Continuum model for radial interface growth

Batchelor, Murray T.; Henry, Bryan R.; Watt, Simon

doi:10.1016/s0378-4371(98)00326-4

Cited by 20 publications

(31 citation statements)

References 11 publications

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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.…”

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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Section: An Homage To Psmentioning

confidence: 99%

A KPZ Cocktail-Shaken, not Stirred...

2015

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“…In order to construct radial growth equations one may invoke the reparametrization invariance principle [21,22], as has already been done a number of times [10,11,[13][14][15][16]. In the case of white and Gaussian fluctuations, the d-dimensional spherical noise is given by…”

Section: Radial Random Depositionmentioning

confidence: 99%

Statistics of interfacial fluctuations of radially growing clusters

Escudero

2011

Phys. Rev. E

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The dynamics of fluctuating radially growing interfaces is approached using the formalism of stochastic growth equations on growing domains. This framework reveals a number of dynamic features arising during surface growth. For fast growth, dilution, which spatially reorders the incoming matter, is responsible for the transmission of correlations. Its effects include the erasing of memory with respect to the initial condition, a partial attenuation of geometrically originated instabilities, and the restoration of universality in some special cases in which the critical exponents depend on the parameters of the equation of motion. In this sense, dilution rends the dynamics more similar to the usual one of planar systems. This fast growth regime is also characterized by the spatial decorrelation of the interface, which, in the case of radially growing interfaces, naturally originates rapid roughening and scale-dependent fractality, and suggests the advent of a self-similar fractal dimension. The center-of-mass fluctuations of growing clusters are also studied, and our analysis suggests the possible nonapplicability of usual scalings to the long-range surface fluctuations of the radial Eden model. In fact, our study points to the fact that this model belongs to a dilution-free universality class.

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“…The shape of radial interfaces was illustrated by means of numerical simulations in [16] and [17]; their deterministic evolution is analogous to that of an elliptic interface of null eccentricity, see Sec.…”

Section: Radial Geometrymentioning

confidence: 99%

“…Subsequent works were devoted to the radial (1 + 1d) KPZ equation [16,17,18], the radial (1 + 1d) and spherical (2 + 1d) Mullins-Herring (MH) equa-tion [19,20], and the general reparametrization invariant formulation of stochastic growth equations [21]. An analytical approach to these equations [18] showed that for short spatial scales and time intervals the dynamics of radial interfaces was equivalent to that of the planar case; however, long time intervals yielded a different output.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of curved interfaces

Escudero

2009

Annals of Physics

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Stochastic growth phenomena on curved interfaces are studied by means of stochastic partial differential equations. These are derived as counterparts of linear planar equations on a curved geometry after a reparametrization invariance principle has been applied. We examine differences and similarities with the classical planar equations. Some characteristic features are the loss of correlation through time and a particular behaviour of the average fluctuations. Dependence on the metric is also explored. The diffusive model that propagates correlations ballistically in the planar situation is particularly interesting, as this propagation becomes nonuniversal in the new regime.

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Continuum model for radial interface growth

Cited by 20 publications

References 11 publications

A KPZ Cocktail-Shaken, not Stirred...

A KPZ Cocktail-Shaken, not Stirred...

Statistics of interfacial fluctuations of radially growing clusters

Dynamics of curved interfaces

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