### Diffusion over a fluctuating barrier in underdamped dynamics

#### Abstract

We apply a Langevin model by imposing additive and multiplicative noises to study thermally activated diffusion over a fluctuating barrier in underdamped dynamics. The barrier fluctuation is characterized by Gaussian colored noise with exponential correlation. We present the exact solutions for the first and second moments. Furthermore, we use direct simulations to calculate the asymptotic probability for a Brownian particle passing over the fluctuating barrier. The results indicate that the correlation of the fluctuating barrier is crucial for barrier crossing dynamics.

DOI:https://doi.org/10.1103/PhysRevE.83.041108

#### Authors & Affiliations

• Department of Physics, Huzhou Teachers College, Huzhou 313000, People’s Republic of China

• *jwmao@hutc.zj.cn

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##### Issue

Vol. 83, Iss. 4 — April 2011

##### Access Options

Article Available via CHORUS

##### Announcement
Physical Review E Scope Description to Include Biological Physics
###### January 14, 2016

The editors of Physical Review E are pleased to announce that the journal’s stated scope has been expanded to explicitly include the term “Biological Physics.”

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#### Images

• ###### Figure 1
Normalized PDF in the presence of additive and multiplicative noises. The initial position ${x}_{0}=-2$ and initial velocity ${v}_{0}=1.5$. $B=\frac{1}{2}m\Omega {}^{2}{x}_{0}^{2}$ is the barrier height measured from the initial position. Here $T/B=0.5$, $\gamma =0.1$, $Q=0.1$, and $\tau =1$. (a) $t=5$. (b) $t=10$.
• ###### Figure 2
A log-log plot of the time-dependent ratio $r=-〈x〉/\sqrt{{\sigma }_{\mathit{xx}}}$ in the system with a static barrier $Q=0$ and fluctuating barriers $Q=0.1$, $\tau =0.1$ and $Q=0.1$, $\tau =1$, respectively. The lines are the analytical results of Eqs. (16) and (17) and the scatters denote numerical integration results. Other parameters are the same as in Fig. 1.
• ###### Figure 3
Temperature dependence of the probability [Eq. (32)] for the weak-friction case of $\gamma =0.1$. Here the critical kinetic energy ${K}_{c}/B=1.1051$. The initial kinetic energies from top to bottom correspond to $K/B=1.5625,1.1051,$ and $0.5625$, respectively. The barrier height $B=2$. For a fluctuating barrier, $Q=0.1$ and $\tau =0.1$.
• ###### Figure 4
Energy regimes with completely different characteristics for the temperature dependence. The solid line is the result of Eq. (33), acting as the boundary of the two regimes. The barrier height $B=2$.
• ###### Figure 5
Probability of passing over a fluctuating barrier as a function of time. The temperature $T/B=0.5$ and friction $\gamma =0.1$ in the simulations. (a) Stationary probability obtained after a time for either small $Q$ or small $\tau$. From bottom to top the three lines indicate $Q=0.1$ and $\tau =0.1$ (small $Q$ and small $\tau$), $Q=0.1$ and $\tau =10$ (small $Q$), and $Q=1.0$ and $\tau =0.1$ (small $\tau$), respectively. The initial kinetic energy $K/B=0.5$. (b) Nonstationary probability for large $Q$ and large $\tau$. Notice that the time scale is much greater than that in (a).
• ###### Figure 6
Asymptotic probability of passing over a fluctuating barrier as a function of the variance $Q$ for three cases. A small correlation time $\tau =0.1$ is taken to ensure a stationary probability within the simulation time. Other parameters are the same as in Fig. 5.
• ###### Figure 7
Semilogarithmic plot of the overpassing probability as a function of the initial kinetic energy. The temperature $T/B=0.5$. The solid line is the result of Eq. (32), which corresponds to the static barrier potential ($Q=0$), and the scatters from bottom to top correspond to the fluctuating barrier with variance $Q=0.1$ and correlation times $\tau =0.1,1.0$, and $10.0$, respectively. The arrow indicates the average barrier height. Inset: Probability at higher energies on a linear scale.
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