Hamiltonian Active Matter in Incompressible Fluid Membranes

TL;DR

This work develops a minimal hydrodynamic and Hamiltonian framework for membrane-embedded force-dipole motors on an incompressible supported membrane. By deriving the real-space Green's tensor for the membrane–subphase system, it identifies a near-field logarithmic 1/r regime and a far-field algebraically screened 1/r^3 regime for dipolar flows, and formulates corresponding stream functions. It shows that the far-field flow is irrotational, yielding an exact position-based Hamiltonian H_far for fixed orientations, while the near-field flow retains vorticity and requires a quenched-orientation limit to define H_near. Simulations reveal a striking kernel-dependent dichotomy: the far-field Hamiltonian promotes rapid aggregation into clusters, whereas the near-field Hamiltonian suppresses collapse and yields extended, non-aggregating configurations. These results establish how hydrodynamic screening dictates collective organization of membrane-bound active matter and provide a foundation for extensions to more complex membranes and experimental tests.

Abstract

Active proteins and membrane-bound motors exert force dipole flows along fluid interfaces and lipid bilayers. We develop a unified hydrodynamic and Hamiltonian framework for the interactions of pusher and puller dipoles embedded in an incompressible two-dimensional membrane supported by a shallow viscous subphase. Beginning from the screened Stokes equations of the membrane-subphase composite, we derive the real-space incompressible Green's tensor, obtain its near- and far-field asymptotics, and construct the resulting dipolar velocity and stream functions. Although generic dipoles reorient under the local membrane vorticity, we show that the far-field dipolar flow is vorticity-free; force-free motors therefore retain fixed orientation and obey a Hamiltonian dynamics in which the positions of $N$ dipoles evolve via an effective Hamiltonian built from the dipolar stream function. In the near field, where the flow possesses finite vorticity, a Hamiltonian formulation is recovered in the quenched-orientation limit. Exploiting this structure, we simulate ensembles of pusher and puller dipoles and compare the dynamics generated by the $1/r$ near-field kernel and the subphase screened $1/r^{3}$ far-field kernel. For identical dipoles, the far-field Hamiltonian produces rapid clustering from random initial conditions, whereas the near-field Hamiltonian suppresses collapse and yields extended, non-aggregating configurations.

Figures (5)

• Figure 1: Schematic of active force--dipole motors confined to a supported fluid membrane. Orange ellipses denote motors with orientations (double arrows). The light blue layer is an incompressible two--dimensional membrane of shear viscosity $\eta_s$, supported by a viscous subphase of viscosity $\eta$ and thickness $h$ above a rigid wall.
• Figure 2: Dynamics of $N=15$ pusher (top row) and puller (bottom row) dipoles in an incompressible supported membrane. Open circles mark initial positions; filled black dots mark final positions. (a–b,e–f) Evolution under the screened far-field Hamiltonian $H_{\rm far}$: particle trajectories (a,e) and mean pair separation $\langle d_{ij}\rangle(t)$ (b,f). Both pushers and pullers exhibit strong mutual attraction and collapse into compact clusters. (c–d,g–h) Evolution under the unscreened near-field Hamiltonian $H_{\rm near}$ in the quenched-orientation limit: trajectories (c,g) and $\langle d_{ij}\rangle(t)$ (d,h). In both cases the ensembles expand and $\langle d_{ij}\rangle(t)$ grows monotonically, demonstrating that near-field interactions suppress the far-field aggregation instability.
• Figure 3: Representative far–zone dynamics of a randomly initialized cluster of twelve pushers in the incompressible membrane with random initial locations (within a disc) and random orientations. Green disks indicate initial positions, red disks indicate final positions, and the grey curves trace the particle trajectories. A soft harmonic repulsion is included to regularize close encounters.
• Figure S1: Far-zone dynamics of co-aligned clusters of twelve pushers (left) and pullers (right) in an incompressible membrane, initialized at random locations within a disc. Initial positions appear in green (pushers) and yellow (pullers), final positions in red and orange, with trajectories in grey. A soft harmonic repulsion is included to prevent particle overlap.
• Figure S2: Representative near-zone dynamics of randomly initialized (within a disc), co-aligned clusters of twelve pushers (left) and pullers (right) in an incompressible membrane. Initial positions are shown in green (pushers) and yellow (pullers); final positions in red and orange, respectively; trajectories are grey. A soft harmonic repulsion is included to regularize close encounters.