L{\'e}vy walks on finite intervals: A step beyond asymptotics

A L{\'e}vy walk of order $\beta$ is studied on an interval of length $L$, driven out of equilibrium by different-density boundary baths. The anomalous current generated under these settings is nonlocally related to the density profile through an integral equation. While the asymptotic solution to this equation is known, its finite-$L$ corrections remain unstudied despite their importance in the study of anomalous transport. Here a perturbative method for computing such corrections is presented and explicitly demonstrated for the leading correction to the asymptotic transport of a L{\'e}vy walk of order $\beta=5/3$, which represents a broad universal class of anomalous transport models. Surprisingly, many other physical problems are described by similar integral equations, to which the method introduced here can be directly applied.