Stationary distribution of the quorum-sensing active particle (QSAP) model

Determine the stationary probability distribution of the quorum-sensing active particle system consisting of N one-dimensional particles in an asymmetric quartic potential U with metastable states at positions a and c, evolving under overdamped Langevin dynamics with thermal noise at inverse temperature β and an interaction kernel η^i(X_t) that assigns an active run-and-tumble velocity of magnitude v0 governed by a telegraph process with rate λ when particle i has no neighbor within cutoff distance ℓ, and otherwise suppresses activity.

Background

To design minimally dissipative control protocols for switching a QSAP system between metastable wells, the authors require source and target distributions. Because the stationary distribution of the active, interacting system under the specified run-and-tumble quorum-sensing rule is unknown, they approximate it using Gaussian expansions of the equilibrium (activity-free) distribution near the well centers.

A precise characterization of the stationary distribution would enable more accurate benchmarking of control protocols and tighter thermodynamic bounds for the active interacting particle system.