Two-Dimensional Active Brownian Particles Crossing a Parabolic Barrier: Transition-Path Times, Survival Probability, and First-Passage Time

Abstract

We derive an analytical expression for the propagator and the transition path time distribution of a two-dimensional active Brownian particle crossing a parabolic barrier with absorbing boundary conditions at both sides. By taking those of a passive Brownian particle as basis states and dealing with the activity as a perturbation, our solution is expressed in terms of the perturbed eigenfunctions and eigenvalues of the associated Fokker-Planck equation once the latter is reduced by taking into account only the coordinate along the direction of the barrier and the self-propulsion angle. We show that transition path times are typically shortened by the self-propulsion of the particle. Our solution also allows us to obtain the survival probability and the first-passage times distribution, which display a strong dependence on the particle's activity, while the rotational diffusivity influences them to a minor extent.

Figures (7)

• Figure 1: Function $\mathcal{X}_{n}(x) := \exp(-\beta k x^2 /4) Y_n(x) / \mathcal{N}_n$ plotted for $n=0,1,\ldots,7$ with $\beta k d^2 = 10$. Absorbing boundary conditions are imposed in $x=\pm d$ which leads to the reported values of $\sigma_n$ (obtained from numerical calculations).
• Figure 2: Numerical eigenvalues $\lambda_{n,s}$ of the Fokker-Planck operator $\mathcal{L} = \mathcal{L}_0+\text{Pe} \, \mathcal{L}_1$ as a function of the Péclet number $\text{Pe}$, for $\beta k d^2 = 10$, $\gamma=2$, $n_{\text{max}}=3$, and $s_{\text{max}}=2$. Transparency of lines and exceptional points highlighted with red circles better show when real components merge and imaginary ones bifurcate.
• Figure 3: Probability distribution at different times $t$ starting with initial condition $x_0 = d/2$ and $\vartheta_0 = \pi$. Comparison between simulations, numerics for $\beta k d^2 =10$, $\text{Pe} = 6$ and $\gamma=2$. For the simulations, statistics has been collected from $10^8$ independent particles. For the numerics, $n_{\text{max}}=7$ and $s_{\text{max}}=6$.
• Figure 4: Probability distribution, integrated over the self-propulsion direction, $\vartheta$, at different times $t$ starting with initial condition $x_0 = d/2$ and $\vartheta_0 = \pi$. Comparison between simulations, numerics for $\beta k d^2 =10$, $\text{Pe} = 6$ and $\gamma=2$. For the simulations, statistics has been collected from $10^7$ independent particles. For the numerics, $n_{\text{max}}=10$ and $s_{\text{max}}=9$.
• Figure 5: Transition-path-time distribution, $P_{\rm TPT}(t) = P_{\rm TPT}(t| \vartheta_0)$, as a function of the direction of the self-propulsion at the left boundary, $\vartheta_0 = 0$. Comparison between simulations (symbols) and numerics (lines) for $\beta k d^2 =10$ and $\gamma=2$ For the simulations, statistics has been collected from $10^6$ independent transition paths defined as paths starting at $x=-d+\varepsilon$ with $\varepsilon = 10^{-7}$ and ending at $x=d$. For the numerics, $n_{\rm max}=32$ and $s_{\rm max}=6$.