TY - JOUR

T1 - Flux in Tilted Potential Systems

T2 - Negative Resistance and Persistence

AU - Baryshnikov, Yuliy

AU - Kvalheim, Matthew D.

N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2023/6

Y1 - 2023/6

N2 - 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 to many applications in physics and biology (e.g., molecular motors) is the net steady-state particle current or “flux” enabled by the noise and an additional driving force. However, this flux cannot usually be calculated analytically. Motivated by this, we investigate the steady-state flux generated by a nondegenerate diffusion process on a general compact manifold; such fluxes are essentially equivalent to the stochastic intersection numbers of Manabe (Osaka Math J 19(2):429–457, 1982). In the case that noise is small and the drift is “gradient-like” in an appropriate sense, we derive a graph-theoretic formula for the small-noise asymptotics of the flux using Freidlin–Wentzell theory. When additionally the drift is a local gradient sufficiently close to a generic global gradient, there is a natural flux equivalent to the entropy production rate—in this case our graph-theoretic formula becomes Morse-theoretic, and the result admits a description in terms of persistent homology. As an application, we provide a mathematically rigorous explanation of the paradoxical “negative resistance” phenomenon in Brownian transport discovered by Cecchi and Magnasco (Phys Rev Lett 76(11):1968, 1996).

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

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U2 - 10.1007/s00220-022-04622-4

DO - 10.1007/s00220-022-04622-4

M3 - Article

AN - SCOPUS:85146614866

SN - 0010-3616

VL - 400

SP - 853

EP - 930

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -