Rapid state-recrossing kinetics in non-Markovian systems

Abstract

The mean first-passage time (MFPT) is one standard measure for the reaction time in thermally activated barrier-crossing processes. While the relationship between MFPTs and phenomenological rate coefficients is known for systems that satisfy Markovian dynamics, it is not clear how to interpret MFPTs for experimental and simulation time-series data generated by non-Markovian systems. Here, we simulate a one-dimensional generalized Langevin equation (GLE) in a bistable potential and compare two related numerical methods for evaluating MFPTs: one that only incorporates information about first arrivals between subsequent states and is equivalent to calculating the waiting time, or dwell time, and one that incorporates information about all first-passages associated with a given barrier-crossing event and is therefore typically employed to enhance numerical sampling. In the Markovian limit, the two methods are equivalent. However, for significant memory times, the two methods suggest dramatically different reaction kinetics. By focusing on first-passage distributions, we systematically reveal the influence of memory-induced rapid state-recrossing on the MFPTs, which we compare to various other numerical or theoretical descriptions of reaction times. Overall, we demonstrate that it is necessary to consider full first-passage distributions, rather than just the mean barrier-crossing kinetics when analyzing non-Markovian time series data.

Figures (5)

• Figure 1: Schematic comparison between barrier-recrossing and state-recrossing. A particle moves on a free energy surface $U(x)$ with a transition state located at $x_0$. Barrier-recrossing occurs when a particle makes multiple crossings of the transition state in a single excursion between the reactant and the product states. State-recrossing occurs when a particle reaches a product state and immediately, or soon thereafter, returns to the previous state. Here, the barrier-recrossing trajectory is generated by GLE simulation with $\tau_{\Gamma}/\tau_{\rm{D}} = 1.0{\times}10^{-3}$ and the state-recrossing trajectory with $\tau_{\Gamma}/\tau_{\rm{D}} = 1.0$.
• Figure 2: (a) A typical trajectory segment for a one-dimensional GLE simulation (Eq. \ref{['GLE']}) in a bistable well ($\tau_{\Gamma}/\tau_{\rm{D}} = 0.1$, $\tau_{\rm{m}}/\tau_{\rm{D}} = 0.001$, and $U_0 = 3\; k_{\rm{B}}T$). (b) Schematic illustration of the two passage-time definitions. The crosses show the initial entries into a new state and the filled circles show all subsequent crossings of the local minimum of that state. $\tau^{\rm{ffp}}_n$ is a single first-to-first passage time, which also includes the time to traverse the transition path $\tau^{\rm{tp}}_n$. $\tau^{\rm{afp}}_{n,i}$ is a sequence of all-to-first passage times corresponding to the single $\tau^{\rm{ffp}}_n$ such that $\tau^{\rm{afp}}_{n,1} = \tau^{\rm{ffp}}_n$. Here, $i=1,2,...,8$.
• Figure 3: Barrier-crossing and transition-path time distributions for systems with memory-dependent friction. (a) Full distributions for the all-to-first passage times $P_{\rm{afp}}(\tau)$, the first-to-first passage times $P_{\rm{ffp}}(\tau)$, and the transition path times $P_{\rm{tp}}(\tau)$. Distributions are accumulated from long equilibrium trajectories (see Fig.\ref{['Fig_1']}(a), total run time $1.25{\times}10^6 \tau_{\text{D}}$ per system) into histograms with exponentially-spaced bin widths. The orange curves show $P_{\rm{afp}}(\tau)$ predicted by mapping from simulation results for $P_{\rm{ffp}}(\tau)$ via Eq. \ref{['Dist_Rel_main']}. $\tau_{\text{m}}/\tau_{\text{D}} = 1{\times}10^{-3}$ for all systems. (b) Replotted distributions from (a) (grey region) with an exponential series (Eq. \ref{['Exp_ffp']}) fit to $P_{\rm{ffp}}(\tau)$, where $N$ is the minimum number of components required to fit a distribution. $N$ increases for increasing $\tau_{\Gamma}$. Exponential components are written here such that $\tau^{\text{exp}}_1>\tau^{\text{exp}}_2>\tau^{\text{exp}}_3>\tau^{\text{exp}}_4$. Black curves are for Eq. \ref{['Exp_afp']}, with $\alpha_n$ and $\tau_n$ taken from fits of Eq. \ref{['Exp_ffp']} to $P_{\rm{ffp}}(\tau)$.
• Figure 4: Comparison of various definitions of barrier-crossing reaction times ($\tau_{\text{react}}$) for systems with memory-dependent friction. $\tau_{\rm{ffp}}$ and $\tau_{\rm{afp}}$ are the mean first-to-first and all-to-first passage times, respectively. $\tau^{\rm{exp}}_1$ is the fitting result for the longest-time-scale exponential components to the distributions $P_{\text{ffp}}(\tau)$ and $P_{\text{afp}}(\tau)$, as given in Eqs. \ref{['Exp_ffp']} and \ref{['Exp_afp']}, and $\tau_{\rm{GH}}$ is the Grote-Hynes prediction for a single component exponential memory kernel. The black- dashed line indicates quadratic scaling.
• Figure 5: (a) Transition-path-time distributions over a range of memory times. The distributions reveal a transition between two modes of behavior, one dominated by overdamped Markovian barrier crossing and one dominated by non-Markovian recrossing where slow energy diffusion effects are strong. The distributions in the intermediate regime are discernibly bi-modal. The values in the legend are for $\tau_{\Gamma}/\tau_{\rm{D}}$. (b) The mean transition-path times $\tau_{\rm{tp}}$ for all memory times. The sigmoidal behavior shows the switching between the two characteristic modes in (a).