Density Profiles in Open Superdiffusive Systems

TL;DR

This paper investigates the steady-state density profiles of Levy random walks in finite one-dimensional systems with sources and boundary reflections, revealing a non-analytic behavior characterized by a meniscus exponent linked to the Levy and reflection parameters.

Contribution

It introduces a novel characterization of boundary singularities in Levy walks through the meniscus exponent and connects this to physical temperature profiles in anomalous heat conduction models.

Findings

Meniscus exponent .5 + r(.5-1) describes boundary singularities.

The model reproduces temperature profiles in anomalous heat conduction.

Negative reflection coefficient corresponds to free-boundary conditions.

Abstract

We numerically solve a discretized model of Levy random walks on a finite one-dimensional domain in the presence of sources and with a reflection coefficient $r$. At the domain boundaries, the steady-state density profile is non-analytic. The meniscus exponent $\mu$, introduced to characterize this singular behavior, uniquely identifies the whole profile. Numerical data suggest that $\mu =\alpha/2 + r(\alpha/2-1)$, where $\alpha$ is the Levy exponent of the step-length distribution. As an application, we show that this model reproduces the temperature profiles obtained for a chain of oscillators displaying anomalous heat conduction. Remarkably, the case of free-boundary conditions in the chain correspond to a Levy walk with negative reflection coefficient.