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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

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 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 slow-down behavior no systematic analytically tractable theory is available. Starting from the Green 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 pre-exponential 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 slow-down regime. We also show that non-Markovian systems are singular in the zero-mass limit, which suggests that the long-memory barrier-crossing slow-down reflects the interplay between mass and memory effects. Thus, physically sound models for non-Markovian systems have to include a finite mass.

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

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 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 slow-down behavior no systematic analytically tractable theory is available. Starting from the Green 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 pre-exponential 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 slow-down regime. We also show that non-Markovian systems are singular in the zero-mass limit, which suggests that the long-memory barrier-crossing slow-down reflects the interplay between mass and memory effects. Thus, physically sound models for non-Markovian systems have to include a finite mass.
Paper Structure (20 sections, 141 equations, 4 figures)

This paper contains 20 sections, 141 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Non-Markovian Green function
  4. From the non-Markovian Green function to the generalized Fokker-Planck equation
  5. Barrier-crossing times from the generalized Fokker-Planck equation
  6. Cumulant expansion of the time-dependent diffusivity $D(t)$
  7. Explicit predictions for different potential shapes
  8. Explicit results for relaxation times
  9. Two-pole analysis for Markovian system: Kramers turnover
  10. Three-pole analysis for non-Markovian system: memory turnover
  11. The mass-less non-Markovian system with single-exponential memory is singular
  12. The zero-memory-time limit of the GLE is time-reversible
  13. Conclusions and Discussion
  14. Derivation of the Green function
  15. Derivation of the generalized Fokker-Planck equation
  16. ...and 5 more sections

Figures (4)

  • Figure 1: Illustration of different potential shapes considered for the calculation of the mean-first passage time according to Eq. (\ref{['MFP12']}). The adsorbing boundary condition is indicated by an abrupt decrease of the potential. a) Harmonic potential. b) Absorbing potential is located on the top of a barrier with vanishing slope. c) Absorbing boundary condition is located to the right of the barrier.
  • Figure 2: Analytical results in the Markovian two-pole limit corresponding to vanishing memory time $\tau=0$. a) Rescaled correlation function $\bar{C}(t)=C(t)/C_0$ according to Eqs. (\ref{['GLE15']}) and (\ref{['GLE16']}) and using the results for the poles in Eq. (\ref{['GLE18']}) for three different rescaled masses $\tau_m/\tau_K=mK/\gamma^2$, where $\tau_m=m/\gamma$ is the inertial relaxation time and $\tau_K=\gamma/K$ is the overdamped relaxation time. Time $t$ is rescaled by the harmonic oscillation period in the frictionless limit $T=2 \pi \sqrt{m/K}$. b) Rescaled relaxation time $\tau_{\rm rel} /T$ according to Eqs. (\ref{['GLE20']}) as function of the rescaled mass $\tau_m/\tau_K=mK/\gamma^2$. The asymptotic scalings in the high-mass limit (high $\tau_m/\tau_K$) $\tau_{\rm rel} \sim m/\gamma$ and in the high-friction limit (small $\tau_m/\tau_K$) $\tau_{\rm rel} \sim \gamma/K$ are indicated by straight broken lines. At the minimum at $\tau_m/\tau_K=mK/\gamma^2=1$ the relaxation time is given by $\tau_{\rm rel} = T/\pi$, which corresponds to the transition-state-theory prediction.
  • Figure 3: Analytical results for the non-Markovian three-pole scenario for finite memory time $\tau$. a) Rescaled correlation function $\bar{C}(t)=C(t)/C_0$ according to Eqs. (\ref{['GLE23']}) and (\ref{['GLE24']}) using the results for the poles in Eq. (\ref{['GLE22']}) for three different rescaled memory times $\tau/\tau_K$, where $\tau_K=\gamma/K$ is the overdamped Markovian relaxation time. Time $t$ is rescaled by the Markovian harmonic oscillation period in the frictionless limit $T=2 \pi \sqrt{m/K}$ and the rescaled mass is set to a small value of $\tau_m/\tau_K=mK/\gamma^2=0.01$. b) Rescaled relaxation time $\tau_{\rm rel} /\tau_K$ in the zero-mass limit according to Eqs. (\ref{['GLE28']}) as function of the rescaled memory time $\tau/\tau_K$. The asymptotic scaling in the long-memory-time limit $\tau_{\rm rel} \sim \tau^2$ is indicated by a straight broken line.
  • Figure 4: Comparison of our analytical results (Eqs. \ref{['final3']}, \ref{['final4']} and \ref{['GLE25']}, black solid lines, Eqs. \ref{['final3']}, \ref{['final4']} and \ref{['GLE28']}, red solid line) and a previous fit of a scaling function to simulation results (Eq. 8 in Lavacchi2025, broken lines) for the well-to-barrier-top mean-first passage time (rescaled by $\tau_K=\gamma/K$) of the non-Markovian model with single-exponential memory, all for a barrier height of $U_0 = 3 k_BT$. Results are shown for three different values of the rescaled mass $\tau_m/\tau_K=mK/\gamma^2=10, 0.1, 0$ (from top to bottom) as a function of the memory time rescaled by $\tau_K=\gamma/K$.