Self-Propelled Particles with Velocity Reversals and Ferromagnetic Alignment: Active Matter Class with Second-Order Transition to Quasi-Long-Range Polar Order

Kinetic Theory of Motility Induced Phase Separation for Active Brownian Particles

When two active Brownian particles collide, they slide along each other until they can continue their free motion. For persistence lengths much larger than the particle diameter, the directors do not change, but the collision can be modeled as producing a net displacement on the particles compared to their free motion in the absence of the encounter. With these elements, a Boltzmann-Enskog-like kinetic theory is built. A linear stability analysis of the homogeneous state predicts a density instability resulting from the effective velocity reduction of tagged particles predicted by the theory.

Logarithmic or algebraic: Roughening of an active Kardar-Parisi-Zhang surface

The Kardar-Parisi-Zhang (KPZ) equation sets the universality class for growing and roughening of nonequilibrium surfaces without any conservation law and nonlocal effects. We argue here that the KPZ equation can be generalized by including a symmetry-permitted nonlocal nonlinear term of active origin that is of the same order as the one included in the KPZ equation. Including this term, the 2D active KPZ equation is stable in some parameter regimes, in which the interface conformation fluctuations exhibit sublogarithmic or superlogarithmic roughness, with nonuniversal exponents, giving positional generalized quasi-long-ranged order. For other parameter choices, the model is unstable, suggesting a perturbatively inaccessible algebraically rough interface or positional short-ranged order. Our model should serve as a paradigmatic nonlocal growth equation.

Active XY model on a substrate: Density fluctuations and phase ordering

We explore the generic long-wavelength properties of an active XY model on a substrate, consisting of a collection of nearly phase-ordered active XY spins in contact with a diffusing, conserved species, as a representative system of active spinners with a conservation law. The spins rotate actively in response to the local density fluctuations and local phase differences, on a solid substrate. We investigate this system by Monte Carlo simulations of an agent-based model, which we set up, complemented by the hydrodynamic theory for the system. We demonstrate that this system can phase-synchronize without any hydrodynamic interactions. Our combined numerical and analytical studies show that this model, when stable, displays hitherto unstudied scaling behavior: As a consequence of the interplay between the mobility, active rotation, and number conservation, such a system can be stable over a wide range of the model parameters characterized by a novel correspondence between the phase and density fluctuations. In different regions of the phase space where the phase-ordered system is stable, it displays generalized quasi-long-range order (QLRO): It shows phase ordering which is generically either logarithmically stronger than the conventional QLRO found in its equilibrium limit, together with "miniscule number fluctuations," or logarithmically weaker than QLRO along with "giant number fluctuations," showing a novel one-to-one correspondence between phase ordering and density fluctuations in the ordered states. Intriguingly, these scaling exponents are found to depend explicitly on the model parameters. We further show that in other parameter regimes there are no stable, ordered phases. Instead, two distinct types of disordered states with short-range phase order are found, characterized by the presence or absence of stable clusters of finite sizes. In a surprising connection, the hydrodynamic theory for this model also describes the fluctuations in a Kardar-Parisi-Zhang (KPZ) surface with a conserved species on it, or an active fluid membrane with a finite tension, without momentum conservation and a conserved species living on it. This implies the existence of stable fluctuating surfaces that are only logarithmically smoother or rougher than the Edward-Wilkinson surface at two dimensions (2D) can exist, in contrast to the 2D pure KPZ-like "rough" surfaces.

Defect dynamics in active polar fluids vs. active nematics

Topological defects play a key role in two-dimensional active nematics, and a transient role in two-dimensional active polar fluids. Using a variational method, we study both the transient and long-time behavior of defects in two-dimensional active polar fluids in the limit of strong order and overdamped, compressible flow, and compare the defect dynamics with the corresponding active nematics model studied recently. One result is non-central interactions between defect pairs for active polar fluids, and by extending our analysis to allow orientation dynamics of defects, we find that the orientation of +1 defects, unlike that of ±1/2 defects in active nematics, is not locked to defect positions and relaxes to asters. Moreover, using a scaling argument, we explain the transient feature of active polar defects and show that in the steady state, active polar fluids are either devoid of defects or consist of a single aster. We argue that for contractile (extensile) active nematic systems, +1 vortices (asters) should emerge as bound states of a pair of +1/2 defects, which has been recently observed. Moreover, unlike the polar case, we show that for active nematics, a linear chain of equally spaced bound states of pairs of +1/2 defects can screen the activity term. A common feature in both models is the appearance of +1 defects (elementary in polar and composite in nematic) in the steady state.

Long-Range Nematic Order in Two-Dimensional Active Matter

Working in two space dimensions, we show that the orientational order emerging from self-propelled polar particles aligning nematically is quasi-long-ranged beyond ℓ_{r}, the scale associated to induced velocity reversals, which is typically extremely large and often cannot even be measured. Below ℓ_{r}, nematic order is long-range. We construct and study a hydrodynamic theory for this de facto phase and show that its structure and symmetries differ from conventional descriptions of active nematics. We check numerically our theoretical predictions, in particular the presence of π-symmetric propagative sound modes, and provide estimates of all scaling exponents governing long-range space-time correlations.

Universality class for a nonequilibrium state of matter: A d=4-ε expansion study of Malthusian flocks

We show that "Malthusian flocks"-i.e., coherently moving collections of self-propelled entities (such as living creatures) which are being "born" and "dying" during their motion-belong to a new universality class in spatial dimensions d>2. We calculate the universal exponents and scaling laws of this new universality class to O(ε) in a d=4-ε expansion and find these are different from the "canonical" exponents previously conjectured to hold for "immortal" flocks (i.e., those without birth and death) and shown to hold for incompressible flocks with spatial dimensions in the range of 2<d≤4. We also obtain a universal amplitude ratio relating the damping of transverse and longitudinal velocity and density fluctuations in these systems. Furthermore, we find a universal separatrix in real space (r) between two regions in which the equal-time density correlation 〈δρ(r,t)δρ(0,t)〉 has opposite signs. Our expansion should be quite accurate in d=3, allowing precise quantitative comparisons between our theory, simulations, and experiments.

Spontaneous Velocity Alignment in Motility-Induced Phase Separation

We study a system of purely repulsive spherical self-propelled particles in the minimal setup inducing motility-induced phase separation (MIPS). We show that, even if explicit alignment interactions are absent, a growing order in the velocities of the clustered particles accompanies MIPS. Particles arrange into aligned or vortexlike domains whose size increases as the persistence of the self-propulsion grows, an effect that is quantified studying the spatial correlation function of the velocities. We explain the velocity alignment by unveiling a hidden alignment interaction of the Vicsek-like form, induced by the interplay between steric interactions and self-propulsion.

Two-dimensional active motion

The diffusion in two dimensions of noninteracting active particles that follow an arbitrary motility pattern is considered for analysis. A Fokker-Planck-like equation is generalized to take into account an arbitrary distribution of scattered angles of the swimming direction, which encompasses the pattern of active motion of particles that move at constant speed. An exact analytical expression for the marginal probability density of finding a particle on a given position at a given instant, independently of its direction of motion, is provided, and a connection with a generalized diffusion equation is unveiled. Exact analytical expressions for the time dependence of the mean-square displacement and of the kurtosis of the distribution of the particle positions are presented. The analysis is focused in the intermediate-time regime, where the effects of the specific pattern of active motion are conspicuous. For this, it is shown that only the expectation value of the first two harmonics of the scattering angle of the direction of motion are needed. The effects of persistence and of circular motion are discussed for different families of distributions of the scattered direction of motion.

Optimizing active work: Dynamical phase transitions, collective motion, and jamming

Active work measures how far the local self-forcing of active particles translates into real motion. Using population Monte Carlo methods, we investigate large deviations in the active work for repulsive active Brownian disks. Minimizing the active work generically results in dynamical arrest; in contrast, despite the lack of aligning interactions, trajectories of high active work correspond to a collectively moving, aligned state. We use heuristic and analytic arguments to explain the origin of dynamical phase transitions separating the arrested, typical, and aligned regimes.