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

Cluster growth in far-from-equilibrium particle models with diffusion, detachment, reattachment and deposition

Monolayer cluster growth in far-from-equilibrium systems is investigated by applying simulation and analytic techniques to minimal hard core particle (exclusion) models. The first model (I), for post-deposition coarsening dynamics, contains mechanisms of diffusion, attachment, and slow activated detachment (at rate epsilon<<1) of particles on a line. Simulation shows three successive regimes of cluster growth: fast attachment of isolated particles; detachment allowing further (epsilon t)^(1/3) coarsening of average cluster size; and t^(-1/2) approach to a saturation size going like epsilon^(-1/2). Model II generalizes the first one in having an additional mechanism of particle deposition into cluster gaps, suppressed for the smallest gaps. This model exhibits early rapid filling, leading to slowing deposition due to the increasing scarcity of deposition sites, and then continued power law (epsilon t)^(1/2) cluster size coarsening through the redistribution allowed by slow detachment. The basic (epsilon t)^(1/3) domain growth laws and epsilon^(-1/2) saturation in model I are explained by a simple scaling picture. A second, fuller approach is presented which employs a mapping of cluster configurations to a column picture and an approximate factorization of the cluster configuration probability within the resulting master equation. This allows quantitative results for the saturation of model I in excellent agreement with the simulation results. For model II, it provides a one-variable scaling function solution for the coarsening probability distribution, and in particular quantitative agreement with the cluster length scaling and its amplitude.Comment: Accepted in Phys. Rev. E; 9 pages with figure

Real-space renormalisation group approach to driven diffusive systems

We introduce a real-space renormalisation group procedure for driven diffusive systems which predicts both steady state and dynamic properties. We apply the method to the boundary driven asymmetric simple exclusion process and recover exact results for the steady state phase diagram, as well as the crossovers in the relaxation dynamics for each phase.Comment: 10 pages, 5 figure

Disordered asymmetric simple exclusion process: mean-field treatment

We provide two complementary approaches to the treatment of disorder in a fundamental nonequilibrium model, the asymmetric simple exclusion process. Firstly, a mean-field steady state mapping is generalized to the disordered case, where it provides a mapping of probability distributions and demonstrates how disorder results in a new flat regime in the steady state current--density plot for periodic boundary conditions. This effect was earlier observed by Tripathy and Barma but we provide treatment for more general distributions of disorder, including both numerical results and analytic expressions for the width $2\Delta_C$ of the flat section. We then apply an argument based on moving shock fronts to show how this leads to an increase in the high current region of the phase diagram for open boundary conditions. Secondly, we show how equivalent results can be obtained easily by taking the continuum limit of the problem and then using a disordered version of the well-known Cole--Hopf mapping to linearize the equation. Within this approach we show that adding disorder induces a localization transformation (verified by numerical scaling), and $\Delta_C$ maps to an inverse localization length, helping to give a new physical interpretation to the problem.Comment: 13 pages, 16 figures. Submitted to Phys. Rev.

Non-universal disordered Glauber dynamics

We consider the one-dimensional Glauber dynamics with coupling disorder in terms of bilinear fermion Hamiltonians. Dynamic exponents embodied in the spectrum gap of these latter are evaluated numerically by averaging over both binary and Gaussian disorder realizations. In the first case, these exponents are found to follow the non-universal values of those of plain dimerized chains. In the second situation their values are still non-universal and sub-diffusive below a critical variance above which, however, the relaxation time is suggested to grow as a stretched exponential of the equilibrium correlation length.Comment: 11 pages, 5 figures, brief addition

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

Limit order market analysis and modelling: on an universal cause for over-diffusive prices

We briefly review data analysis of the Island order book, part of NASDAQ, which suggests a framework to which all limit order markets should comply. Using a simple exclusion particle model, we argue that short-time price over-diffusion in limit order markets is due to the non-equilibrium of order placement, cancellation and execution rates, which is an inherent feature of real limit order markets.Comment: 6 pages, 3 figures. Contribution to the proceedings of Econophysics Bali Conference 200