Density profiles in open superdiffusive systems

We numerically solve a discretized model of Levy random walks on a finite one-dimensional domain with a reflection coefficient r and in the presence of sources. At the domain boundaries, the steady-state density profile is nonanalytic. 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 corresponds to a Levy walk with negative reflection coefficient.