![Transport in porous media. (A) Geometric criterion for optimal transport of active agents in porous media. Effective diffusivities Deff obtained from simulations of self-propelled polymers in 3D porous media (see right panel for a simulation snapshot) and compared to a theory. Reproduced from [12]. CC BY 4.0. (B) Corner flow in unsaturated porous media generated by surfactant-producing bacteria, which communicate via quorum sensing and where the surfactant changes wettability of the surface to drive spreading. Reproduced with permission from [13].](https://www.researchgate.net/publication/388269377/figure/fig1/AS:11431281310904705@1740027151977/Transport-in-porous-media-A-Geometric-criterion-for-optimal-transport-of-active-agents_Q320.jpg)
![From entanglement to chemotaxis. (A) Entangled blob of California blackworms dissolves rapidly under environmental stress (scale bar is 3 mm). From [14]. Reprinted with permission from AAAS. (B) Crowding-enhanced diffusion of self-propelled filaments in a highly-entangled environment of other active agents. Mean-square displacements ⟨(Δr(t))2⟩ for different reduced number densities n∗as a function of time t. Reprinted figure with permission from [16], Copyright (2020) by the American Physical Society. (C) Suppression of MIPS due to chemotaxis. Here, α0=M0/Dc measures the relative importance of the active diffusivity M0 and the chemical diffusivity Dc and PeC=χ0/M0 is the ratio of the chemotactic coefficient χ0 and M0. Reprinted figure with permission from [17], Copyright (2023) by the American Physical Society.](https://www.researchgate.net/publication/388269377/figure/fig2/AS:11431281310857504@1740027152170/From-entanglement-to-chemotaxis-A-Entangled-blob-of-California-blackworms-dissolves_Q320.jpg)
![Light-controlled micromotors powered by bacteria: (a) array of 16 microfabricated rotors; (b) fluorescence image showing bacteria cell bodies, which tend to occupy microchambers distributed around the gears; (c), (d) zoomed-in view of one of the rotors. Reproduced from [21]. The Author(s). CC BY 4.0.](https://www.researchgate.net/publication/388269377/figure/fig3/AS:11431281310877094@1740027152351/Light-controlled-micromotors-powered-by-bacteria-a-array-of-16-microfabricated-rotors_Q320.jpg)
![Aggregation dynamics in a monolayer of passive sticky colloids is accelerated by swimming E. coli bacteria and gives rise to aggregate morphologies that are unlike those observed in thermal systems. Reproduced from [23]. The Author(s). CC BY 4.0.](https://www.researchgate.net/publication/388269377/figure/fig4/AS:11431281310921964@1740027152592/Aggregation-dynamics-in-a-monolayer-of-passive-sticky-colloids-is-accelerated-by-swimming_Q320.jpg)
![Simulations snapshots that display spontaneous trapping of active particles moving through (a) a random distribution of obstacles. Reprinted figure with permission from [34], Copyright (2013) by the American Physical Society. (b ) A random stress field. Reprinted figure with permission from [35], Copyright (2018) by the American Physical Society.](https://www.researchgate.net/publication/388269377/figure/fig5/AS:11431281310886692@1740027152795/Simulations-snapshots-that-display-spontaneous-trapping-of-active-particles-moving_Q320.jpg)
February 2025
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7 Citations
Activity and autonomous motion are fundamental aspects of many living and engineering systems. Here, the scale of biological agents covers a wide range, from nanomotors, cytoskeleton, and cells, to insects, fish, birds, and people. Inspired by biological active systems, various types of autonomous synthetic nano- and micromachines have been designed, which provide the basis for multifunctional, highly responsive, intelligent active materials. A major challenge for understanding and designing active matter is their inherent non-equilibrium nature due to persistent energy consumption, which invalidates equilibrium concepts such as free energy, detailed balance, and time-reversal symmetry. Furthermore, interactions in ensembles of active agents are often non-additive and non-reciprocal. An important aspect of biological agents is their ability to sense the environment, process this information, and adjust their motion accordingly. It is an important goal for the engineering of micro-robotic systems to achieve similar functionality. Many fundamental properties of motile active matter are by now reasonably well understood and under control. Thus, the ground is now prepared for the study of physical aspects and mechanisms of motion in complex environments, the behavior of systems with new physical features like chirality, the development of novel micromachines and microbots, the emergent collective behavior and swarming of intelligent self-propelled particles, and particular features of microbial systems. The vast complexity of phenomena and mechanisms involved in the self-organization and dynamics of motile active matter poses major challenges, which can only be addressed by a truly interdisciplinary effort involving scientists from biology, chemistry, ecology, engineering, mathematics, and physics. The 2025 motile active matter roadmap of Journal of Physics: Condensed Matter reviews the current state of the art of the field and provides guidance for further progress in this fascinating research area.
![FIG. 2. (a) Experimental (solid lines) protocols for the motion of an optical trap that is displaced by λ f ¼ 2 μm within the time t f ¼ 1 s through a viscous fluid. Different protocols are characterized by their jump height Δλ. The dashed lines correspond to protocols with ideal infinitely fast jumps at the start and end of the protocol. The optimal protocol (green) λ Ã corresponds to Δλ Ã ¼ 0.4 μm [see Eq. (C2) in Appendix C]. (b) Experimentally measured mean work hWi (black) and asymmetry parameter hA x i (orange) as functions of Δλ (symbols). The minima of hWi and hA x i are positioned at the optimal protocol λ Ã , in agreement with theoretical predictions (solid lines). The fact that hA x i ≈ 0 for λ Ã suggests that both λ and x are time symmetric for optimal control. Error bars correspond to the standard error of the mean (SEM). (c) Average particle trajectories x ¼ hXi normalized by the position at the end of the protocol xðt f Þ corresponding to the protocols shown in panel (a).](https://www.researchgate.net/publication/380859871/figure/fig2/AS:11431281247131469@1716661355559/a-Experimental-solid-lines-protocols-for-the-motion-of-an-optical-trap-that-is_Q320.jpg)
![FIG. 3. Diagram showing implications (⇒) and nonimplications (⇏) between time-reversal symmetry [defined in Eq. (3)] of the protocol λ and mean particle trajectory hXi ¼ x, and optimality with respect to the work required, for control processes in linear media. Optimal control is characterized by time-reversal symmetry of both λ and hXi.](https://www.researchgate.net/publication/380859871/figure/fig3/AS:11431281247131470@1716661355719/Diagram-showing-implications-and-nonimplications-between-time-reversal-symmetry_Q320.jpg)
![FIG. 4. (a) Sketch visualizing the used Maxwell model with a fictitious bath particle with friction γ b , coupled to the tracer particle with friction γ with a linear spring with stiffness κ. Only the tracer feels forces from the optical potential. (b) Different time-reversal symmetric control protocols λðtÞ executed, which are a linear combination of a tanh function (blue line), the predicted optimal protocol (green line), and a steplike function (purple line) weighed by a parameter α (for details, see Appendix B). We depict the corresponding particle trajectories in Fig. 8 in Appendix B. (c) Experimentally measured mean work hWi (black symbols) showing a distinct minimum close to α ¼ 0.5 matching the predicted optimal protocol. Theoretical predictions (black line) using the Maxwell model [Eqs. (9) and (10)], which are offset by −290k B T, are in good qualitative agreement with the experimental data. The asymmetry parameter A x [orange, Eq. (4)] quantifies the deviation of the mean trajectory x from time-reversal symmetry and exhibits a minimum at the same position as hWi. Error bars again show the SEM. (d) Measured time resolved power hdW=dti traces for the protocols shown in panel (b). Individual protocols are offset to negative values by 200k B T for better readability; gray lines indicate the corresponding y-axis origin. Protocols including a jump show strong peaks at the beginning and end. The work trace for the optimal protocol (green line) shows a complex nonlinear and nonmonotonic shape. The matching curve from numerically solving the equation of motion (black line) is in good agreement with the experiment.](https://www.researchgate.net/publication/380859871/figure/fig4/AS:11431281247131471@1716661355849/a-Sketch-visualizing-the-used-Maxwell-model-with-a-fictitious-bath-particle-with_Q320.jpg)
![FIG. 7. Mean squared displacement (MSD) of the particle in the stationary trap to experimentally determine the friction coefficient γ 0 . Symbols show the ensemble-averaged data from experiments; error bars correspond to the SEM. The solid line shows the analytical expression for the MSD of a 2.73-μm colloid in a harmonic trap plotted using the fitted parameters. The trap stiffness κ is fitted from the positional distribution in equilibrium, and γ 0 is derived afterward from the initial slope of the MSD [57].](https://www.researchgate.net/publication/380859871/figure/fig5/AS:11431281247216114@1716661356009/Mean-squared-displacement-MSD-of-the-particle-in-the-stationary-trap-to-experimentally_Q320.jpg)
![(a) Schematic drawing of the experimental setup. An oscillating mirror mounted onto a piezo-driven post is deflecting a laser beam through an objective into the sample cell where it creates an extended optical trap. (b) Snapshots of a dumbbell at the start t=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=0$$\end{document} and the end of a recoil experiment (t∼20s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\sim {20}\,\hbox {s}$$\end{document}). The initial angle of the dumbbell’s long axis θ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0$$\end{document} decreases during the recoil by δθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \theta$$\end{document}. (c) Corresponding trajectories of the single particles forming the dumbbell particles demonstrate the complex particle motion during recoil. The horizontal dotted line is along the direction of shear.](https://www.researchgate.net/publication/374719674/figure/fig1/AS:11431281205488819@1700286810731/a-Schematic-drawing-of-the-experimental-setup-An-oscillating-mirror-mounted-onto-a_Q320.jpg)
![Temporal evolution of angle θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta$$\end{document} made by the axis of the dumbbell with recoil direction for different recoil runs (colored lines) and the average curve (thick black line) for θ0∼40∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0\sim 40^\circ$$\end{document}, v=0.3μm/s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v = {0.3}\,{\upmu \hbox {m}/\hbox {s}}$$\end{document} and tsh=50s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_{\textrm{sh}} = {50}\,\hbox {s}$$\end{document}. The angle made by the dumbbell after release (t>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t>0$$\end{document}) deviates significantly from the initial angle θ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0$$\end{document} at t<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t<0$$\end{document}. The black dashed line denotes the standard deviation in the spread of the trajectories.](https://www.researchgate.net/publication/374719674/figure/fig2/AS:11431281205488820@1700286810864/Temporal-evolution-of-angle-thdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
![(a) Amplitudes of MIA for fixed shear time tsh=50s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_{\textrm{sh}}={50}\,\hbox {s}$$\end{document} and two different shear velocities, v=0.2μm/s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v={0.2}\,{\upmu \hbox {m}/\hbox {s}}$$\end{document} (dark blue) and v=0.3μm/s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v={0.3}\,{\upmu \hbox {m}/\hbox {s}}$$\end{document} (light blue), as a function of initial angle θ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0$$\end{document}. Open symbols correspond to experimental data, the solid line shows simulation results of the model introduced in Fig. 4b. The curves show a maximum around θ0=45∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0=45^\circ$$\end{document} and decrease to 0 towards θ0=0∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0=0^\circ$$\end{document} and 90∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$90^\circ$$\end{document}. (b) and (c) Show individual recoils (colored lines) and their mean (thick black curve) for θ0=0∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0=0^\circ$$\end{document} and 90∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$90^\circ$$\end{document}, for fixed shear time tsh=50s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_{\textrm{sh}}={50}\,\hbox {s}$$\end{document} and shear velocity v=0.3μm/s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v={0.3}\,{\upmu \hbox {m}/\hbox {s}}$$\end{document}. Even though the mean curve remains constant in both cases, the individual trajectories show a huge spread at θ0=90∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0 = 90^\circ$$\end{document} compared to θ0=0∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0 = 0^\circ$$\end{document} signaling an instability when the dumbbell axis is perpendicular to the recoil axis. (d) Sketch of a director (top) and a vector (bottom). While driving left or right is equivalent for a director (green arrows), this is not the case for a vector (green and red arrows lead to different scenarios).](https://www.researchgate.net/publication/374719674/figure/fig3/AS:11431281205488821@1700286811007/a-Amplitudes-of-MIA-for-fixed-shear-time-tsh50sdocumentclass12ptminimal_Q320.jpg)
![(a) Amplitudes of MIA and translational (inset) recoils as a function of shear velocity v for experiments (open symbols) and simulations (solid line) with fixed initial angle θ0∼40∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0 {{\sim }} 40^\circ$$\end{document} and shear time tsh=50s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_{\textrm{sh}}={50}\,\hbox {s}$$\end{document}. While MIA increases quadratically with v, translational recoil is linear in v. (b) Sketch of the microscopic model: The colloidal dumbbell is modeled as a rod of length l (gray), coupled to a bath rod (green) of length lb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_b$$\end{document} via a nonlinear spring.](https://www.researchgate.net/publication/374719674/figure/fig4/AS:11431281205488822@1700286811149/a-Amplitudes-of-MIA-and-translational-inset-recoils-as-a-function-of-shear-velocity-v_Q320.jpg)
![Amplitudes of MIA and translational (inset) recoils for fixed initial angle θ0∼40∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0 {{\sim }} 40^\circ$$\end{document} and shear velocity v=0.3μm/s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v={0.3}\,{\upmu \hbox {m}/\hbox {s}}$$\end{document} as a function of shear time tsh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_{\textrm{sh}}$$\end{document}. Open symbols correspond to experimental data and solid lines to simulations. For short times, the MIA amplitude scales ∝tsh2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\propto t_{\textrm{sh}}^2$$\end{document}, while the translational recoil scales ∝tsh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\propto t_{\textrm{sh}}$$\end{document}.](https://www.researchgate.net/publication/374719674/figure/fig5/AS:11431281205451374@1700286811263/Amplitudes-of-MIA-and-translational-inset-recoils-for-fixed-initial-angle_Q320.jpg)
