Hamiltonian Active Particles in Incompressible Fluid Membranes

Abstract

Active proteins and membrane-bound motors exert force dipole flows along fluid interfaces and lipid bilayers. We develop a Hamiltonian framework for the interactions of pusher and puller dipoles embedded in an \emph{incompressible} two-dimensional membrane supported by a shallow viscous subphase. Beginning from the Brinkman-regularized Stokes equations of the membrane-subphase system, we construct the near-field and far-field dipolar velocity fields and associated stream functions. For two quenched (fixed orientation) dipoles, we obtain exact analytic solutions in both the near- and far-field regimes. Although generic dipoles reorient under local membrane vorticity, we show that the far-field dipolar flow is vorticity-free; force-free motors therefore retain fixed orientations and obey a position-based 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. For identical dipoles, the far-field Hamiltonian produces rapid clustering from random initial conditions, whereas the near-field Hamiltonian suppresses collapse leading to non-aggregating configurations. The phase portraits reveal that the many-body collapse observed in the time evolution of the far field Hamiltonian arises from angular transport across phase space enabled by the screened hydrodynamic interaction. Our work thus provides a concrete realization of a position-based Hamiltonian formulation for active particles in incompressible fluid membranes and shows that hydrodynamic screening can reorganize both the dynamical phase space and the collective organization of active dipoles.

Some exact solutions for pair dynamics added

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