Two-Dimensional Active Brownian Particles Crossing a Parabolic Barrier: Finite Rectangular Domain with Absorbing Boundary Conditions

Abstract

We solve the time-dependent Fokker-Planck equation for a two-dimensional active Brownian particle exploring a rectangular domain with absorbing boundary and in the presence of a parabolic barrier along one direction. By taking those of a passive Brownian particle as basis states and dealing with the activity as a perturbation, we provide a matrix representation of the Fokker-Planck operator and express the propagator in terms of the perturbed eigenvalues and eigenfunctions. Our solution also allows us to obtain the survival probability and the first-passage-time distribution. The non-equilibrium character of the dynamics induces a strong dependence of the latter quantities on the particle's activity, while the rotational diffusivity influences them to a minor extent.

Figures (8)

• Figure 1: Numerical eigenvalues $\lambda_{n,m,s}$ 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 $\beta k d^2 = 10$, $\gamma=2$, $\alpha=1.5$, $n_{\text{max}}=2$, $m_{\text{max}}=2$, and $s_{\text{max}}=1$. Transparency of lines and exceptional points highlighted with red circles better show when real components merge and imaginary ones bifurcate.
• Figure 2: Spatial probability distribution at different times $t$ starting with initial condition $x_0 = -d/2$, $y_0 = 0.5d$, and $\vartheta_0 = \pi/4$. Comparison between simulations, numerics for $\beta k d^2 =10$, $\alpha=1.5$, $\text{Pe} = 6$ and $\gamma=0.2$. For the simulations, statistics has been collected from $10^8$ independent particles. For the numerics, $n_{\text{max}}=6$, $m_{\text{max}}=7$, and $s_{\text{max}}=5$. In both cases, the distributions were obtained by binning each spatial direction with a resolution of $0.05d$, yielding $41$ and $61$ bins along the $x$ and $y$ axes, respectively.
• Figure 3: Survival probability, $S(t) = S(t| \boldsymbol{\mathrm{r}}_0,\vartheta_0)$, as a function of time for different $\text{Pe}$ and with initial condition $x_0=-d/2$, $y_0=\alpha d$, and $\vartheta_0 = 0$. Comparison between simulations (symbols) and numerics (lines) for $\beta k d^2 =10$, $\alpha=1.5$, and $\gamma=0.4$ For the simulations, statistics has been collected from $10^5$ independent particles. For the numerics, $n_{\rm max}=16$, $m_{\rm max}=14$, and $s_{\rm max}=12$. Inset: Survival probability for different $\alpha$ at $\text{Pe}=4$, $\beta k d^2 =10$, and $\gamma=0.4$, and with initial condition $x_0=-d/2$, $y_0=\alpha d$, and $\vartheta_0 = 0$. The curve for $\alpha=\infty$ is obtained from the theory developed in Ref. Caraglio2025.
• Figure 4: For different Péclet numbers, difference between the survival probability, $S(t) = S(t| \boldsymbol{\mathrm{r}}_0,\vartheta_0)$ computed numerically according to Eq. \ref{['eq:survival_probability2']}, and its value at the leading order in $\text{Pe}$, $S^{(0)}(t) + \text{Pe} \, S^{(1)}(t)$, as given in Eqs. \ref{['eq:survival_probability_first_order_0']} and \ref{['eq:survival_probability_first_order_1']}. Initial condition: $x_0=-d/2$, $y_0=\alpha d$, and $\vartheta_0 = 0$. Other parameters: $\beta k d^2 =10$, $\gamma=0.4$, and $\alpha=1.5$. For the numerics, $n_{\rm max}=16$, $m_{\rm max}=14$, and $s_{\rm max}=12$.
• Figure 5: First-passage-time distribution, $F(t) = F(t| \boldsymbol{\mathrm{r}}_0,\vartheta_0)$ for different $\text{Pe}$ and $\gamma$ and with initial condition $x_0=-d/2$, $y_0=\alpha d$, and $\vartheta_0 = 0$. Comparison between simulations (symbols) and numerics (lines) for $\beta k d^2 =10$ and $\alpha=1.5$. For the simulations, statistics has been collected from $10^7$ independent particles. For the numerics, $n_{\rm max}=32$, $m_{\rm max}=30$, and $s_{\rm max}=6$. In the inset the same quantity is reported for $\text{Pe}=100$ and $\gamma=5$. Here, $m_{\rm max}=12$, $s_{\rm max}=6$, and $n_{\rm max}$ changes according to the caption.