Strong density fluctuations in active particles with local alignment interactions

1. Strong density fluctuations in active particles with local alignment interactions Hugues Chaté Francesco Ginelli Fernando Peruani Shradha Mishra Sriram Ramaswamy Please request permission to use any of this material to: hugues.chate@cea.fr Thank you!

2. Collective motion at all scales • From the largest mammals to bacteria, and even within the cell ..collective motion in the presence of noise/fluctuations/turbulence • Large groups without leaders, without ordering field, without global interaction • Underlying universal properties?

3. One of the best examples: starling flocks at twilight

4. Starling flocks in Rome…

5. Starling flocks in Rome • the Starflag project • understanding how, not why • confronting 3D data to predictions of simple models (to start) • beyond birds, general properties of a fluid of active, self-propelled particles?

6. B(ird)oids: what most models do Alignment Attraction-repulsion ...and no surrounding fluid…

7. Here:Minimal microscopic models with no fluid nor cohesion (think of shaken anisotropic granular particles) • Minimality: best framework to capture universal features, to increase numerical efficiency, and perhaps ease analytical approaches • Microscopic level: generic fluctuations included, full nonlinear character, do not rely on large-scale approximation or symmetry argument • NB: no consensus on macroscopic or mesoscopic descriptions

8. Absolutely minimal: Vicsek-style models • point particles move off-lattice in driven-overdamped dynamics: • fixed velocity , no inertia, parallel updating at discrete timesteps • strictly local interaction range • alignment according to local order parameter in neighborhood • noise source: random angle or random force

9. More explicitly, in 2D: Calculation of new orientation with angular noise (operator Θ returns direction/axis of order parameter) Interactions: polar or apolar (k particles in neighborhood of j particle) Streaming: polar or apolar

10. 3 interesting cases: • polar case: (original Vicsek model) • apolar case: • mixed case:

11. Phase diagram in density/noise parameter plane • zero noise: perfect order (if finite density ρ) • strong noise: perfect random walks • transition for sure, but at finite noise level σ? Yes! • transition line in (ρ,σ)plane: for polar case at low density: for nematic case

12. Main results • polar case: • transition to collective motion is discontinuous • fast domain growth leading to high-density/high order solitary bands/sheets (2D/3D), then giant density fluctuations • apolar case • KT transition to quasi-long-range nematic order (2D) • slow domain growth leading to high-density/high order macroscopic cluster with giant density fluctuations • mixed case (in progress) • discontinuous transition to true long-range nematic order • segregation to large cluster with giant density fluctuations

13. Part I: Polar particles with polar interactions (Vicsek model)

14. Discontinuous transition to collective motion at large enough size, discontinuous variation of order parameter

15. 3D polar particles without cohesion:discontinuous transition Near threshold, at moderate sizes: flip-flop dynamics of order parameter leading to bimodal distribution at large enough size, discontinuous transition order parameter z y x total noise strength time

16. Ordered phase: fast domain growth Quench into ordered phase (coarse-grained density field) L=16384, ρ=1/8 (32M boids)

17. Ordered phase: fast domain growth Hydrodynamic, Model H-like growth: ξ~t Unusual correlation fonctions with apparent algebraic decay Linear growth of lengthscale extracted from exponential tail of two-point correlation function of coarse-grained density field

18. 2D ordered state in a finite box: Traveling high-density high-order solitary band(s) coarse-grained density field density and order parameter profiles

19. 3D ordered state in a finite box: Traveling high-density high-order solitary sheet(s) density profiles color code: local order

20. 2D: starting from ordered, homogenous-density, configuration much later… short times… time

21. starting from ordered, homogenous-density, configuration: instability of trivial solution conclusion: not a wave train, but solitary structures late configuration atypical growth late spectrum early spectrum

22. Bands disappear at low noise,leaving anomalous density fluctuations Band-train profile widths Weaker bands: typical profiles No band region: typical profiles No band region: « giant » density fluctuations

23. Part II: Apolar particles with nematic interactions

24. 2D: Kosterlitz-Thouless transition to QLROorder parameter scaling • order parameter curves at various sizes do not cross each other • power law decay of order parameter with system size in (quasi-)ordered phase • crossover to normal decay (slope -1/2) in disordered phase • variation of exponent with noise strength; at estimated threshold, expected equilibrium value

25. “Normal” phase ordering: single lengthscale coarse-grained density L=256, 131072 particles growth of density and orientation lengthscale

26. Deviations from Porod’s law: short-distance cusp " fluctuations-dominated coarsening " C(r) with b~0.5

27. Highly segregated yet fluctuating ordered phase • Time series of scalar order parameter (note the time scale) • Typical state • During a global rearrangement • Another typical state

28. Giant density fluctuations in 2D In (quasi-) ordered phase, giant density fluctuations: rms Δn scales like n (in 2D)

29. Recent experiment: vibrated rods Vijay Narayan, Narayanan Menon and Sriram Ramaswamy

30. Part III: polar particles with nematic interactions

31. True long range nematic order and discontinuous transition No polar order, isotropic-nematic transition OP vs noise at different sizes Time series of OP near transition

32. True long range nematic order and discontinuous transition algebraic dependence of critical noise with densitydiscontinuous transition for all densities?

33. Asymptotic state: large, macroscopic, band L=512 L=256

34. Asymptotic state: large, macroscopic, band First:coarsening to nematic orderwith colliding polar packetsSecond:emergence of single macroscopic band coarse-grained density

35. Asymptotic state: giant density fluctuations

36. Part IV: In progress: beyond numerics

37. A mesoscopic description derived from the microscopics(here pure nematic case) order parameter/density coupling (includes non-equilibrium current) multiplicative and conserved noise giant number fluctuations predicted

38. without either of them, no giant density fluctuations without right noise, no segregation Both terms are necessary for a faithful description:

39. Summary/conclusions/perspectives • nature of ordered phase • true LRO in polar and mixed case, QLRO in nematic case (2D) • strong segregation between high-density/high order and low-density/low order and/or giant density fluctuations • connection with condensation/ZRP ? • order of transition • discontinuous in polar and mixed case, continuous in nematic case • mesoscopic description (in progress) • better numerics for low-density regions, analytically? • possibility of deterministic description?