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

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

Logarithmic coarsening and glassy behavior in a polymer model with mass-dependent diffusion

We present a model of polymer growth and diffusion with frustration mechanisms for density increase and with diffusion rates of Arrhenius form with mass-dependent energy barriers Gamma(m) ~ (m-1)^gamma. It shows non-universal logarithmic coarsening involving the exponent gamma. Strong-glass behavior is found in the typical times for disappearance of all polymers up to a given length, without reference to the equilibrium states of the macroscopic system. These features are predicted by numerical simulations, scaling theories and an analytic solution of the master equation within an independent interval approximation, which also provides the cluster size distribution.Comment: 16 pages, including 7 figures. To be published in Phys. Rev.

Diffusion-annihilation dynamics in one spatial dimension

We discuss a reaction-diffusion model in one dimension subjected to an external driving force. Each lattice site may be occupied by at most one particle. The particles hop with asymmetric rates (the sum of which is one) to the right or left nearest neighbour site if it is vacant, and annihilate with rate one if it is occupied. We compute the long time behaviour of the space dependent average density in states where the initial density profiles are step functions. We also compute the exact time dependence of the particle density for uncorrelated random initial conditions. The representation of the uncorrelated random initial state and also of the step function profile in terms of free fermions allows the calculation of time-dependent higher order correlation functions. We outline the procedure using a field theoretic approach.Comment: 26 pages, 1 Postscript figure, uses epsf.st

Analyzing and modelling 1+1d markets

We report a statistical analysis of the Island ECN (NASDAQ) order book. We determine the static and dynamic properties of this system, and then analyze them from a physicist's viewpoint using an equivalent particle system obtained by treating orders as massive particles and price as position. We identify the fundamental dynamical processes, test existing particles models of such markets against our findings, and introduce a new model of limit order markets.Comment: 17 pages, 14 figures, small typos correctio

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