Non-Markovian barrier crossing with two-time-scale memory is dominated by the faster memory component

Abstract

We investigate non-Markovian barrier-crossing kinetics of a massive particle in one dimension in the presence of a memory function that is the sum of two exponentials with different memory times τ1 and τ2. Our Langevin simulations for the special case where both exponentials contribute equally to the total friction show that the barrier crossing time becomes independent of the longer memory time if at least one of the two memory times is larger than the intrinsic diffusion time. When we associate memory effects with coupled degrees of freedom that are orthogonal to a one-dimensional reaction coordinate, this counterintuitive result shows that the faster orthogonal degrees of freedom dominate barrier-crossing kinetics in the non-Markovian limit and that the slower orthogonal degrees become negligible, quite contrary to the standard time-scale separation assumption and with important consequences for the proper setup of coarse-graining procedures in the non-Markovian case. By asymptotic matching and symmetry arguments, we construct a crossover formula for the barrier crossing time that is valid for general multi-exponential memory kernels. This formula can be used to estimate barrier-crossing times for general memory functions for high friction, i.e. in the overdamped regime, as well as for low friction, i.e. in the inertial regime. Typical examples where our results are important include protein folding in the high-friction limit and chemical reactions such as proton-transfer reactions in the low-friction limit.

Non-Markovian modeling of protein folding

“…1 , one obtains ns, which is of the order of the longest memory time . This places the system in the so-called memory-acceleration regime, where memory effects are relevant and significantly accelerate barrier crossing ( 36 – 38 ).…”

“…This stays true even when the friction coefficient is allowed to depend on the reaction coordinate. As predicted by the Grote–Hynes theory, memory typically accelerates barrier crossing, where the acceleration magnitude depends primarily on the ratio of the memory time and the distance between the minimum and the barrier in reaction coordinate space ( 33 – 38 ). This memory-induced speedup of folding and unfolding is found to be accompanied by pronounced anomalous diffusion in reaction coordinate space.…”

Properties of a nonlinear bath: experiments, theory, and a stochastic Prandtl–Tomlinson model

“…where V 0 is the height of the potential barrier between the wells, and d 0 is half the distance between their minima. Note that for a single particle moving in such a potential interesting barrier-crossing kinetics were found for a Langevin equation with bi-exponential memory only recently [61].…”

Negative friction memory induces persistent motion

“…The generalized Langevin equation (GLE) offers a convenient framework for the description of non-Markovian dynamics by introducing a memory term in the equation of motion. It has been successfully applied to passive microrheology [4][5][6], the modeling of (bio-)molecular systems [14,[26][27][28][29][30] and for the investigation of non-Markovian effects on barrier-crossing dynamics [31,32]. In one dimension, the GLEs for the underdamped (UD) and overdamped (OD) cases read [33]:…”