Growing interfaces in quenched media: stochastic differential equation

We present the stochastic differential equation with quenched noise for the Tang and Leschhorn model (Phys. Rev. A 45 (1992) R8309). The equation derived from the microscopic rules using regularization procedure predicts accurately the roughness, the dynamical and velocity exponents of the directed percolation depinning models and the quenched Kardar–Parisi–Zhang equation. In order to prove the close relationship existing between the microscopic equation and the continuous differential equation, we express the latter by means of two additive contributions: the substratum and the lateral one. The macroscopic behaviour of these contributions leads us to a deeper explanation of the intrinsic structure of the stochastic differential equation.