Active particles in power-law potentials: steady state distributions and shape transitions

TL;DR

This work analyzes the nonequilibrium stationary states of two canonical active-particle models, ABP and RTP, confined in two-dimensional power-law potentials U(r) ∝ r^n (n even, ≥2) with vanishing translational diffusion. By employing a Gaussian orientation ansatz, the authors derive a closed radial equation for the scaled distribution Ψ(ξ) (ξ = r/R) and obtain an explicit ABP radial solution Ψ_n(ξ) along with a large-n approximation, revealing a shape transition in the positional distribution as trap strength varies; the transition is continuous for n = 2 and discontinuous for n > 2, with a detailed bifurcation structure in the angular distribution. For RTP, they extend the formalism to the harmonic case (n = 2), obtaining an exact Ψ_2(ξ) with a β′-modified critical point β′_c = 1/3 and showing a continuous transition, while simulations indicate similar transition behavior for higher n. The study illuminates how activity, confinement, and orientational dynamics conspire to produce boundary-accumulation and orbiting phenomena, providing a framework to understand orientational bifurcations and shape transitions in active matter under confining potentials.

Abstract

We study the stationary states of an active Brownian particle (ABP) and run-and-tumble particle (RTP) in two dimensional power-law potentials, in the limit where translational diffusion is negligible. The potential energy of the particle has the form $U(r)\propto r^{n}$, where $n\geq 2$ and even. In two dimensions, we derive the exact equations for the positional probability distribution $φ({\bf r})$ of ABP ($n\geq 2)$ and RTP ($n=2$), whose solutions are obtained under the assumption that the particle's orientation angle is Gaussian. Both analytical and numerical results show that, in all cases, $φ({\bf r})$ has compact support and undergoes a phase transition-like shape change as a function of the trap strength. For ABP, our theory predicts a continuous transition in shape for $n=2$ and a discontinuous transition for $n>2$, both of which agree with the simulation results. Simulations suggest the existence of both types of shape transition in the case of RTP as well. For ABP, in the strongly active regime, the orientational probability distribution is unimodal near the outer boundary but becomes bimodal towards the interior, signifying a transition from predominantly radial orientation to orbiting motion. In RTP, the analogous shape transition in the orientational distribution is almost absent.

Figures (12)

• Figure 1: Schematic figure showing the relevant dynamical variables of an active particle in two dimensions. Here, ($r, \, \phi$) are the polar coordinates for the particle's position while $\hat{\mathbf{u}}$ is a unit vector indicating the direction of self-propulsion.
• Figure 2: The positional distribution $\Psi_n(\xi)$ for the ABP, predicted by theory (Eq. \ref{['expression_of_psi']}) is plotted against $\xi$ for different values of $\beta$ and $n$, as indicated. To ensure series convergence, the upper limit of the summation over $s$ in Eq. \ref{['expression_of_psi']} is set as 500. In each figure, $n$ is fixed (as $2,4,8$ and $20$ for $\textbf{a},\textbf{b}, \textbf{c}$ and $\textbf{d}$ panels respectively), and the different colors (or shapes) correspond to different values of $\beta$.
• Figure 3: This figure demonstrates how the shape transition is different for $n=4$, from $n=2$. In both (a) and (b), the peak, denoted by the symbol red $P$ is the highest point of the distribution, while the red star (max) and blue circle (min) indicate local maxima and minima respectively. The diamond-shaped marker stands for a higher-order stationary point (HOSP), which is colored red if it corresponds to a peak.
• Figure 4: Shape transition in $\Psi_n(\xi)$ as a function of $n$ for a fixed $\beta$ is shown. Star (red) and circle (blue) shaped points indicate position of the maximum (or peak here) and minimum respectively, while the diamond shaped point represents a higher order stationary point (HOSP), which is colored red if it corresponds to a peak. The shape transition is discontinuous, as discussed in the text.
• Figure 5: In both panels, $\Psi_n(\xi)$ obtained in simulations (gray circles) is plotted against $\xi$ for two different $\beta$ ($\beta=0.1$ in the top panel and $\beta=1.0$ in the bottom panel) and four different values of $n$ (2,4,8 and 20). As $n$ is increased, there is a shift in the position of the peak of the distribution function $\Psi_n(\xi)$ from the origin towards $\xi=1$ as predicted. The dashed lines (red) are theoretical fits using the expression in Eq. \ref{['expression_of_psi']} and solid lines (dark blue) are approximate closed form expression of $\Psi_n$ for large $n$ (Eq. \ref{['closed_form_expression_large_n']}). Note that both the theoretical predictions closely align with the numerical results.