Lévy walks on finite intervals: A step beyond asymptotics

Area of Science:

• Statistical Physics
• Non-equilibrium Systems
• Anomalous Transport

Background:

• Lévy walks exhibit anomalous transport, characterized by a current nonlocally related to the density profile via an integral equation.
• Existing research focuses on asymptotic solutions, neglecting finite-size effects crucial for real-world applications.
• The importance of finite-length corrections in anomalous transport phenomena is widely recognized but remains theoretically underdeveloped.

Purpose of the Study:

• To develop a perturbative method for calculating finite-length corrections to anomalous transport in Lévy walks.
• To explicitly demonstrate the method by computing the leading correction for a Lévy walk of order β=5/3.
• To highlight the broad applicability of the developed method to other physical systems described by similar integral equations.

Main Methods:

• A novel perturbative approach is introduced to systematically derive finite-length corrections.
• The method is applied to a specific case: a Lévy walk of order β=5/3.
• The derived integral equation and its asymptotic solution are utilized to compute corrections.

Main Results:

• A perturbative method for computing finite-length corrections to anomalous transport is successfully developed.
• The leading correction to the asymptotic transport for a Lévy walk of order β=5/3 is explicitly calculated.
• The study demonstrates that the method is applicable to a wide range of physical problems involving similar integral equations.

Conclusions:

• The developed perturbative method provides a powerful tool for analyzing finite-size effects in anomalous transport.
• The findings are significant for understanding transport phenomena in systems with boundaries or finite dimensions.
• The method's generalizability suggests potential applications across diverse fields of physics.