Collective dynamics of polar and apolar self-propelled particles

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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_at_cea.fr Thank you!
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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?

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One of the best examples starling flocks at
twilight
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Starling flocks in Rome
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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?

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

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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

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

• polar case
• (original Vicsek model)
• apolar case
• mixed case

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Phase diagram in density/noise parameter plane

• zero noise perfect order (if finite density ?)
• strong noise perfect random walks
• transition for sure, but at finite noise level s
  ? Yes!
• transition line in (?,s) plane

for polar case
at low density
for nematic case
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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

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Part I Polar particles with polar interactions
(Vicsek model)
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Discontinuous transition to collective motion
at large enough size, discontinuous variation of
order parameter
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3D polar particles without cohesiondiscontinuous
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
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Ordered phase fast domain growth
Quench into ordered phase (coarse-grained
density field) L16384, ?1/8 (32M boids)
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Ordered phase fast domain growth
Hydrodynamic, Model H-like growth ?t
Linear growth of lengthscale extracted from
exponential tail of two-point correlation
function of coarse-grained density field
Unusual correlation fonctions with apparent
algebraic decay
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2D ordered state in a finite box
Traveling high-density high-order solitary
band(s)
coarse-grained density field
density and order parameter profiles
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3D ordered state in a finite box
Traveling high-density high-order solitary
sheet(s)
color code local order
density profiles
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2D starting from ordered, homogenous-density,
configuration
much later
short times
time
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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
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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
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Part II Apolar particles with nematic
interactions
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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

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Normal phase ordering single lengthscale
coarse-grained density L256, 131072 particles
growth of density and orientation lengthscale
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Deviations from Porods law short-distance cusp
" fluctuations-dominated coarsening "
C(r)
with b0.5
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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

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Giant density fluctuations in 2D
In (quasi-) ordered phase, giant density
fluctuations rms ?n scales like n (in 2D)
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Recent experiment vibrated rods
Vijay Narayan, Narayanan Menon and Sriram
Ramaswamy
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Part III polar particles with nematic
interactions
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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
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True long range nematic order and discontinuous
transition
algebraic dependence of critical noise with
densitydiscontinuous transition for all
densities?
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Asymptotic state large, macroscopic, band
L512
L256
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Asymptotic state large, macroscopic, band
Firstcoarsening to nematic orderwith colliding
polar packetsSecondemergence of single
macroscopic band
coarse-grained density
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Asymptotic state giant density fluctuations
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Part IV In progress beyond numerics
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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
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Both terms are necessary for a faithful
description

• without either of them, no giant density
  fluctuations

• without right noise, no segregation

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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?