Coarsening in the two-dimensional incompressible Toner-Tu equation: Signatures of turbulence

Coarsening dynamics of aster defects in model polar active matter

We numerically study the dynamics of topological defects in 2D polar active matter coupled to a conserved density field, which shows anomalous kinetics and defect distribution. The initial many-defect state relaxes by pair-annihilation of defects, which behave like Ostwald ripening on short timescales. However, defect coarsening is arrested at long timescales, and the relaxation kinetics becomes anomalously slow compared to the equilibrium state. Specifically, the number of defects in the active system approaches a steady state, following a power-law dependence in the rate of change of the inverse density. In contrast, in thermal equilibrium, the decay is exponential. Finally, we show that the anomalous coarsening of defects leads to unique patterns in the coupled density field, which is consistent with patterns observed in experiments on the actin cytoskeleton. These patterns can act as cell signaling platforms and may have important biological consequences.

Spectral energy transfers in domain growth problems

In the domain growth process, small structures gradually vanish, leaving behind larger ones. We investigate spectral energy transfers in two standard models for domain growth: (a) the Cahn-Hilliard (CH) equation with conserved dynamics and (b) the time-dependent Ginzburg-Landau (TDGL) equation with nonconserved dynamics. The nonlinear terms in these equations dissipate fluctuations and facilitate energy transfers among Fourier modes. In the TDGL equation, only the ϕ(k=0,t) mode survives, and the order parameter ϕ(r,t) approaches a uniform state with ϕ=+1 or -1. On the other hand, there is no dynamics of the ϕ(k=0,t) mode in the CH equation due to the conservation law, highlighting the different dynamics of these equations.

Defect turbulence in a dense suspension of polar, active swimmers

We study the effects of inertia in dense suspensions of polar swimmers. The hydrodynamic velocity field and the polar order parameter field describe the dynamics of the suspension. We show that a dimensionless parameter R (ratio of the swimmer self-advection speed to the active stress invasion speed [Phys. Rev. X 11, 031063 (2021)2160-330810.1103/PhysRevX.11.031063]) controls the stability of an ordered swimmer suspension. For R smaller than a threshold R_{1}, perturbations grow at a rate proportional to their wave number q. Beyond R_{1} we show that the growth rate is O(q^{2}) until a second threshold R=R_{2} is reached. The suspension is stable for R>R_{2}. We perform direct numerical simulations to characterize the steady-state properties and observe defect turbulence for R

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Affiliation(s)

Navdeep Rana Max Planck Institute for Dynamics and Self-Organization (MPIDS), D-37077 Göttingen, Germany
Rayan Chatterjee Stanford Medicine, Stanford University, Stanford, California 94305, USA
Sunghan Ro Department of Physics, Technion-Israel Institute of Technology, Haifa 3200003, Israel
Dov Levine Department of Physics, Technion-Israel Institute of Technology, Haifa 3200003, Israel
Sriram Ramaswamy Department of Physics, Indian Institute of Science, Bengaluru 560 012, India
Prasad Perlekar Tata Institute of Fundamental Research, Hyderabad 500046, India

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Flocking turbulence of microswimmers in confined domains

We extensively study the Toner-Tu-Swift-Hohenberg model of motile active matter by means of direct numerical simulations in a two-dimensional confined domain. By exploring the space of parameters of the model we investigate the emergence of a new state of active turbulence which occurs when the aligning interactions and the self-propulsion of the swimmers are strong. This regime of flocking turbulence is characterized by a population of few strong vortices, each surrounded by an island of coherent flocking motion. The energy spectrum of flocking turbulence displays a power-law scaling with an exponent which depends weakly on the model parameters. By increasing the confinement we observe that the system, after a long transient characterized by power-law-distributed transition times, switches to the ordered state of a single giant vortex.

Nonlinear energy dissipation and transfers in coarsening systems

During coarsening, small structures disappear, leaving behind only large ones. Here we study the spectral energy transfers in Model A, where the order parameter ϕ evolves via nonconserved dynamics. We show that the nonlinear interactions dissipate fluctuations and facilitate energy transfers among the Fourier modes so that only ϕ(k=0), where k is the wave number, survives at the end and approaches the asymptotic value +1 or -1. We contrast the coarsening evolution for the initial conditions with 〈ϕ(x,t=0)〉=0 and with uniformly positive or negative ϕ(x,t=0).

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.

Phase ordering, topological defects, and turbulence in the three-dimensional incompressible Toner-Tu equation

We investigate the phase-ordering dynamics of the incompressible Toner-Tu equation in three dimensions. We show that the phase ordering proceeds via defect merger events and the dynamics is controlled by the Reynolds number Re. At low Re, the dynamics is similar to that of the Ginzburg-Landau equation. At high Re, turbulence controls phase ordering. In particular, we observe a forward energy cascade from the coarsening length scale to the dissipation scale, clustering of defects, and multiscaling in velocity correlations.