Kinetics of rare events for non-Markovian stationary processes and application to polymer dynamics

How much time does it take for a fluctuating system, such as a polymer chain, to reach a target configuration that is rarely visited—typically because of a high energy cost? This question generally amounts to the determination of the first-passage time statistics to a target zone in phase space with lower occupation probability. Here, we present an analytical method to determine the mean first-passage time of a generic non-Markovian random walker to a rarely visited threshold, which goes beyond existing weak-noise theories. We apply our method to polymer systems, to determine (i) the first time for a flexible polymer to reach a large extension, and (ii) the first closure time of a stiff inextensible wormlike chain. Our results are in excellent agreement with numerical simulations and provide explicit asymptotic laws for the mean first-passage times to rarely visited configurations.

Figure 1

What is the mean first time to reach a rare configuration? This Rapid Communication investigates this question in the case of (a) an attached flexible polymer, for which we compute the time that a large extension is reached, and (b) a stiff wormlike chain, for which we compute the time that the extremities get into contact.

Figure 2

(a) Mean FPT for a flexible chain to reach an extension $z=3.5l_0\sqrt{3N}$, corresponding to a fixed energy cost $U=18.4k_{B}T$. Symbols: Simulations of Ref. [35]. Different curves correspond to different theories, obtained (from top to bottom) via a mapping over 1D dynamics (Milner-McLeish reptation theory [39, 40], upper dashed line), the minimal action path method [35], the pseudo-Markovian (Wilemski-Fixman [52]) approximation, the non-Markovian theory (this work, black thick line), asymptotic expansions of the non-Markovian theory [dashed blue line, Eq. (9), this work], and the weak-noise result $T\to 0$, fixed $N$ [30, 35]. Details on all theories can be found in SM [49]. (b) Mean FPT in rescaled variables, with supplemental simulation data of Ref. [35] (symbols). Lines share the same color code as in (a). (c) Rescaled average trajectory $\mu \left(t\right)$ in the future of the FPT for a scale invariant process with $H=1/4$. The dashed red line would be the future trajectory by assuming equilibrium at initial time.

Figure 3

Mean closure time for wormlike chains as a function of capture radius $a$, shown in rescaled variables. Symbols: Simulations of Ref. [48], rescaled by $p_s\left(r\right)$ given in Ref. [63]. Continuous black line: Pseudo-Markovian approximation [Eq. (11)], with asymptotic regimes indicated by the dashed black lines. Dashed-dotted orange line: Non-Markovian theory for $L^2/l_p\ll a\ll L$, Eq. (12).