Statistics of circular interface fluctuations in an off-lattice Eden model

Takeuchi, Kazumasa A.

doi:10.1088/1742-5468/2012/05/p05007

Cited by 49 publications

(104 citation statements)

References 55 publications

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“…Similar results were also obtained for the off-lattice Eden model simulations in 2-d [286]. These radial exponents are thus considerably smaller than the corresponding exponents in the (1 + 1) dimensions with a flat substrate [280]: θ + 0 = 1.18 ± 0.08 and θ − 0 = 1.64 ± 0.08.…”

Section: Persistence Properties In Flat Versus Radial Geometrysupporting

confidence: 82%

Persistence and first-passage properties in nonequilibrium systems

Bray

Majumdar

Schehr

2013

Advances in Physics

530

669

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In this review we discuss the persistence and the related first-passage properties in extended many-body nonequilibrium systems. Starting with simple systems with one or few degrees of freedom, such as random walk and random acceleration problems, we progressively discuss the persistence properties in systems with many degrees of freedom. These systems include spins models undergoing phase ordering dynamics, diffusion equation, fluctuating interfaces etc. Persistence properties are nontrivial in these systems as the effective underlying stochastic process is non-Markovian. Several exact and approximate methods have been developed to compute the persistence of such non-Markov processes over the last two decades, as reviewed in this article. We also discuss various generalisations of the local site persistence probability. Persistence in systems with quenched disorder is discussed briefly. Although the main emphasis of this review is on the theoretical developments on persistence, we briefly touch upon various experimental systems as well.

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Section: Persistence Properties In Flat Versus Radial Geometrysupporting

confidence: 82%

Persistence and first-passage properties in nonequilibrium systems

Bray

Majumdar

Schehr

2013

Advances in Physics

530

669

View full text Add to dashboard Cite

show abstract

“…In particular, for the BD model in d = 1 + 1, exponents in agreement with the KPZ ones were obtained through appropriated extrapolations of effective exponents [29] and, more recently, from extremely large-scale simulations accessing the regimes where corrections become negligible [30]. Moreover, recent studies of height distributions have given additional proofs of the KPZ universality of Eden and BD models in d = 1 + 1 [16][17][18][19]. For Eden models, scaling exponents and height distributions consistent with KPZ class were also found in d = 2 + 1 [24,31].…”

Section: Introductionmentioning

confidence: 87%

“…(1)], Γ is a nonuniversal constant associated to the amplitude of the interface fluctuations, β is the growth exponent, and χ is a stochastic quantity given by Tracy-Widom [13] distributions. This conjecture was confirmed in distinct KPZ systems [14][15][16][17][18][19] besides exact solutions of KPZ equation [20][21][22][23]. Recent numerical simulations have shown that the KPZ ansatz can be generalized to 2+1 [24][25][26] and higher [27] dimensions, but the exact forms of the asymptotic distributions of χ are yet not known.…”

Section: Introductionmentioning

confidence: 89%

Origins of scaling corrections in ballistic growth models

2014

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We study the ballistic deposition and the grain deposition models on two-dimensional substrates. Using the Kardar-Parisi-Zhang (KPZ) ansatz for height fluctuations, we show that the main contribution to the intrinsic width, which causes strong corrections to the scaling, comes from the fluctuations in the height increments along deposition events. Accounting for this correction in the scaling analysis, we obtained scaling exponents in excellent agreement with the KPZ class. We also propose a method to suppress these corrections, which consists in divide the surface in bins of size ε and use only the maximal height inside each bin to do the statistics. Again, scaling exponents in remarkable agreement with the KPZ class were found. The binning method allowed the accurate determination of the height distributions of the ballistic models in both growth and steady state regimes, providing the universal underlying fluctuations foreseen for KPZ class in 2+1 dimensions. Our results provide complete and conclusive evidences that the ballistic model belongs to the KPZ universality class in 2 + 1 dimensions. Potential applications of the methods developed here, in both numerics and experiments, are discussed.

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“…We mention that, aside from some early works [101,102,103], and more recently [104], there has been little direct effort on numerical integration of 2d KPZ equation in polar coordinates; see, too [105,106,107]. Indeed, all work on this subclass, aside from radial Eden model simulations [108,109], have resorted to difficult, somewhat frustrating pt-pt simulations of various KPZ/DPRM models [110,65] in what is, effectively, constrained wedge geometries. The frustration arises because, in contrast to simulations for the flat KPZ subclass where all substrate points contribute to the ensemble average, the pt-pt Monte Carlo yields a few datum only per realization.…”

Section: An Homage To Psmentioning

confidence: 99%

A KPZ Cocktail-Shaken, not Stirred...

2015

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The stochastic partial differential equation proposed nearly three decades ago by Kardar, Parisi and Zhang (KPZ) continues to inspire, intrigue and confound its many admirers. Here, we i) pay debts to heroic predecessors, ii) highlight additional, experimentally relevant aspects of the recently solved 1+1 KPZ problem, iii) use an expanding substrates formalism to gain access to the 3d radial KPZ equation and, lastly, iv) examining extremal paths on disordered hierarchical lattices, set our gaze upon the fate of d=∞ KPZ. Clearly, there remains ample unexplored territory within the realm of KPZ and, for the hearty, much work to be done, especially in higher dimensions, where numerical and renormalization group methods are providing a deeper understanding of this iconic equation.Keywords Nonequilibrium Growth · Extremal Paths · Universal Limit Distributions In a NutshellThe history of physics has been punctuated at seminal moments by the appearance of certain fundamental equations (and associated models) which have vigorously propelled the enterprise forward, serving as an explosively rich departure point, generating a myriad of alternative perspectives, creative insights, surprising connections and, given its sustained impregnability, often remain for many years a sacred object of fascination and obsession to its dedicated disciples. Recent, obvious suspects in this regard include the quantum mechanical Schrödinger equation (equally well, its flipside-Feynman's path integral formulation), or the wonderfully elusive Navier-Stokes equation governing fluid mechanics, by which may be gleaned the scaling secrets of turbulent flow, dynamically encoded in the whirling eddies drawn by da Vinci centuries ago. Within the domain of equilibrium statistical physics, the 2d Ising Model, with its tour-de-force algebraic solution by Onsager, followed by combinatoric, graphical, Grassmannian, Monte Carlo, as well as full-blown field-theoretic, scaling, and renormalization group treatments, represents an extraordinary legacy that continues unabated to this very day. Arguably, a non-equilibrium statistical mechanical analogue to Ising/Onsager is the iconic equation [1] proposed a generation ago by Kardar, Parisi, and Zhang (KPZ); it captures the statistical fluctuations of a kinetically-roughened scalar height field h(x, t):

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Statistics of circular interface fluctuations in an off-lattice Eden model

Cited by 49 publications

References 55 publications

Persistence and first-passage properties in nonequilibrium systems

Persistence and first-passage properties in nonequilibrium systems

Origins of scaling corrections in ballistic growth models

A KPZ Cocktail-Shaken, not Stirred...

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