Additive-Functional Approach to Transport in Periodic and Tilted Periodic Potentials

TL;DR

This paper presents a rigorous additive-functional framework to understand effective transport in periodic and tilted periodic potentials, clarifying the roles of bounded and unbounded motion components in Brownian dynamics.

Contribution

It introduces a novel additive-functional approach that rigorously separates bounded and unbounded motion, deriving classical transport formulas as natural outcomes.

Findings

Effective drift and diffusion depend solely on unbounded motion.

Classical formulas like Lifson-Jackson are derived from the new framework.

Extension to higher dimensions recovers standard homogenized tensors.

Abstract

In this Letter, we clarify the physical origin of effective transport in periodic and tilted periodic systems. When Brownian dynamics is examined on the scale of a single period, the particle displacement admits a natural separation into a bounded part associated with recurrent motion within the periodic landscape, and an unbounded stochastic part that grows in time and carries the net transport. We show that effective drift and diffusion are governed entirely by this unbounded component, while local potential-induced fluctuations contribute only bounded corrections. Treating the displacement as an additive functional of the stochastic dynamics provides a rigorous formulation of this separation and leads to a corrector-martingale representation at the trajectory level. Within this framework, classical results-including the Lifson-Jackson formula for unbiased periodic systems and the Stratonovich expressions for tilted periodic potentials-follow as direct consequences of the same underlying structure. The same perspective extends naturally to higher-dimensional periodic environments, recovering the standard homogenized transport tensors.