Feedback control and delayed interactions in active matter

Abstract

Feedback control plays a central role in active matter, yet it is inevitably accompanied by noise and finite perception--action delays. This Perspective reviews recent advances on active systems with delayed interactions, showing how time delay can induce activity, chirality, transport, and collective pattern formation, and can act as an effective control parameter for switching between dynamical states. We discuss representative single-particle and many-body systems, highlight key experimental realizations, and argue that time delay constitutes an underexplored dimension of morphological intelligence--where intrinsic response dynamics, rather than explicit sensors or computation, enable functional behavior in active matter.

Figures (5)

• Figure 1: All living systems and most artificial active-matter systems rely on feedback, which inevitably introduces delays into their dynamics. Together with noise, these delays can strongly influence both stationary and dynamical properties. They underlie phenomena such as phantom traffic jams bando1998analysisTrafficJamsReview, bacterial chemotaxis deGennes2004, and the emergence of optimal propulsion speeds in feedback-controlled active Brownian microswimmers muinos2021reinforcement.
• Figure 2: Density, $\rho$, (a,c) and polarization, $p$, (b,d), corresponding to the one-dimensional velocity profile $v$ shown in panels (a,c), for vanishing delay (a,b) and a delay of $184ms$ (c,d). Experimental results obtained with thermophoretic microswimmers Franzl2021 are shown as symbols, the blue dashed line denotes theoretical predictions in Eqs. \ref{['eq:rho']} and \ref{['eq:p']}, and the solid black line corresponds to simulations. Parameters: $D_\theta = 2.9\,rad\squared\per s$, $D = 0.04\,µm\squared\per s$. Data from Ref. holubec2025delayedactiveswimmervelocity.
• Figure 3: (a) A particle repelled from its own past position and subject to noise undergoes a symmetry-breaking transition, resulting in an active state. Figure reprinted with permission from R. A. Kopp and S. H. L. Klapp, Phys. Rev. E, 107, 024611 (2023). Copyright (2023) by the American Physical Society Klapp2023. (b–e) Formation of bands in a system of sterically interacting, passive Brownian particles experiencing strong delayed repulsion from their past positions. Panels (b–e) show system snapshots at times $t = 0, 10, 50$, and $100$, respectively. Simulation results reprinted with permission from S. Tarama, S. U. Egelhaaf, and H. Löwen, Phys. Rev. E, 100, 022609 (2019). Copyright (2019) by the American Physical Society Lowen2019.
• Figure 4: Retarded attraction induces chirality. (a) A particle locates the target at time $t-\delta t$, and (b) begins swimming toward it with fixed speed $v_0$ at time $t$. (c) When the product $\delta t v_0$ is sufficiently large, the particle transiently rotates around the target. (d) Multiple particles attracted to the same target synchronize their rotation due to steric and hydrodynamic interactions. Figures a--d reprinted from X. Wang et al., Spontaneous vortex formation by microswimmers with retarded attractions. Nature Communications 14, 56 (2023). Copyright (2023) by The Authors. Licensed under https://creativecommons.org/licenses/by/4.0/Wang2023. (e) For large numbers of particles interacting only via steric forces, increasing the delay at fixed density $\rho$ generates a rich sequence of dynamical phases. The system transitions from a static crystallite to a series of rotating states: crystallite, crystallite with shear bands, circle, yin–yang, and blob. Figure reprinted from P.-C. Chen et al., Active particles with delayed attractions form quaking crystallites. EPL 142, 67003 (2023). Copyright (2023) by The Authors. Licensed under https://creativecommons.org/licenses/by/4.0/Chen2023.
• Figure 5: (a) Phase diagram of the delayed Vicsek model as a function of the noise strength $\eta$ and the time delay $\tau$, rescaled by the ratio of agent speed $v_0$ to interaction radius $R$, based on data from R. Horton and V. Holubec, Order-disorder transition and phase separation in delay Vicsek model. New. J. Phys. 27, 094402 (2025). Copyright (2025) by The Authors. Licensed under https://creativecommons.org/licenses/by/4.0/horton2025. (b) Phase diagram of a modified delayed Vicsek model as a function of the mean delay $\mu_\tau$ and the strength of repulsive interactions $r_{\mathrm{rep}}$, using data digitized from Fig. 6 of Physica A 634, F. Pakpour and T. Vicsek, Delay-induced phase transitions in active matter, 129453, Copyright (2024), with permission from Elsevier pakpour2023delays.