Braunstein, Buceta, and Giovambattista Reply:

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

“…is called the activity function [13] and Θ(x) is the unit step function defined as Θ(x) = 1 for x ≥ 0 and equals to 0 otherwise, p is the microscopic driving force and g i (h i + a) is the quenched noise just above the interface distributed in the interval [0, 1]. Notice that the activity function F is the competition between the driving force and the quenched noise, so F is also a "noise".…”

“…where τ is the mean lapse between successive election of any site and G i [13] are the microscopic growing rules for the evolution of the height at this site due that a site j is chosen at time t. Here η i is a Gaussian "thermal" noise with zero mean and covariance…”

“…Notice that all the heights are in units of a in order to keep the arguments of the step function without units. For W i±1 [13] the δ Kronecker function has been taken as…”

“…and F ≡ F (x, h) as was defined in Eq. (13). Notice that µ( F ) is now the effective competition between the driving force and the quenched noise.…”

“…Notice that µ( F ) is now the effective competition between the driving force and the quenched noise. Equation (14) shows that the nonlinearity arises naturally as a consequence of the microscopic model. This explains the previous numerical results obtained by Amaral et al [4], that studied the effects of an effective coefficient λ ef f from a tilted interface showing that the nonlinear term must exist.…”