Growing interfaces in quenched disordered media

Abstract

We present the microscopic equation of growing interface with quenched noise for the Tang and Leschhorn model [{\em Phys. Rev.} {\bf A 45}, R8309 (1992)]. The evolution equations for the mean heigth and the roughness are reached in a simple way. Also, an equation for the interface activity density (i.e. interface density of free sites) as function of time is obtained. The microscopic equation allows us to express these equations in two contributions: the diffusion and the substratum one. All the equation shows the strong interplay between both contributions in the dynamics. A macroscopic evolution equation for the roughness is presented for this model for the critical pressure $p=0.461$. The dynamical exponent $\beta=0.629$ is analitically obtained in a simple way. Theoretical results are in excellent agreement with the Monte Carlo simulation.Comment: 6 pages and 3 figures. Conference on Percolation and disordered systems: theory and applications, Giessen, Germany, (July, 1998

Directed percolation depinning models: Evolution equations

“…We present here the microscopic equation for the TL model [4,5]. The time evolution equation for the interface height, in a time step δt, is…”

“…Several authors have been focused their attention on scaling properties and relationships between the dynamical and statics exponent for these models. The Tang and Leschhorn (TL) model has been recently reviewed by Braunstein et al [4,5] from a different point of view than the traditional one.…”

A Review of Growing Interfaces in Quenched Disordered Media

“…The cross mechanism between both contributions is not taken into account in this equation because the noise is additive. This cross mechanism is due to the fact that the quenched noise is coupled to the dynamic of the interface as was shown by Braunstein et al [19,20,21]. In order to explain the origin of this coupling let us introduce the passage to the continuum of our microscopic equation.…”

“…Equation (25) shows that the nonlinearity arises naturally as a consequence of the microscopic model. As we approach to the critical value, in the dynamical regime, the density of active sites f = G goes asymptotically to zero [20] and the coefficient of the nonlinear term becomes relevant. In these case the main responsible of the nonlinearities is the lateral contribution [see Eq.…”

“…A powerful method to derive the macroscopic behavior of these models is to attempt to write the microscopic equation for the evolution of the local height in these models. This is done via the master equation approach [18] or directly by writing the microscopic rules for the evolution of the local height [19,20,21]. These two methods have been proved to be equivalent [23].…”