The Roughness Exponent and the Parallel Correlation Length in the Edwards‐Wilkinson Model with Quenched Disorder

The equation of motion of a driven interface in isotropic random media at the depinning transition can be described by the Edwards-Wilkinson equation with quenched disorder 1] @h @t = r 2 h + (x; h) + F (1) where the term r 2 h, reeecting the surface tension, smoothes the interface while the noise roughens the interface and F represents an applied driving force. As usual the interfaces are characterized by the roughness exponent and their dynamic exponent z which can be obtained, for instance, from the height correlation function C(r; t) = h(x + r; t) ? h(x; t)] 2 ? k (t) 2 g r k (t) (2) with k (t) / t 1=z denoting the time dependent correlation length and g denoting the scaling function which has the properties g(y) const for y 1 and g(y) / y 2 for y 1. In recent numerical investigations 2] of Eq.(1) a value of the roughness exponent larger than one was obtained in the interface dimension D = 1 contrary to the renormalization group (RG) prediction = (4 ? D)=3 3] and our numerical and theoretical results 4] which connrm the RG result. Note, however, that for the case > 1 the small-r behavour of the height correlation function change from C(r k (t); t) / r 2 to C(r k (t); t) / r 2 , see 5] for a detailed discussion. As a consequence, from the r dependence of the height correlation function it is not possible to obtain the roughness exponent in this case. Thus other methods have to be taken into account to determine , for instance the measurement of the steady state structure factor. In this note we want to propose an alternative method for the determination of the roughness exponent. In the steady state time regime the correlation length k converges to a driving eld dependent value which diverges at the depinning transition k (F) / jFC ? F j ? (3) where the exponent is given by the exact exponent relation = 1=(2 ?) 3]. Thus the measurement of the driving eld dependence of k leads to an indirect determination of the roughness exponent. Therefore we integrated numerically the spatially discretized Eq.(1) for diierent driving elds F < FC and 10 diierent realizations of the disorder within an Euler-scheme with dt = 0:01 until the interfaces gets pinned. For the system …