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

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.