Compact Waves in Microscopic Nonlinear Diffusion

Abstract

We analyze the spread of a localized peak of energy into vacuum for nonlinear diffusive processes. In contrast with standard diffusion, the nonlinearity results in a compact wave with a sharp front separating the perturbed region from vacuum. In $d$ spatial dimensions, the front advances as $t^{1/(2+da)}$ according to hydrodynamics, with $a$ the nonlinearity exponent. We show that fluctuations in the front position grow as $\sim t^{\mu }\eta$, where $\mu <1/(2+da)$ is a new exponent that we measure and $\eta$ is a random variable whose distribution we characterize. Fluctuating corrections to hydrodynamic profiles give rise to an excess penetration into vacuum, revealing scaling behaviors and robust features. We also examine the discharge of a nonlinear rarefaction wave into vacuum. Our results suggest the existence of universal scaling behaviors at the fluctuating level in nonlinear diffusion.

5 pages, 4 figures

References in corpus (5)

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Cited by in corpus (11)

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• Weak additivity principle for current statistics in d-dimensions
• Lattice models for granular-like velocity fields: Hydrodynamic limit
• Lattice models for granular-like velocity fields: Finite-size effects
• The kinetic exclusion process: a tale of two fields
• Reaction-Diffusion Processes with Nonlinear Diffusion
• Assortative Exchange Processes