Density Profiles in Open Superdiffusive Systems

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 $μ$, introduced to characterize this singular behavior, uniquely identifies the whole profile. Numerical data suggest that $μ=α/2 + r(α/2-1)$, where $α$ 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.

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