Emergent Dynamics of Active Systems on Curved Environments

TL;DR

This work addresses how curvature shapes active matter dynamics on curved two-dimensional manifolds embedded in three-dimensional space. It develops a covariant, surface-constrained model using the spin connection and zweibein to derive a geometric torque $\Omega_t$ that steers particle orientation, yielding geodesic motion in the noiseless, non-interacting limit. The authors show curvature-induced lensing and quasi-trapping for single particles, and that curvature generally disrupts flock coherence unless a strong alignment torque $J$ is present; on a sphere, a rotating-slice model captures band-like states that are validated by bead-chain simulations, though full dynamics reveal the band as liquid-like rather than rigid. The framework provides a minimal, covariant basis for interpreting curvature–activity interactions with potential relevance to cellular morphogenesis and tissue navigation.

Abstract

Curvature plays a central role in the proper function of many biological processes. With active matter being a standard framework for understanding many aspects of the physics of life, it is natural to ask what effect curvature has on the collective behaviour of active matter. In this paper, we use the classical theory of surfaces to explore the active motion of self-propelled agents confined to move on a smooth curved two-dimensional surface embedded in Euclidean space. Even without interactions and alignment, the motion is non-trivially affected by the presence of curvature, leading to effects akin, e.g.\ to gravitational lensing and tidal forces. Such effects can lead to intermittent trapping of particles and profoundly affect their flocking behaviour. We show that these effects are governed by a geometric torque that, in the absence of noise and interactions, compels particles to move along geodesics.

Figures (5)

• Figure 1: (a) Active agents (yellow) are confined to move on the surface $\mathcal{S}$ (pale orange) embedded in $\mathbb{R}^3$. Each agent is endowed with a unit-length vector $\mathbf{n}_i$ (purple) tangent to $\mathcal{S}$ at every point; it sets the direction of the self-propulsion. Tangent planes at every point are spanned by a pair of tangent vectors defined as $\mathbf{e}_\alpha=\partial_\alpha\mathbf{r}$ (green and blue). We use subscripts $P$ and $Q$ to denote that the tangent vectors depend on the point on the surface. Vector $\mathbf{N}$ (red) is the unit normal to the surface, which also depends on the position. (b) The coordinate basis $\{\mathbf{e}_1,\mathbf{e}_2\}$ (green and blue) in the tangent space $T_P\mathcal{S}$ is not necessarily orthonormal. One can, however, choose an orthonormal (i.e. frame) basis $\{\mathbf{E}_1,\mathbf{E}_2\}$ (black). Vectors in tangent spaces $T_P\mathcal{S}$ and $T_P\mathcal{S}$ are related via Levi-Civita connection ($\nabla_\alpha$). Expressed in the frame bases, connection coefficients are known as the spin connection ($\tensor{\omega}{_\alpha^a_b}$). It acts to rotate the frame basis around the surface normal as one moves along $\mathcal{S}$.
• Figure 2: Scattering and quasi-trapping of active particles by a Gaussian bump with $A/\sigma=50$. Panel (a) shows the scenario where the Gaussian bump scatters an active particle. Panels (b) and (c) show two cases where an active particle circles around the centre of the bump for an extended time. While there are no closed geodesics on a Gaussian bump, some can wind multiple times, corresponding to the particle being effectively trapped. Shades of grey show the elevation of the Gaussian bump. The parula colours along the trajectory indicate the height which the particle reached.
• Figure 3: Jacobi field $\mathcal{J}$ as a function of arc-length parameter $s$ for six geodesics. Each geodesic starts at $x_0=-3$ and $y_0\in\{0,0.2,0.4,0.6,0.8,1.0\}$. The Jacobi field is calculated for the "pair" geodesic initially at the distance $0.1$ from the corresponding primary geodesic, with both of them initially pointing along $\mathbf{e}_\mathrm{x}$, i.e. $\mathcal{J}(0)=0.1$ and $\mathcal{J}'(0)=0$. The Jacobi field curves are coloured by the local value of the Gaussian curvature, $K$. The black dot denotes a conjugate point. $A/\sigma=1$. Inset: Gaussian bump coloured by the local value of Gaussian curvature $K$. Note that for $r<\sigma$ ($r>\sigma$), $K>0$ ($K<0$). The $K<0$ region is, however, exponentially suppressed and has a much weaker effect than the $K>0$ region.
• Figure 4: A flock of active agents moving along the Gaussian bump with $A/\sigma=3$. (a) $J=0$ (i.e. a non-interacting flock) where particles move purely under the influence of geometric torque and follow geodesics. (b) Intermediate alignment ($J=1.0$) -- particles no longer follow geodesics, and the flock scatters as it crosses the bump. (c) Strong alignment ($J=10.0$), where external torque dominates, resulting in a significantly more compact flock. Coloured triangles represent individual particles before ($x/\sigma = -10$) and after ($x/\sigma = 10$) crossing the bump. The flock is initially composed of $11$ particles, each separated by a distance of $0.5$. Shades of grey represent the height profile of the Gaussian bump. Particle trajectories are coloured according to the "elevation" they reach.
• Figure 5: Angle $\alpha$ between $\mathbf{n}_i$ and the $\mathbf{E}_\phi$ axis of the local orthonormal frame vs. the latitude $\psi$ for the numerical solution (blue) and the simulation (red) of the toy model for interacting active agents on a sphere of radius $R=10$, with $J=0.5$ and $v_0=0.5$, $k=100$, $\sigma=0.25$. Bottom inset: Sketch of the toy model of the band. Blue discs represent particles and grey coils represent the soft-core potential between nearest neighbours. Top inset: Snapshot of the simulation showing positions of agents (yellow) and directions of the vectors $\mathbf{n}_i$ (red arrows).