Theoretical continuous equation derived from the microscopic dynamics for growing interfaces in quenched media

Abstract

We present an analytical continuous equation for the Tang and Leschhorn model [Phys. Rev. A 45, R8309 (1992)] derived from their microscopic rules using a regularization procedure. As well in this approach, the nonlinear term (nablah)(2) arises naturally from the microscopic dynamics even if the continuous equation is not the Kardar-Parisi-Zhang equation [Phys. Rev. Lett. 56, 889 (1986)] with quenched noise (QKPZ). Our equation is similar to a QKPZ equation but with multiplicative quenched and thermal noise. The numerical integration of our equation reproduces all the scaling exponents of the directed percolation depinning model.

Model for domain wall avalanches in ferromagnetic thin films

“…For a given parameter p ∈ [0, 1], if ξ(i, k) ≤ p the node (i, k) is active, otherwise is inactive. To characterize the disorder in the lattice we use the activity function [23] where θ(x) is the unit step function defined as θ(x) = 1 for x ≥ 0 and θ(x) = 0 for x < 0. A dipole located at the nodes of the DW has perpendicular component to the easy direction.…”

Evolution equation for a model of surface relaxation in complex networks

“…where G i represents the deterministic growth rules that cause evolution of the node i, τ = Nδt is the mean time to grow a layer of the interface, and η i is a Gaussian noise with zero mean and covariance given by [18,19] …”

Adsorption–desorption kinetics of the monomer–dimer mixture

“…The master equation approach is the usual methodology to calculate the kinetic equations. However, this formulation cannot always be applied in a straightforward way, particularly for irreversible kinetics like the irreversible growth models [15,16].…”

“…This technique, the so-called local evolution rules, has been used in different systems in the past, as for example in the irreversible growth models [15,16], adsorption-desorption kinetics of molecules with multisite occupancy [17][18][19], models of surface reactions [20], etc. Taking into account the characteristics of the system, we describe the evolution of a generic site ''i'', in the following way: at time t a given site i can be empty, occupied by a monomer or by the left/right part of a dimer.…”

“…This approach is the so-called local evolution rules, and was used to describe the irreversible growth models, particularly to derive the Langevin equations in (1 + 1)-dimensional systems [15,16], as well as, the dimer kinetics in a one-dimensional lattice [17]. The method is based on the definition of the local Hamiltonian and after monitoring the time evolution of a chosen site and averaging adequately, a hierarchy of coupled differential equations is obtained for the coverage and higher correlators.…”