TY - JOUR

T1 - Flux in Tilted Potential Systems

T2 - Negative Resistance and Persistence

AU - Baryshnikov, Yuliy

AU - Kvalheim, Matthew D.

N1 - Funding Information:
This work is supported in part by the Army Research Office (ARO) under the SLICE Multidisciplinary University Research Initiatives (MURI) Program, award W911NF1810327. The authors gratefully acknowledge helpful conversations with Maria K. Cameron, J. Diego Caporale, Wei-Hsi Chen, Matthias Heymann, Daniel E. Koditschek, and Shai Revzen. The authors also thank the two anonymous reviewers for valuable comments and suggestions.
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 -