Control protocols for harmonically confined run-and-tumble particles

Run-and-tumble particles constitute one of the simplest models of self-propelled active matter, and provide an ideal playground to the understanding of out-of-equilibrium systems. We consider an idealized setup where one such particle is subject to a harmonic confining potential, and an external agent can vary in time the tumbling rate and the strength of the trap. We search for time-dependent control protocols steering the system between assigned end states, in a prescribed time interval. To this aim, we propose a description of the dynamics, alternative to the usual ones, in the form of an infinite set of ordinary differential equations. Solutions based on a suitable closure of such hierarchy, which we expect to hold true in the limit of long protocol duration, are discussed and compared with numerical simulations. We also look for the protocol completing the task with the minimal work, on average: the problem can be tackled analytically, again in the regime of slow (but not quasi-static) transformations. The solution provides insightful intuition on the optimal strategies for the control of active matter systems.

💡 Research Summary

The paper addresses the problem of optimally steering a single run‑and‑tumble particle (RTP) confined in a one‑dimensional harmonic trap from a prescribed initial nonequilibrium steady state to a desired final steady state within a finite time. The RTP dynamics is defined by the overdamped equation (\dot{x}= -\mu U’(x)+v_{0}\sigma(t)) where the orientation (\sigma=\pm1) follows a random telegraph process with tumbling rate (\alpha/2). No thermal noise is present (temperature set to zero), so all fluctuations are generated by the active propulsion. The harmonic potential (U(x)=\frac{1}{2}\kappa x^{2}) yields a stationary density that is a Beta distribution on the finite interval (|x|<1/\kappa). The shape of this distribution is controlled by the dimensionless parameter (\beta=\alpha/(2\kappa)-1); (\beta>0) corresponds to a passive‑like, centrally peaked profile, while (\beta<0) produces an active‑like accumulation at the boundaries.

Because the full probability density obeys coupled partial differential equations for the density (\rho(x,t)) and the magnetization (m(x,t)), solving the control problem directly is analytically intractable. The authors therefore introduce an alternative representation: they expand the density as an exponential of an infinite series of even‑power monomials with time‑dependent coefficients (\beta_{n}(t)) and a time‑dependent front factor (e^{\kappa(t)}). The magnetization is written as (m(x,t)=\gamma(x,t)\rho(x,t)x) with (\gamma) expanded in even powers with coefficients (c_{n}(t)). This mapping translates the original PDE system into an infinite hierarchy of ordinary differential equations (Eqs. 12a–12e) linking the control parameters (\kappa(t)) and (\alpha(t)) to the state variables ({e^{\kappa},\beta_{n},c_{n}}).

To construct feasible “shortcuts to adiabaticity” (STA) for the RTP, the authors first consider the case where both initial and final states are stationary. They prescribe smooth interpolation functions for the primary parameters, \