Distinctive Fluctuations in a Confined Geometry

Degawa, Masashi; Stasevich, Timothy J.; Cullen, William; Pimpinelli, Alberto; Einstein, T. L.; Williams, Ellen D.

doi:10.1103/physrevlett.97.080601

Cited by 30 publications

(45 citation statements)

References 43 publications

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“…Their observations caught the imagination of many and the paper deserves the full credit of having triggered a large number of investigations. Examples of physical phenomena modeled by the KPZ class include turbulent liquid crystals [159], crystal growth on a thin film [171], facet boundaries [52], bacteria colony growth [172,122], paper wetting [109], crack formation [67], and burning fronts [118,116,119].…”

Section: Kardar Parisi Andmentioning

confidence: 99%

The Kardar–parisi–zhang Equation and Universality Class

Corwin

2012

Random Matrices: Theory Appl.

722

824

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Abstract. Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past twenty five years a new universality class has emerged to describe a host of important physical and probabilistic models (including one dimensional interface growth processes, interacting particle systems and polymers in random environments) which display characteristic, though unusual, scalings and new statistics. This class is called the Kardar-Parisi-Zhang (KPZ) universality class and underlying it is, again, a continuum object -a non-linear stochastic partial differential equation -known as the KPZ equation.The purpose of this survey is to explain the context for, as well as the content of a number of mathematical breakthroughs which have culminated in the derivation of the exact formula for the distribution function of the KPZ equation started with narrow wedge initial data. In particular we emphasize three topics: (1) The approximation of the KPZ equation through the weakly asymmetric simple exclusion process; (2) The derivation of the exact one-point distribution of the solution to the KPZ equation with narrow wedge initial data; (3) Connections with directed polymers in random media.As the purpose of this article is to survey and review, we make precise statements but provide only heuristic arguments with indications of the technical complexities necessary to make such arguments mathematically rigorous.

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Section: Kardar Parisi Andmentioning

confidence: 99%

The Kardar–parisi–zhang Equation and Universality Class

Corwin

2012

Random Matrices: Theory Appl.

722

824

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“…Despite the usefulness of the Cartesian representation in many cases, there are some growth profiles that can not be described according to it. Physical settings such as fluid flow in porous media [1], grain-grain displacement in Hele-Shaw cells [2], fracture dynamics [3], adatom and vacancy islands on crystal surfaces [4], and atomic ledges bordering crystalline facets [5,6] present interfaces that violate the hypothesis of the Cartesian representation. Biological systems are also characterized by an approximate spherical symmetry: bacterial colonies [7], fungi [8], epithelial cells [9], and cauliflowers [10] develop rough surfaces which are not describable from a planar reference frame.…”

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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“…Not only biological systems, but also physical settings such as fluid flow in porous media [1], grain-grain displacement in Hele-Shaw cells [6], adatom and vacancy islands on surfaces [7], and atomic ledges bordering crystalline facets [8,9] present interfaces that either become larger as time evolves or have a non-Euclidean geometry. For this reason, this problem has been considered as one of the main open questions of scaling analysis [10].…”

mentioning

confidence: 99%

Dynamic Scaling of Non-Euclidean Interfaces

Escudero

2008

Phys. Rev. Lett.

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The dynamic scaling of curved interfaces presents features that are strikingly different from those of the planar ones. Spherical surfaces above one dimension are flat because the noise is irrelevant in such cases. Kinetic roughening is thus a one-dimensional phenomenon characterized by a marginal logarithmic amplitude of the fluctuations. Models characterized by a planar dynamical exponent z > 1, which include the most common stochastic growth equations, suffer a loss of correlation along the interface, and their dynamics reduce to that of the radial random deposition model in the long time limit. The consequences in several applications are discussed, and we conclude that it is necessary to reexamine some experimental results in which standard scaling analysis was applied.

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Distinctive Fluctuations in a Confined Geometry

Cited by 30 publications

References 43 publications

The Kardar–parisi–zhang Equation and Universality Class

The Kardar–parisi–zhang Equation and Universality Class

Dynamics of curved interfaces

Dynamic Scaling of Non-Euclidean Interfaces

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