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

## TL;DR

This paper develops a perturbative method to compute finite-size corrections for Lévy walks on finite intervals, advancing understanding of anomalous transport beyond asymptotic solutions and applicable to various physical models.

## Contribution

It introduces a novel perturbative approach to calculate finite-L corrections for Lévy walks, extending analysis beyond known asymptotic solutions.

## Key findings

- Derived explicit finite-L correction for β=5/3 Lévy walk.
- Method applicable to a broad class of anomalous transport models.
- Highlights importance of finite-size effects in non-equilibrium systems.

## Abstract

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.