Metastability of Discrete-Symmetry Flocks.

We study the stability of the ordered phase of flocking models with a scalar order parameter. Using both the active Ising model and a hydrodynamic description, we show that droplets of particles moving in the direction opposite to that of the ordered phase nucleate and grow. We characterize analytically this self-similar growth and demonstrate that droplets spread ballistically in all directions. Our results imply that, in the thermodynamic limit, discrete-symmetry flocks-and, by extension, continuous-symmetry flocks with rotational anisotropy-are metastable in all dimensions.

Flocking in one dimension: Asters and reversals.

We study the one-dimensional active Ising model in which aligning particles undergo diffusion biased by the signs of their spins. The phase diagram obtained varying the density of particles, their hopping rate, and the temperature controlling the alignment shows a homogeneous disordered phase but no homogeneous ordered one, as well as two phases with localized dense structures. In the flocking phase, large ordered aggregates move ballistically and stochastically reverse their direction of motion. In what we termed the "aster" phase, dense immobile aggregates of opposite magnetization face each other, exchanging particles, without any net motion of the aggregates. Using a combination of numerical simulations and mean-field theory, we study the evolution of the shapes of the flocks, the statistics of their reversal times, and their coarsening dynamics. Solving exactly for the zero-temperature dynamics of an aster allows us to understand their coarsening, which shows extremal dynamics, while mean-field equations account for their shape.

Susceptibility of Polar Flocks to Spatial Anisotropy.

We study the effect of spatial anisotropy on polar flocks by investigating active q-state clock models in two dimensions. In contrast to the equilibrium case, we find that any amount of anisotropy is asymptotically relevant, drastically altering the phenomenology from that of the rotationally invariant case. All of the well-known physics of the Vicsek model, from giant density fluctuations to microphase separation, is replaced by that of the active Ising model, with short-range correlations and complete phase separation. These changes appear beyond a length scale that diverges in the q→∞ limit, so that the Vicsek-model phenomenology is observed in finite systems for weak enough anisotropy, i.e., sufficiently high q. We provide a scaling argument which explains why anisotropy has such different effects in the passive and active cases.

Dynamic clustering of passive colloids in dense suspensions of motile bacteria.

Mixtures of active and passive particles are predicted to exhibit a variety of nonequilibrium phases. Here we report a dynamic clustering phase in mixtures of colloids and motile bacteria. We show that colloidal clustering results from a balance between bond breaking due to persistent active motion and bond stabilization due to torques that align active particle velocity tangentially to the passive particle surface. Furthermore, dynamic clustering spans a broad regime between diffusivity-based and motility-induced phase separation that subsumes typical bacterial motility parameters.

Metastability of Constant-Density Flocks.

We study numerically the Toner-Tu field theory where the density field is maintained constant, a limit case of "Malthusian" flocks for which the asymptotic scaling of correlation functions in the ordered phase is known exactly. While we confirm these scaling laws, we also show that such constant-density flocks are metastable to the nucleation of a specific defect configuration, and are replaced by a globally disordered phase consisting of asters surrounded by shock lines that constantly evolves and remodels itself. We demonstrate that the main source of disorder lies along shock lines, rendering this active foam fundamentally different from the corresponding equilibrium system. We thus show that in the context of active matter also, a result obtained at all orders of perturbation theory can be superseded by nonperturbative effects, calling for a different approach.

Fluctuation-Induced Phase Separation in Metric and Topological Models of Collective Motion.

We study the role of noise on the nature of the transition to collective motion in dry active matter. Starting from field theories that predict a continuous transition at the deterministic level, we show that fluctuations induce a density-dependent shift of the onset of order, which in turn changes the nature of the transition into a phase-separation scenario. Our results apply to a range of systems, including models in which particles interact with their "topological" neighbors that have been believed so far to exhibit a continuous onset of order. Our analytical predictions are confirmed by numerical simulations of fluctuating hydrodynamics and microscopic models.

Self-Organized Critical Coexistence Phase in Repulsive Active Particles.

We revisit motility-induced phase separation in two models of active particles interacting by pairwise repulsion and uncover new qualitative features: the resulting dense phase contains gas bubbles distributed algebraically up to a typically extremely large cutoff scale. At large enough system size and/or global density, all the gas may be contained inside the bubbles, at which point the system is microphase separated with a finite cutoff bubble scale. We further observe that the ordering is clearly anomalous, with different dynamics for the coarsening of the dense phase and of the gas bubbles. This self-organized critical phenomenology is reproduced by a "reduced bubble model" that implements the basic idea of reverse Ostwald ripening put forward in Tjhung et al. [Phys. Rev. X 8, 031080 (2018)PRXHAE2160-330810.1103/PhysRevX.8.031080].

The 2020 motile active matter roadmap.

Activity and autonomous motion are fundamental in living and engineering systems. This has stimulated the new field of 'active matter' in recent years, which focuses on the physical aspects of propulsion mechanisms, and on motility-induced emergent collective behavior of a larger number of identical agents. The scale of agents ranges from nanomotors and microswimmers, to cells, fish, birds, and people. Inspired by biological microswimmers, various designs of autonomous synthetic nano- and micromachines have been proposed. Such machines provide the basis for multifunctional, highly responsive, intelligent (artificial) active materials, which exhibit emergent behavior and the ability to perform tasks in response to external stimuli. A major challenge for understanding and designing active matter is their inherent nonequilibrium nature due to persistent energy consumption, which invalidates equilibrium concepts such as free energy, detailed balance, and time-reversal symmetry. Unraveling, predicting, and controlling the behavior of active matter is a truly interdisciplinary endeavor at the interface of biology, chemistry, ecology, engineering, mathematics, and physics. The vast complexity of phenomena and mechanisms involved in the self-organization and dynamics of motile active matter comprises a major challenge. Hence, to advance, and eventually reach a comprehensive understanding, this important research area requires a concerted, synergetic approach of the various disciplines. The 2020 motile active matter roadmap of Journal of Physics: Condensed Matter addresses the current state of the art of the field and provides guidance for both students as well as established scientists in their efforts to advance this fascinating area.

Ramifications of disorder on active particles in one dimension.

The effects of quenched disorder on a single and many active run-and-tumble particles are studied in one dimension. For a single particle, we consider both the steady-state distribution and the particle's dynamics subject to disorder in three parameters: a bounded external potential, the particle's speed, and its tumbling rate. We show that in the case of a disordered potential, the behavior is like an equilibrium particle diffusing on a random force landscape, implying a dynamics that is logarithmically slow in time. In the situations of disorder in the speed or tumbling rate, we find that the particle generically exhibits diffusive motion, although particular choices of the disorder may lead to anomalous diffusion. Based on the single-particle results, we find that in a system with many interacting particles, disorder in the potential leads to strong clustering. We characterize the clustering in two different regimes depending on the system size and show that the mean cluster size scales with the system size, in contrast to nondisordered systems.

Response of active Brownian particles to shear flow.

We study the linear response of interacting active Brownian particles in an external potential to simple shear flow. Using a path integral approach, we derive the linear response of any state observable to initiating shear in terms of correlation functions evaluated in the unperturbed system. For systems and observables which are symmetric under exchange of the x and y coordinates, the response formula can be drastically simplified to a form containing only state variables in the corresponding correlation functions (compared to the generic formula containing also time derivatives). In general, the shear couples to the particles by translational as well as rotational advection, but in the aforementioned case of xy symmetry, only translational advection is relevant in the linear regime. We apply the response formulas analytically in solvable cases and numerically in a specific setup. In particular, we investigate the effect of a shear flow on the morphology and the stress of N confined active particles in interaction, where we find that the activity as well as additional alignment interactions generally increase the response.

Nonequilibrium forces following quenches in active and thermal matter.

Nonequilibrium systems with conserved quantities like density or momentum are known to exhibit long-ranged correlations. This, in turn, leads to long-ranged fluctuation-induced (Casimir) forces, predicted to arise in a variety of nonequilibrium settings. Here, we study such forces, which arise transiently between parallel plates or compact inclusions in a gas of particles, following a change ("quench") in temperature or activity of the medium. Analytical calculations, as well as numerical simulations of passive or active Brownian particles, indicate two distinct forces: (i) The immediate effect of the quench is adsorption or desorption of particles of the medium to the immersed objects, which in turn initiates a front of relaxing (mean) density. This leads to time-dependent density-induced forces. (ii) A long-term effect of the quench is that density fluctuations are modified, manifested as transient (long-ranged) (pair-)correlations that relax diffusively to their (short-ranged) steady-state limit. As a result, transient fluctuation-induced forces emerge. We discuss the properties of fluctuation-induced and density-induced forces as regards universality, relaxation as a function of time, and scaling with distance between objects. Their distinct signatures allow us to distinguish the two types of forces in simulation data. Our simulations also show that a quench of the effective temperature of an active medium gives rise to qualitatively similar effects to a temperature quench in a passive medium. Based on this insight, we propose several scenarios for the experimental observation of the forces described here.

Phase Transition in Protocols Minimizing Work Fluctuations.

For two canonical examples of driven mesoscopic systems-a harmonically trapped Brownian particle and a quantum dot-we numerically determine the finite-time protocols that optimize the compromise between the standard deviation and the mean of the dissipated work. In the case of the oscillator, we observe a collection of protocols that smoothly trade off between average work and its fluctuations. However, for the quantum dot, we find that as we shift the weight of our optimization objective from average work to work standard deviation, there is an analog of a first-order phase transition in protocol space: two distinct protocols exchange global optimality with mixed protocols akin to phase coexistence. As a result, the two types of protocols possess qualitatively different properties and remain distinct even in the infinite duration limit: optimal-work-fluctuation protocols never coalesce with the minimal-work protocols, which therefore never become quasistatic.

Generalized thermodynamics of phase equilibria in scalar active matter.

Motility-induced phase separation (MIPS) arises generically in fluids of self-propelled particles when interactions lead to a kinetic slowdown at high densities. Starting from a continuum description of scalar active matter akin to a generalized Cahn-Hilliard equation, we give a general prescription for the mean densities of coexisting phases in flux-free steady states that amounts, at a hydrodynamics scale, to extremizing an effective free energy. We illustrate our approach on two well-known models: self-propelled particles interacting either through a density-dependent propulsion speed or via direct pairwise forces. Our theory accounts quantitatively for their phase diagrams, providing a unified description of MIPS.

Stresses in non-equilibrium fluids: Exact formulation and coarse-grained theory.

Starting from the stochastic equation for the density operator, we formulate the exact (instantaneous) stress tensor for interacting Brownian particles and show that its average value agrees with expressions derived previously. We analyze the relation between the stress tensor and forces due to external potentials and observe that, out of equilibrium, particle currents give rise to extra forces. Next, we derive the stress tensor for a Landau-Ginzburg theory in generic, non-equilibrium situations, finding an expression analogous to that of the exact microscopic stress tensor, and discuss the computation of out-of-equilibrium (classical) Casimir forces. Subsequently, we give a general form for the stress tensor which is valid for a large variety of energy functionals and which reproduces the two mentioned cases. We then use these relations to study the spatio-temporal correlations of the stress tensor in a Brownian fluid, which we compute to leading order in the interaction potential strength. We observe that, after integration over time, the spatial correlations generally decay as power laws in space. These are expected to be of importance for driven confined systems. We also show that divergence-free parts of the stress tensor do not contribute to the Green-Kubo relation for the viscosity.

Generic Long-Range Interactions Between Passive Bodies in an Active Fluid.

A single nonspherical body placed in an active fluid generates currents via breaking of time-reversal symmetry. We show that, when two or more passive bodies are placed in an active fluid, these currents lead to long-range interactions. Using a multipole expansion, we characterize their leading-order behaviors in terms of single-body properties and show that they decay as a power law with the distance between the bodies, are anisotropic, and do not obey an action-reaction principle. The interactions lead to rich dynamics of the bodies, illustrated by the spontaneous synchronized rotation of pinned nonchiral bodies and the formation of traveling bound pairs. The occurrence of these phenomena depends on tunable properties of the bodies, thus opening new possibilities for self-assembly mediated by active fluids.

Active Particles with Soft and Curved Walls: Equation of State, Ratchets, and Instabilities.

We study, from first principles, the pressure exerted by an active fluid of spherical particles on general boundaries in two dimensions. We show that, despite the nonuniform pressure along curved walls, an equation of state is recovered upon a proper spatial averaging. This holds even in the presence of pairwise interactions between particles or when asymmetric walls induce ratchet currents, which are accompanied by spontaneous shear stresses on the walls. For flexible obstacles, the pressure inhomogeneities lead to a modulational instability as well as to the spontaneous motion of short semiflexible filaments. Finally, we relate the force exerted on objects immersed in active baths to the particle flux they generate around them.

Pressure and phase equilibria in interacting active brownian spheres.

We derive a microscopic expression for the mechanical pressure P in a system of spherical active Brownian particles at density ρ. Our exact result relates P, defined as the force per unit area on a bounding wall, to bulk correlation functions evaluated far away from the wall. It shows that (i) P(ρ) is a state function, independent of the particle-wall interaction; (ii) interactions contribute two terms to P, one encoding the slow-down that drives motility-induced phase separation, and the other a direct contribution well known for passive systems; and (iii) P is equal in coexisting phases. We discuss the consequences of these results for the motility-induced phase separation of active Brownian particles and show that the densities at coexistence do not satisfy a Maxwell construction on P.

From phase to microphase separation in flocking models: the essential role of nonequilibrium fluctuations.

We show that the flocking transition in the Vicsek model is best understood as a liquid-gas transition, rather than an order-disorder one. The full phase separation observed in flocking models with Z(2) rotational symmetry is, however, replaced by a microphase separation leading to a smectic arrangement of traveling ordered bands. Remarkably, continuous deterministic descriptions do not account for this difference, which is only recovered at the fluctuating hydrodynamics level. Scalar and vectorial order parameters indeed produce different types of number fluctuations, which we show to be essential in selecting the inhomogeneous patterns. This highlights an unexpected role of fluctuations in the selection of flock shapes.

Pattern formation in flocking models: A hydrodynamic description.

We study in detail the hydrodynamic theories describing the transition to collective motion in polar active matter, exemplified by the Vicsek and active Ising models. Using a simple phenomenological theory, we show the existence of an infinity of propagative solutions, describing both phase and microphase separation, that we fully characterize. We also show that the same results hold specifically in the hydrodynamic equations derived in the literature for the active Ising model and for a simplified version of the Vicsek model. We then study numerically the linear stability of these solutions. We show that stable ones constitute only a small fraction of them, which, however, includes all existing types. We further argue that, in practice, a coarsening mechanism leads towards phase-separated solutions. Finally, we construct the phase diagrams of the hydrodynamic equations proposed to qualitatively describe the Vicsek and active Ising models and connect our results to the phenomenology of the corresponding microscopic models.

Emergent spatial structures in flocking models: a dynamical system insight.

We show that hydrodynamic theories of polar active matter generically possess inhomogeneous traveling solutions. We introduce a unifying dynamical-system framework to establish the shape of these intrinsically nonlinear patterns, and show that they correspond to those hitherto observed in experiments and numerical simulation: periodic density waves, and solitonic bands, or polar-liquid droplets both cruising in isotropic phases. We elucidate their respective multiplicity and mutual relations, as well as their existence domain.