact probability function for bulk density and current in the asymmetric exclusion process

We examine the asymmetric simple exclusion process with open boundaries, a paradigm of driven diffusive systems, having a nonequilibrium steady state transition. We provide a full derivation and expanded discussion and digression on results previously reported briefly in M. Depken and R. Stinchcombe, Phys. Rev. Lett. {\bf 93}, 040602, (2004). In particular we derive an exact form for the joint probability function for the bulk density and current, both for finite systems, and also in the thermodynamic limit. The resulting distribution is non-Gaussian, and while the fluctuations in the current are continuous at the continuous phase transitions, the density fluctuations are discontinuous. The derivations are done by using the standard operator algebraic techniques, and by introducing a modified version of the original operator algebra. As a byproduct of these considerations we also arrive at a novel and very simple way of calculating the normalization constant appearing in the standard treatment with the operator algebra. Like the partition function in equilibrium systems, this normalization constant is shown to completely characterize the fluctuations, albeit in a very different manner.Comment: 10 pages, 4 figure