Duality between relaxation and first passage in reversible Markov dynamics: rugged energy landscapes disentangled

Abstract

Relaxation and first passage processes are the pillars of kinetics in condensed matter, polymeric and single-molecule systems. Yet, an explicit connection between relaxation and first passage time-scales so far remained elusive. Here we prove a duality between them in the form of an interlacing of spectra. In the basic form the duality holds for reversible Markov processes to effectively one-dimensional targets. The exploration of a triple-well potential is analyzed to demonstrate how the duality allows for an intuitive understanding of first passage trajectories in terms of relaxational eigenmodes. More generally, we provide a comprehensive explanation of the full statistics of reactive trajectories in rugged potentials, incl. the so-called 'few-encounter limit'. Our results are required for explaining quantitatively the occurrence of diseases triggered by protein misfolding. 1 l -, the longest relaxation time, and the mean first passage time to surmount the highest barrier in the landscape [7][8][9][10][11]56]. However, in spite of the

Extreme value statistics of ergodic Markov processes from first passage times in the large deviation limit

“…(9) for ℘ a in the following manner (see also Refs. [38,39] for more details). First, we Laplace-transform ‡ the renewal theorem (9) in time (t → s), which converts the convolution to a product,P (a, s|x 0 ) =P (a, s|a)℘ a (s|x 0 ), implying…”

“…In this work we deepen this connection between the first passage and the extremum functional, which allows us to obtain the statistics of extreme values in finite time for Markovian diffusion processes in confining potentials on time-scales, where consecutive time-steps remain correlated. Exploiting further a duality between first passage processes and ensemble propagation [38,39] we derive a chain of inequalities, which bound the long time asymptotics (i.e., the large deviation limit) of the probability densities of extrema both from above and from below. As we will show, the large deviation limit approximates the probability density of extreme values surprisingly well even on relatively short timescales.…”

“…2 we recapitulate the well-known connection between distributions of extrema and first passage time densities. We then utilize recent findings on the large time asymptotics of first passage time densities [38,39] to determine distributions of extrema in the large deviation limit. The usefulness of our general results is demonstrated in Sec.…”

Spectral theory of imperfect diffusion-controlled reactions on heterogeneous catalytic surfaces

“…More recent works investigated how the mean reaction time is affected by a finite lifetime of diffusing particles [59][60][61], by partial reactivity and interactions [63,64], by target aspect ratio [65], by reversible target-binding kinetics [66,67] and surface-mediated diffusion [68][69][70][71], by heterogeneous diffusivity [72], and by rapid re-arrangments of the medium [73][74][75]. Some of the related effects onto the whole distribution of reaction times were analyzed [76][77][78][79][80]. However, the current understanding of diffusion-controlled reactions on catalytic surfaces with continuously varying heterogeneous reactivity remains episodic.…”

Manifestations of Projection-Induced Memory: General Theory and the Tilted Single File

“…An important field of applications of the spectral-theoretic ideas developed here is the field of statistical kinetics in the context of first passage concepts (e.g. [90,91,92]), where general results for non-Markovian dynamics are quite sparse [93,94,95,49,96,97,98,99] and will be the subject of our future studies.…”