On the automaton representation of the Edwards–Wilkinson model with quenched disorder

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Abstract

We reinvestigate the automaton version of an Edwards–Wilkinson-type model with quenched disorder introduced recently by Leschhorn [Leschhorn, Physica A 195 (1993) 324] in interface dimensions D=1,2,3 at the depinning transition. Our numerical results suggest that the dynamic exponent z has a D-independent value z=32 while the roughness exponent α has the value α=(4−D)/3.

Section snippets

Introduction and model

The problem of an interface driven through random media is of great interest for more than one decade since it appears in a variety of technological important areas like the oil recovery where one fluid (the oil) is replaced by another fluid (water) or in random magnets, see for an overview Ref. [1]. Typically, the interface moves with a finite velocity if an applied driving force (in the above examples, the pressure of the water or a homogenous magnetic field, respectively) is larger than a

D=1

To study the scaling behavior at the depinning transition we simulate systems with sizes up to L=262144 for g=1. By a study of the p-dependence of the velocity for the largest system, it is found that the depinning transition occurs at approximately pC≃0.8007, a value slightly larger than the one obtained by Leschhorn [5] (pC,Lh≃0.8004).

Starting from an initially flat interface the local pinning forces η roughen the interface. The evolution of this roughness can be analyzed with dynamic scaling

D=2, D=3

Following Leschhorn, we choose g=2D for the strength of the pinning forces in D>1. For the largest simulated system (L=1024) in D=2 we found pC≃0.6418 which is again slightly larger than the value Leschhorn obtained in his simulation. Measuring the height correlation function we obtain αHCF=0.7±0.01 for the value of the roughness exponent (Fig. 5) at the depinning transition and measuring the structure factor Eq. (8)in the steady-state time regime we obtain the value αSF=0.68±0.02 (Fig. 6)

Concluding remarks

To summarize our numerical results, we found strong evidence that the morphology of an interface defined by , in the interface dimension D at the depinning transition can be described by the roughness exponent α=(4−D)/3. This value for α has also been found for the EW equation with quenched disorder, Eq. (1).

For the dynamic exponent we obtain the D-independent value z=32. Thus, contrary to the agreement of the roughness exponent of the automaton model and the Edwards–Wilkinson model with

Acknowledgements

This work was supported by the Deutsche Forschungsgemeinschaft through Graduiertenkolleg Struktur und Dynamik heterogener Systeme.

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