Transition path dynamics across rough inverted parabolic potential barrier

Abstract

Transition path dynamics have been widely studied in chemical, physical, and technological systems. Mostly, the transition path dynamics is obtained for smooth barrier potentials, for instance, generic inverse-parabolic shapes. We here present analytical results for the mean transition path time, the distribution of transition path times, the mean transition path velocity, and the mean transition path shape in a rough inverted parabolic potential function under the driving of Gaussian white noise. These are validated against extensive simulations using the forward flux sampling scheme in parallel computations. We observe how precisely the potential roughness, the barrier height, and the noise intensity contribute to the particle transition in the rough inverted barrier potential.

Appendix A: The transition path probability density function

According to Eq. (1), the reactive trajectories from $x_\mathrm{A}$ to $x_\mathrm{B}$ should satisfy the stochastic differential equation [21]

$$\begin{aligned} \mathrm {d}x=\left( f(x)+\frac{2D\mathrm{d}\phi _\mathrm{B}(x)/\mathrm {d}x}{\phi _\mathrm{B}(x)}\right) \mathrm{d}t+\sqrt{2D}\mathrm{d}W(t), \end{aligned}$$

(A1)

where W(t) is the unit Wiener process, $\phi _\mathrm{B}(x)$ is the committor function as shown in Eq. (11). Then, the probability density function for the transition paths, $P_{\mathrm {TP}}(x,t)$ from $x_\mathrm{A}$ to $x_\mathrm{B}$, satisfies

$$\begin{aligned} \frac{{\partial } P_{\mathrm {TP}}(x,t)}{{\partial } t}=-\frac{{\partial }}{{\partial } x} \left( f(x)+\frac{2D\mathrm{d}\phi _\mathrm{B}(x)/\mathrm {d}x}{\phi _\mathrm{B}(x)}\right) P_{\mathrm {TP}}(x,t)+D\frac{ {\partial }^2}{{\partial } x^2}P_{\mathrm {TP}}(x,t). \end{aligned}$$

(A2)

With $f(x)=-\mathrm{d}V(x)/\mathrm {d}x=-(1/D)\mathrm{d}G(x)/\mathrm {d}x$,

(A3)

Defining $M(x)=\exp \left( G(x)/D\right)$, we obtain

(A4)

Hence, Eq. (A2) is recast in the Smoluchowski equation [72].

(A5)

This is Eq. (9) in the main text.