Diffusion in time-dependent random media and the Kardar-Parisi-Zhang equation

Diffusion in time-dependent random media and the Kardar-Parisi-Zhang equation Although time-dependent random media with short-range correlations lead to (possibly biased) normal tracer diffusion, anomalous fluctuations occur away from the most probable direction. This was pointed out recently in one-dimensional (1D) lattice random walks, where statistics related to the 1D Kardar-Parisi-Zhang (KPZ) universality class, i.e., the Gaussian unitary ensemble Tracy-Widom distribution, were shown to arise. Here, we provide a simple picture for this correspondence, directly in the continuum, which allows one to study arbitrary space dimensions and to predict a variety of universal distributions. In d=1, we predict and verify numerically the emergence of the Gaussian orthogonal ensemble Tracy-Widom distribution for fluctuations of the transition probability. In d=3, we predict a phase transition from Gaussian fluctuations to three-dimensional KPZ-type fluctuations as the bias is increased. We predict KPZ universal distributions for the arrival time of a first particle from a cloud diffusing in such media. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Physical Review E American Physical Society (APS)

Diffusion in time-dependent random media and the Kardar-Parisi-Zhang equation

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Diffusion in time-dependent random media and the Kardar-Parisi-Zhang equation

Abstract

Although time-dependent random media with short-range correlations lead to (possibly biased) normal tracer diffusion, anomalous fluctuations occur away from the most probable direction. This was pointed out recently in one-dimensional (1D) lattice random walks, where statistics related to the 1D Kardar-Parisi-Zhang (KPZ) universality class, i.e., the Gaussian unitary ensemble Tracy-Widom distribution, were shown to arise. Here, we provide a simple picture for this correspondence, directly in the continuum, which allows one to study arbitrary space dimensions and to predict a variety of universal distributions. In d=1, we predict and verify numerically the emergence of the Gaussian orthogonal ensemble Tracy-Widom distribution for fluctuations of the transition probability. In d=3, we predict a phase transition from Gaussian fluctuations to three-dimensional KPZ-type fluctuations as the bias is increased. We predict KPZ universal distributions for the arrival time of a first particle from a cloud diffusing in such media.
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Publisher
The American Physical Society
Copyright
Copyright © ©2017 American Physical Society
ISSN
1539-3755
eISSN
550-2376
D.O.I.
10.1103/PhysRevE.96.010102
Publisher site
See Article on Publisher Site

Abstract

Although time-dependent random media with short-range correlations lead to (possibly biased) normal tracer diffusion, anomalous fluctuations occur away from the most probable direction. This was pointed out recently in one-dimensional (1D) lattice random walks, where statistics related to the 1D Kardar-Parisi-Zhang (KPZ) universality class, i.e., the Gaussian unitary ensemble Tracy-Widom distribution, were shown to arise. Here, we provide a simple picture for this correspondence, directly in the continuum, which allows one to study arbitrary space dimensions and to predict a variety of universal distributions. In d=1, we predict and verify numerically the emergence of the Gaussian orthogonal ensemble Tracy-Widom distribution for fluctuations of the transition probability. In d=3, we predict a phase transition from Gaussian fluctuations to three-dimensional KPZ-type fluctuations as the bias is increased. We predict KPZ universal distributions for the arrival time of a first particle from a cloud diffusing in such media.

Journal

Physical Review EAmerican Physical Society (APS)

Published: Jul 10, 2017

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