Flux in Tilted Potential Systems: Negative Resistance and Persistence

Many real-world systems are well-modeled by Brownian particles subject to gradient dynamics plus noise arising, e.g., from the thermal fluctuations of a heat bath. Of central importance many applications in physics and biology (e.g., molecular motors) is net steady-state particle current or “flux” enabled an additional driving force. However, this flux cannot usually be calculated analytically. Motivated this, we investigate generated nondegenerate diffusion process on general compact manifold; such fluxes essentially equivalent stochastic intersection numbers Manabe (Osaka Math J 19(2):429–457, 1982). In case that small drift “gradient-like” appropriate sense, derive graph-theoretic formula for small-noise asymptotics using Freidlin–Wentzell theory. When additionally local sufficiently close generic global gradient, there natural entropy production rate—in our becomes Morse-theoretic, result admits description terms persistent homology. As application, provide mathematically rigorous explanation paradoxical “negative resistance” phenomenon transport discovered Cecchi Magnasco (Phys Rev Lett 76(11):1968, 1996).