Time-reversed dielectric-breakdown model for erosion phenomena.

Abstract

A time-reversed dielectric-breakdown model in which the annihilating probability of a particle on the surface site (x,h) depends on Laplacian field phi(x,h,t) as P(x,h,t)=| inverted Delta phi(x,h,t)|(kappa)/ Sigma(x,h)| inverted Delta phi(x,h,t)|(kappa) is suggested. This model is shown to be a theoretical model that covers a variety of eroding surfaces from the linear phenomena with dynamic exponent z=1 to those showing nonlinear behavior. phi(x,t) is defined to satisfy the Laplace equation inverted Delta (2)phi=0 with the boundary condition phi=0 on the material and phi=1 far from the material. The model with 0.5 < or = kappa < or = 2 is found to follow the linear growth equation with z=1 as the diffusion-limited erosion, which is also a time-reversed version of diffusion-limited deposition. For small kappa, the dynamical scaling property of the eroding surface belongs to the Kardar-Parisi-Zhang universal class as the time-reversed Eden model. The model with kappa >2.5 does not show any surface roughening behavior.