Strong density fluctuations in active particles with local alignment interactionsHugues Chat

Francesco GinelliFernando PeruaniShradha MishraSriram Ramaswamy

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Collective motion at all scalesFrom the largest mammals to bacteria, and even within the cell ..collective motion in the presence of noise/fluctuations/turbulenceLarge groups without leaders, without ordering field, without global interactionUnderlying universal properties?

One of the best examples: starling flocks at twilight

Starling flocks in Rome

Starling flocks in Romethe Starflag projectunderstanding how, not whyconfronting 3D data to predictions of simple models (to start)beyond birds, general properties of a fluid of active, self-propelled particles?

B(ird)oids: what most models doAlignmentAttraction-repulsion...and no surrounding fluid

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

Absolutely minimal: Vicsek-style modelspoint 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 neighborhoodnoise source: random angle or random force

More explicitly, in 2D:(operator returns direction/axis of order parameter)Calculation of new orientation with angular noiseInteractions: polar or apolarStreaming: polar or apolar(k particles in neighborhood of j particle)

3 interesting cases:polar case:(original Vicsek model)

apolar case:

mixed case:

Phase diagram in density/noise parameter planezero noise: perfect order (if finite density )strong noise: perfect random walkstransition for sure, but at finite noise level ? Yes!transition line in (,) plane:

for polar casefor nematic caseat low density:

Main resultspolar case:transition to collective motion is discontinuousfast domain growth leading to high-density/high order solitary bands/sheets (2D/3D), then giant density fluctuationsapolar case KT transition to quasi-long-range nematic order (2D)slow domain growth leading to high-density/high order macroscopic cluster with giant density fluctuationsmixed case (in progress)discontinuous transition to true long-range nematic order segregation to large cluster with giant density fluctuations

Part I:

Polar particles with polar interactions (Vicsek model)

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

3D polar particles without cohesion:discontinuous transitionat large enough size, discontinuous transitionNear threshold, at moderate sizes: flip-flop dynamics of order parameter leading to bimodal distribution z y x totaltime noise strength order parameter

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

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

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

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

2D: starting from ordered, homogenous-density, configurationshort timesmuch latertime

starting from ordered, homogenous-density, configuration: instability of trivial solutionlateconfigurationatypicalgrowthlatespectrumearlyspectrumconclusion: not a wave train, but solitary structures

Bands disappear at low noise,leaving anomalous density fluctuationsBand-train profile widthsNo band region: giant density fluctuationsNo band region: typical profilesWeaker bands: typical profiles

Part II:

Apolar particles with nematic interactions

2D: Kosterlitz-Thouless transition to QLROorder parameter scalingorder parameter curves at various sizes do not cross each otherpower law decay of order parameter with system size in (quasi-)ordered phasecrossover to normal decay (slope -1/2) in disordered phasevariation of exponent with noise strength; at estimated threshold, expected equilibrium value

Normal phase ordering: single lengthscalecoarse-grained densityL=256, 131072 particlesgrowth of density and orientation lengthscale

Deviations from Porods law: short-distance cuspC(r)with b~0.5"fluctuations-dominated coarsening "

Highly segregated yet fluctuating ordered phaseTime series of scalar order parameter (note the time scale)Typical stateDuring a global rearrangementAnother typical state

Giant density fluctuations in 2DIn (quasi-) ordered phase, giant density fluctuations: rms n scales like n (in 2D)

Recent experiment: vibrated rodsVijay Narayan, Narayanan Menon and Sriram Ramaswamy

Part III:

polar particles with nematic interactions

True long range nematic order and discontinuous transitionNo polar order, isotropic-nematic transitionOP vs noise at different sizesTime series of OP near transition

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

Asymptotic state: large, macroscopic, bandL=512L=256

Asymptotic state: large, macroscopic, bandcoarse-grained densityFirst:coarsening to nematic orderwith colliding polar packets

Second:emergence of single macroscopic band

Asymptotic state: giant density fluctuations

Part IV:

In progress: beyond numerics

A mesoscopic description derived from the microscopics(here pure nematic case)order parameter/density coupling (includes non-equilibrium current)multiplicative andconserved noisegiant numberfluctuations predicted

Both terms are necessary for a faithful description: without either of them, no giant density fluctuations

without right noise, no segregation

Summary/conclusions/perspectivesnature of ordered phasetrue 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 fluctuationsconnection with condensation/ZRP ?order of transitiondiscontinuous in polar and mixed case, continuous in nematic casemesoscopic description (in progress)better numerics for low-density regions, analytically?possibility of deterministic description?