Dynamics of an Interface Driven through Random Media: The Effect of a Spatially Correlated Noise

Abstract

We studied the quenched Edwards-Wilkinson (QEW) equation with power-law type of a correlated noise near the depinning threshold. We solved analytically the QEW equation by using a functional renormalization group method. We obtained the critical exponents characterizing the depinning transition. We found that the value of the roughness exponent increases from 1 to 1.46341 in one spatial dimension as does from 0 to 0.5, where is a constant characterizing the degree of the correlation.Interface motion in random media has been a popular research topic in the field of nonequilibrium statistical physics 1) because it is related to many physical phenomena such as fluid invasion in porous media, 2,3) depinning of charge density waves, 4) fluid imbibition in paper, 5) driven flux motion in type-II superconductors, 6,7) etc. Interface motion driven through random media can be explained by the quenched Kardar-Parisi-Zhang (QKPZ) equation, 8) @Hðx; tÞ @twhere Hðx; tÞ is the interface height of a site x on the substrate at time t. ðx; HÞ denotes a quenched noise with the following properties:ÁðzÞ is assumed to be a monotonically decreasing function of z for z > 0, and to decay exponentially to zero over a finite distance a ? . f ðx À x 0 Þ is usually assumed to be a delta function d ðx À x 0 Þ, where d is a substrate dimension. It is known that in eq. (1) is nonzero for the interface driven in anisotropic random media but zero for the interface driven in isotropic random media when the velocity of a driven interface is almost 0. 9,10) Equation (1) is called the quenched Edwards-Wilkinson (QEW) equation in case of ¼ 0. 11) The interface in eq. (1) is pinned when the driving force F is smaller than F c . However, the interface moves with a constant velocity when F > F c . This phenomenon is called the pinning-depinning (PD) transition. The velocity of the interface follows v $ ðF À F c Þ close to the transition point, where is called the velocity exponent. When F ) F c , the dynamics of the driven interface show the same scaling behavior as in the case ðx; HÞ $ ðx; tÞ. 12) Then, if there do not exist spatial and temporal correlation in the noise, the noise ðx; tÞ has the following properties:Equation (1) is called the Kardar-Parisi-Zhang (KPZ) equation if 6 ¼ 0 and ðx; HÞ can be described as ðx; tÞ because of F ) F c . The width of the interface driven through random media shows a nontrivial dynamical scaling behavior near the depinning threshold,where h i ðtÞ is the height of a site i on the substrate at time t. " h h and L denote the mean height and system size, respectively. The symbol hÁ Á Ái stands for the statistical average. The scaling function gðxÞ approaches a constant for x ) 1, and gðxÞ $ x for x ( 1 with z ¼ =. The exponents , , z are called the roughness, growth, and dynamic exponent, respectively. The scaling exponents for the QKPZ equation are ¼ 0:63 and ¼ 0:63 in d ¼ 1. 13,14) The exponents for the QEW equation are ¼ 1 and ¼ 0:75 in d ¼ 1. 15,16) When one considers the continuum equation for a driven interface...

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