Boundary-Mediated Phases of Self-Propelled Kuramoto Particles

Abstract

Active agents can transfer energy to their environment through collective motion, generating accumulation patterns near confining obstacles. Here we investigate how the nature of the microscopic drive-self-propulsion or velocity alignment-selects distinct accumulation patterns, leading to either delocalized or compact clustered states. We first characterize the dynamical regimes emerging from the interplay of these two driving mechanisms under perfectly reflective or smooth boundary conditions. We then introduce boundary friction and observe a drastic change in the accumulation patterns, with new dynamical phases that are absent in the previous case. By connecting emergent macroscopic structures to their underlying microscopic interactions, this work provides a practical route to infer the dominant interaction ruling boundary-mediated collective behavior, with applications ranging from single-cell migration to bio-inspired robotics.

Figures (5)

• Figure 1: (a) A particle before and after collision with a smooth boundary. (b) Particle-monomer interaction in a rough boundary with radial $f_r$ and tangential $f_\theta$ components.
• Figure 2: Phases of SPKPs confined by a smooth boundary at fixed persistence time $\tau_p=10^3$. Values of $\tau_p$ and $\tau_K$ are reported in units of $\tau_i$. (a) Gas (G) phase at $\tau_K = 7.1\times10^1$. (b) Delocalized clustered (DC) phase at $\tau_K = 2.4$. (c) Localized clustered (LC) phase at $\tau_K = 7.1\times10^{-4}$. Particles and velocity arrows are color coded according to $\varphi_v$. (d) Diagrams in the plane $(\tau_p, \tau_K)$ color coded by $\phi_r^C$, $\tilde{I}$ and $\tilde{L}$. The vertical line at $\tau_p=10^3$ and the horizontal line at $\tau_K = 7.1\times10^1$ refer to the coordinate axes of Fig. \ref{['fig:moments']}a. Dashed and dotted lines distinguish phases, while squares indicate the configurations shown above.
• Figure 3: Dynamical regimes of SPKPs confined by a rough boundary at fixed persistence time $\tau_p=10^3$. (a) Trapped gas (TG) phase at $\tau_K = 7.1\times10^1$. (b) Coexistence between partially delocalized clustered (PDC) phase and gas at $\tau_K = 2.4$. (c) PDC phase at $\tau_K = 7.1\times10^{-2}$. (d) Diagrams in the plane $(\tau_p, \tau_K)$ color coded by $\phi_r^C$, $\tilde{I}$ and $\tilde{L}$. The vertical line at $\tau_p=10^3$ and the horizontal line at $\tau_K = 7.1\times10^1$ refer to the coordinate axes of Fig. \ref{['fig:moments']}a. Dashed and dotted lines distinguish phases, while squares indicate the configurations shown above.
• Figure 4: (a) Time averages of $\tilde{I}$ and $\tilde{L}$ as functions of $\tau_K$ and $\tau_p$ for smooth (gray circles) and rough (black squares) boundaries. Left panels correspond to $\tau_p = 10^3$ and right panels to $\tau_K = 7.1\times10^1$, representing two cuts of the diagrams in Fig. \ref{['fig:smooth-phases']}d and \ref{['fig:rough-phases']}d indicated by the horizontal and vertical lines. The dashed line in the top panels correspond to the value of $\tilde{I}$ for an annulus of thickness $10\sigma$. The background is colored according to the color maps used in Fig. \ref{['fig:smooth-phases']}d and \ref{['fig:rough-phases']}d. (b) Dependence of $\tilde{L}$ on $\sigma_m/\sigma$ for the phases observed in rough boundary conditions at $\tau_p = 10^3$: LC (upward triangles, $\tau_K = 7.1\times10^{-4}$), PDC (squares, $\tau_K = 7.1\times10^{-2}$), coexistence between PDC and gas phases (circles, $\tau_K = 2.4$), and TG (downward triangles, $\tau_K = 7.1\times10^1$).
• Figure 5: Time averages of the number of clusters, $\langle N_c \rangle$, and the cluster fraction, $\langle N_p \rangle/N$, as functions of $\tau_K$ and $\tau_p$ for smooth (gray circles) and rough (black squares) boundaries. Data are obtained from the same simulation runs as those reported in Fig. \ref{['fig:moments']}.