Origins of scale invariance in growth processes

Krug, Joachim

doi:10.1080/00018739700101498

Cited by 976 publications

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References 398 publications

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“…There is also a great interest in modeling interface growth [6,7], traffic flow [8], temperature dependent catalytic reactions [9], etc. Nonequilibrium magnetic systems, with a well defined hamiltonian, have been also studied in the context of nonequilibrium processes [10,11] as well.…”

Section: Introductionmentioning

confidence: 99%

Catalysis with competitive reactions: static and dynamical critical behavior

Costa

Figueiredo²

2003

Braz. J. Phys.

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We studied in this work a competitive reaction model between monomers on a catalyst. The catalyst is represented by hypercubic lattices in d = 1, 2 and 3 dimensions. The model is described by the following reactions: A + A → A2 and A + B → AB, where A and B are two monomers that arrive at the surface with probabilities y A and y B , respectively. The model is studied in the adsorption controlled limit where the reaction rate is infinitely larger than the adsorption rate. We employ site and pair mean-field approximations as well as static and dynamical Monte Carlo simulations. We show that, for all d, the model exhibits a continuous phase transition between an active steady state and a B-absorbing state, when the parameter y A is varied through a critical value. Monte Carlo simulations and finite-size scaling analysis near the critical point are used to determine the static critical exponents β, ν ⊥ and the dynamical critical exponents ν || , δ, η and z. The results found for this competitive reaction model are in accordance with the conjecture of Grassberger, which states that any system undergoing a continuous phase transition from an active steady state to a single absorbing state, exhibits the same critical behavior of the directed percolation universality class. I IntroductionIn the course of the last decade the statistical mechanics community has made great progress in the study of nonequilibrium phenomena. Until now, we do not have a complete theory accounting for the nonequilibrium systems. The fundamental concept of a Gibbsian distribution of states in equilibrium has no counterpart in the nonequilibrium situation. This happens because many of these systems do not present even an hamiltonian function and, if it is possible to define an hamiltonian, the detailed balance would be violated.Examples of recent problems on nonequilibrium processes include markets [1, 2], rain precipitation [3], sandpiles [4] and conserved contact process [5]. There is also a great interest in modeling interface growth [6,7], traffic flow [8], temperature dependent catalytic reactions [9], etc. Nonequilibrium magnetic systems, with a well defined hamiltonian, have been also studied in the context of nonequilibrium processes [10,11] as well.For the equilibrium systems we can induce phase transitions by changing some external parameters. Usually, the temperature is the selected control parameter to study phase transitions between equilibrium states. In the case of continuous phase transitions, at the critical point, long range correlations are established inside the system and a set of critical exponents can be defined to describe the critical behavior of some thermodynamic properties. The renormalization group theory [12] is a well known theory that allows the calculation of these critical exponents.We can also consider external constraints for the nonequilibrium systems that can drive the dynamical behavior of the system. The nature of the external parameter depends on the nature of the system. For instance, in an epidemic mode...

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Section: Introductionmentioning

confidence: 99%

Catalysis with competitive reactions: static and dynamical critical behavior

Costa

Figueiredo²

2003

Braz. J. Phys.

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“…The latter case is realized during growth at relatively low temperature, where desorption of adatoms can be neglected and therefore the growth rate is completely determined by the external deposition flux [41]. A generic model that incorporates the strongly and weakly conserved situation is given by…”

Section: Strongly and Weakly Conserved Step Dynamicsmentioning

confidence: 99%

Nonlinear Dynamics of Surface Steps

Krug

2010

Nonlinear Dynamics of Nanosystems

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“…Its main effects led to the proposal of fourth order linear and nonlinear growth equations, which are reviewed in Refs. [2] and [3].…”

Section: A Interface Growthmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

Depinning transitions in interface growth models

Reis

2003

Braz. J. Phys.

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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...

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Origins of scale invariance in growth processes

Cited by 976 publications

References 398 publications

Catalysis with competitive reactions: static and dynamical critical behavior

Catalysis with competitive reactions: static and dynamical critical behavior

Nonlinear Dynamics of Surface Steps

Depinning transitions in interface growth models

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