Exact solution of a Lévy walk model for anomalous heat transport

The Lévy walk model is studied in the context of the anomalous heat conduction of one-dimensional systems. In this model, the heat carriers execute Lévy walks instead of normal diffusion as expected in systems where Fourier's law holds. Here we calculate exactly the average heat current, the large deviation function of its fluctuations, and the temperature profile of the Lévy walk model maintained in a steady state by contact with two heat baths (the open geometry). We find that the current is nonlocally connected to the temperature gradient. As observed in recent simulations of mechanical models, all the cumulants of the current fluctuations have the same system-size dependence in the open geometry. For the ring geometry, we argue that a size-dependent cutoff time is necessary for the Lévy walk model to behave like mechanical models. This modification does not affect the results on transport in the open geometry for large enough system sizes.