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Barrier-crossing and energy relaxation dynamics of non-Markovian inertial systems connected via analytical Green-Fokker-Planck approach

Roland R. Netz*

  • *Contact author: rnetz@physik.fu-berlin.de
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Phys. Rev. Research 8, 033311 – Published 14 September, 2026

DOI: https://doi.org/10.1103/vrtr-9dby

Abstract

From numerical simulations, it is known that the barrier-crossing time of a non-Markovian one-dimensional reaction coordinate with a single exponentially decaying memory function exhibits a memory turnover: For intermediate values of the memory decay time, the barrier-crossing time is reduced compared to the Markovian limit, and for long memory times, it increases quadratically with the memory time when keeping the total integrated friction and the mass constant. The intermediate memory acceleration regime is accurately predicted by Grote-Hynes theory; for the asymptotic long-memory slowdown behavior, no systematic analytically tractable theory is available. Starting from the Green's function for a general inertial (i.e., finite-mass) non-Markovian Gaussian reaction coordinate in a harmonic well, we derive by an exact mapping a generalized Fokker-Planck equation with a time-dependent effective diffusion constant. To first order in a systematic cumulant expansion, we derive an analytical Arrhenius expression for the barrier-crossing time with the preexponential factor given by the energy relaxation time, which can be used to robustly predict barrier-crossing times from simulation or experimental trajectory data of general non-Markovian inertial systems without the need to extract memory functions. For a single-exponential memory kernel, we give a closed-form expression for the barrier-crossing time, which reproduces the Kramers turnover between the high-friction and high-mass limits as well as the memory turnover from the intermediate memory acceleration to the asymptotic long-memory slowdown regime. We also demonstrate that a non-Markovian system with single-exponential memory is singular in the zero-mass limit and that the long-memory barrier-crossing slowdown reflects the interplay between mass and memory effects. Thus, physically sound models for non-Markovian systems should include a finite mass.

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Phys. Rev. Research 8, 033311– Published 14 September, 2026
  • Received 16 January 2026
  • Accepted 31 July 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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