Control protocols for harmonically confined run-and-tumble particles

TL;DR

The paper addresses steering a one-dimensional run-and-tumble particle in a harmonic trap between nonequilibrium steady states in finite time by introducing a cumulant-like expansion that yields an infinite hierarchy of ODEs for time-dependent state parameters. In the slow-driving limit, it derives tractable reductions (two-parameter truncations) and exact expressions for control fields $(\kappa(t),\alpha(t))$ that implement shortcuts and minimize the average work, revealing abrupt initial changes in confinement and activity as a common feature. The key contributions are the reformulation of active RTP dynamics into a parameter-driven hierarchy, the development of recursive and long-duration shortcuts, and the demonstration of explicit optimal protocols with activity-dependent strategies, offering a general route to optimal control in nonequilibrium active matter. The results provide conceptual and practical guidance for designing energy-efficient control in active systems and point to extensions to broader classes of active matter models.

Abstract

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.

Figures (9)

• Figure 1: Stationary distribution of the RTP process in the presence of harmonic confinement, Eq. \ref{['eq:rhos-harmonic']}. The red line corresponds to the active case ($\beta<0$), while the green and blue lines to the passive case ($\beta>0$). The stiffness is set to $\kappa=1$.
• Figure 2: Shortcut in the long-duration regime. In panel (a) the analytical prediction (red) and numerical simulations (blue) for the variance are compared. Panel (b) shows the assigned distribution parameter $\widetilde{\kappa}(t)$ (solid red) defined by Eqs. \ref{['eq:simuk']} and \ref{['eq:gf']}, and the control $\kappa(t)$ (dashed green), analytically computed in Eq. \ref{['eq:kappapert']}. Panel (c) displays the distribution parameter $\widetilde{\alpha}_1(t)$ (solid red), defined by Eq. \ref{['eq:a1']}, where $\beta_1(t)$ is given by Eqs. \ref{['eq:simub']} and \ref{['eq:gf']}. The parameter $\widetilde{\alpha}_2$ (solid yellow), computed analytically by means of Eqs. \ref{['eq:a2']} and \ref{['eq:closure2']}, and the control $\alpha$ (dashed green) defined by Eq. \ref{['eq:alphapert']} are also shown. Finally, in panels (d)-(h) the instantaneous distribution of the process is plotted at different times, both for the analytical prediction \ref{['eq:rhopert']} (red solid line) and the numerical simulation (blue histogram). Parameters: $t_f=5$, $\widetilde{\kappa}_i=1$, $\widetilde{\kappa}_f=1.5$, $\beta_i=0.5$, $\beta_f=-0.5$.
• Figure 3: Shortcut in the long-duration regime. Same as in Fig. \ref{['fig:nonopt1']}, with $t_f=20$.
• Figure 4: Optimal shortcut in the long-duration regime. The plotted quantities and the color code are the same as in Fig. \ref{['fig:nonopt1']}, but this time the protocol has been obtained by solving (numerically) the Euler-Lagrange equations \ref{['eq:eulag']}. Parameters: $t_f=4$, $\widetilde{\kappa}_i=1$, $\widetilde{\kappa}_f=1.1$, $\beta_i=0.05$, $\beta_f=-0.05$.
• Figure 5: Nonoptimal shortcut in the long-duration regime. The protocol is found with the same procedure discussed in relation to Fig. \ref{['fig:nonopt1']}. The boundary conditions are the same as in Fig. \ref{['fig:opt']}.