Driven Depinning in Anisotropic Media

Tang, Lei-Han; Kardar, Mehran; Dhar, Deepak

doi:10.1103/physrevlett.74.920

Cited by 122 publications

(157 citation statements)

References 23 publications

Supporting

Mentioning

146

Contrasting

Unclassified

Order By: Relevance

“…They must obey rotational invariance, as discussed in Ref. [37][38][39] , which prevents the additional KPZ term λ(∇ x u xt ) 2 to be generated at f = f + c . There is always a KPZ term generated at v > 0 from the broken symmetry x → −x, but λ can vanish or not as v → 0 + , depending on whether rotational invariance is broken or not.…”

Section: B Model Scaling and Fluctuationsmentioning

confidence: 99%

“…This means that the starting model has sufficient rotational invariance, as discussed below, which guarantees that additional Kardar-Parisi-Zhang terms are absent. A general discussion of the various universality classes can be found in 37,38 and an application of our non-analytic field theory (NAFT) methods to the case of "anisotropic depinning" will be presented in 39 .…”

Section: Introduction a Overviewmentioning

confidence: 99%

See 1 more Smart Citation

Two-loop functional renormalization group theory of the depinning transition

2002

View full text Add to dashboard Cite

We construct the field theory which describes the universal properties of the quasi-static isotropic depinning transition for interfaces and elastic periodic systems at zero temperature, taking properly into account the nonanalytic form of the dynamical action. This cures the inability of the 1-loop flow-equations to distinguish between statics and quasi-static depinning, and thus to account for the irreversibility of the latter. We prove two-loop renormalizability, obtain the 2-loop β-function and show the generation of "irreversible" anomalous terms, originating from the non-analytic nature of the theory, which cause the statics and driven dynamics to differ at 2-loop order. We obtain the roughness exponent ζ and dynamical exponent z to order ǫ 2 . This allows to test several previous conjectures made on the basis of the 1-loop result. First it demonstrates that randomfield disorder does indeed attract all disorder of shorter range. It also shows that the conjecture ζ = ǫ/3 is incorrect, and allows to compute the violations, as ζ = ǫ 3(1 + 0.14331ǫ), ǫ = 4 − d. This solves a longstanding discrepancy with simulations. For long-range elasticity it yields ζ = ǫ 3(1 + 0.39735ǫ), ǫ = 2 − d (vs. the standard prediction ζ = 1/3 for d = 1), in reasonable agreement with the most recent simulations. The high value of ζ ≈ 0.5 found in experiments both on the contact line depinning of liquid Helium and on slow crack fronts is discussed.

show abstract

Section: B Model Scaling and Fluctuationsmentioning

confidence: 99%

Section: Introduction a Overviewmentioning

confidence: 99%

Two-loop functional renormalization group theory of the depinning transition

2002

View full text Add to dashboard Cite

show abstract

“…The d-dimensional DK model is equivalent to a (d − 2)-dimensional interface growing in time. As in the case of depinning transition of a driven tilted interface [17], the height-height correlation function in d-dimensions will have the scaling form…”

mentioning

confidence: 99%

An Exactly Solvable Anisotropic Directed Percolation Model in Three Dimensions

Rajesh

Dhar

1998

Phys. Rev. Lett.

Self Cite

View full text Add to dashboard Cite

We solve exactly a special case of the anisotropic directed bond percolation problem in threedimensions, in which the occupation probability is 1 along two spatial directions, by mapping it to a 5-vertex model. We determine the asymptotic shape of the infinite cluster and hence the direction dependent critical probability. The exponents characterizing the fluctuations of the boundary of the wetted cluster in d-dimensions are related to those of the (d − 2)-dimensional KPZ equation.It is well-known that the critical behaviour of a very large class of reaction-diffusion systems is describable in terms of the critical exponents of directed percolation (DP) [1]. Examples include heterogeneous catalysis [2], contact processes and epidemic models, self organized criticality [3] and the driven depinning transition [4]. Though rather precise estimates of critical parameters are known from various numerical techniques such as series expansions and Monte Carlo simulations [5], an exact solution has not been possible so far, even in the simple case of 1+1 dimensions. The only analytically tractable case known is the anisotropic bond percolation on a square lattice, solved by Domany and Kinzel (DK). They addressed the problem in which horizontal and vertical bonds are present with different concentrations, and solved exactly the special case when one of these is set equal to 1 [6]. In this case, the wetted cluster has no holes, and this fact helps in reducing the problem to that of a simple random walk in one dimension [7].In this paper, we solve the three dimensional generalization of DK model. We consider the DP on a three dimensional simple cubic lattice with bond concentration in the three directions being p x , p y and p z , and solve the case p x = p y = 1, p z arbitrary. We study the properties of the infinite cluster and obtain the macroscopic shape of the cluster and the scaling form for the fluctuations of its boundary . By universality, the behaviour of fluctuations of the outer boundary of the infinite cluster at large length scales in this special soluble case is expected to be the same as in the more general case above percolation threshold with p x , p y , and p z arbitrary.In addition to the usual critical exponents of the DP problem, which are defined in terms of the power-law behaviour of various quantities near the critical point, there are universal power-law prefactors to the exponential decay of correlation functions away from the critical point. For example, for p < p c , the probability of a finite cluster having s sites varies as s −θ exp(−As) as s → ∞. For p > p c , the probability varies as s −θ ′ exp(−Bs d−1 d ) for large s. The exponents θ and θ ′ are examples of offcritical exponents [8]. We find that the fluctuations of the boundary of the infinite cluster in d-dimensions are in the same universality class as those of a (d-2)-dimensional interface moving in a (d-1)-dimensional space. The latter are described by the well-known KPZ equation [9]. Thus we identify the critical exponents of KPZ equ...

show abstract

“…(14) in the case λ = 0 is called quenched EW (QEW) equation. These distinct critical behaviors reflect the symmetries of the underlying media [40]. Before going further into the discussion of continuum theories, we will introduce the main lattice models.…”

Section: Growing Interfaces In Disordered Mediamentioning

confidence: 99%

“…On a phenomenological level, it is believed that the QKPZ equation (14) provides their continuum description at criticality [40,41], with anisotropy being responsible for the presence of a non-vanishing nonlinear term. It contrasts to the usual kinetic origin of nonlinearities in the growth equations (λ ∼ v) [19], which would give λ = 0 at the transition.…”

Section: Growing Interfaces In Disordered Mediamentioning

confidence: 99%

Depinning transitions in interface growth models

Reis

2003

Braz. J. Phys.

View full text Add to dashboard Cite

Pinning-depinning transitions are roughening transitions separating a growing phase and pinned (or blocked) one, and are frequently connected to transitions into absorbing states. In this review, we discuss lattice growth models exhibiting this type of dynamic transition. Driven growth in media with impurities, the competition between deposition and desorption and deposition of poisoning species are some of the physical mechanisms responsible for the transitions, leading to different types of stochastic growth rules. The growth models are classified according to the those mechanisms and possible applications are shown, which include suggestions of experimental realizations of directed percolation transitions. I IntroductionThe study of surface and interface growth processes attracts much interest from the technological point of view mainly because it helps to understand the growth mechanisms of nanostructures and, consequently, their physical properties [1]. From the theoretical point of view, they motivated significant advances in non-equilibrium Statistical Mechanics and related fields [2,3]. Theoretical modeling is usually based on stochastic differential equations or on discrete atomistic models. In various models, changing one parameter leads to a roughening transition between a rough and a smooth phase. Two well known examples are the transition in the Kardar-Parisi-Zhang equation in dimensions d ≥ 3, separating regimes with linear (smooth) and nonlinear growth, and temperature-induced roughening transitions in equilibrium conditions [2]. On the other hand, in the so-called depinning transitions, the interface propagates only in the rough phase, being pinned in the smooth phase. Several mechanisms may be responsible for this type of transition, such as the presence of impurities in the growth media, competition between deposition and evaporation or formation of poisoning species at the growing surface. One of the interesting features of depinning transitions is that they frequently can be mapped onto transitions into absorbing states, whose most prominent example is directed percolation (DP). Different connections of depinning transitions to DP are found in discrete and continuous growth models, as well as connections with other classes of transitions to absorbing states [4] and, eventually, with statistical equilibrium transitions. Besides the fundamental interest of these systems, the large variety of interface growth phenomena observed in nature suggests them as strong candidates to experimental realizations of models of dynamic transitions, such as DP (see e. g. Ref.[5]).The aim of the present work is to review the basic features of lattice growth models exhibiting depinning transitions. Before introducing these systems, we will briefly summarize the theory and phenomenology concerning interface growth and phase transitions into absorbing states (Sec. II). Then we will discuss the problem of growing interfaces in disordered media with quenched disorder, in which transitions in the DP class or in the cl...

show abstract

Driven Depinning in Anisotropic Media

Cited by 122 publications

References 23 publications

Two-loop functional renormalization group theory of the depinning transition

Two-loop functional renormalization group theory of the depinning transition

An Exactly Solvable Anisotropic Directed Percolation Model in Three Dimensions

Depinning transitions in interface growth models

Contact Info

Product

Resources

About