FLOCKS, COLLECTIVE MOTION

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1 FLOCKS, COLLECTIVE MOTION AND DRY ACTIVE MATTER Gareth Alexander Department of Physics and Centre for Complexity Science, University of Warwick Nonequilibrium statistical mechanics & active matter school and workshop Beijing, 8th-20th August 2016

2 Outline lecture 1 Vicsek & Zafeiras, Collective Motion, Phys Rep 517, 71 (2012) Vicsek et al., Novel Type of Phase Transition in a System of Self-Driven Particles, Phys Rev Lett 75, 1226 (1995) lecture 2 Ramaswamy, The Mechanics and Statistics of Active Matter, Annu Rev Condens Matter Phys 1, 323 (2010) Marchetti et al., Hydrodynamics of soft active matter, Rev Mod Phys 85, 1143 (2013) lecture 3 Cates & Tailleur, Motility-Induced Phase Separation, Annu Rev Condens Matter Phys 6, 219 (2015)

3 Flocking a) Wingless Locusts b) ants c) golden rays d) fish e) starlings f) zebra g) people h) sheep

4 Collective Nouns flock of birds, shoal of fish, herd of cattle, crowd of people, army of ants, gaggle of geese, pack of wolves, brood of chickens, pod of dolphins, troop of monkeys, plague of locusts, colony of penguins, company of parrots, pride of lions, mob of kangaroos, caravan of camels, convocation of eagles, parliament of owls, knot of toads, bed of worms, congregation of alligators, thunder of hippos, murder of crows, unkindness of ravens, murmuration of starlings

5 starlings, Aberystwyth, th December 2013

6 Starlings average angular density of neighbours Starflag project

7 Swimming Ducks surf scoters reconstructed velocity profile

8 Motility Assays Filaments of actin slide over a carpet of molecular motors (myosin) stuck to a substrate They move independently at low density, but form cohesively moving bands at high density

9 Dividing Escherichia coli 60 mins 90 mins 138 mins

10 Dry Active Matter a) b) c) d) a) vibrated discs b) fish keratocytes c) human melanocytes d) myxobacteria

11 Dry Active Matter Can we understand collective motion? Is every system different, or are there typical classes? Are there model systems that capture some of the richness and diversity? This involves a blend of the familiar with the unfamiliar familiar unfamiliar flocks are collections of individuals that move around and interact through collisions the collision rule does not conserve momentum Active matter may be viewed as a material that possesses a new type of mechanics; it represents a new paradigm for non-equilibrium phenomena We would like to identify its key organising principles and develop a unified description across many length scales

12 Dry Active Matter active = each particle drives its own motion dry = momentum non-conserving equivalently, neglect the fluid environment

13 Vicsek Model Birds (or boids) fly with some speed and align with their neighbours, subject to noise R

14 Vicsek Model Birds (or boids) fly with some speed and align with their neighbours, subject to noise step 1: propagation R step 2: alignment x i (t + t) =x i (t)+v i (t) t v i = v 0 cos i e x +sin i e y i (t + t) = j (t) x i x j <R + i(t) i (t) 2 [, ] angular noise uniform random variable

15 Vicsek Model Birds (or boids) fly with some speed and align with their neighbours, subject to noise step 1: propagation R step 2: alignment x i (t + t) =x i (t)+v i (t) t v i = v 0 cos i e x +sin i e y i (t + t) = j (t) x i x j <R + i(t) i (t) 2 [, ] angular noise uniform random variable Relevant variables: flying speed interaction range number density noise strength v 0 R = N/L 2 unimportant unimportant

16 Vicsek Model low density high noise low density low noise high density high noise high density low noise

17 Vicsek Model To quantify the coherence, one can look at the average velocity in the flock Order parameter: U = 1 Nv 0 X i v i (t) U! 0 U! 1 disordered aligned flock Look for a flocking transition as we vary the relevant parameters relevant variables: number density ρ noise strength η at zero noise the birds will align perfectly at low density interactions will be too few to generate widespread alignment

18 Vicsek Model Look for a flocking transition as we vary the relevant parameters increase size critical noise strength critical number density Reminiscent of a continuous transition with finite size effects

19 Vicsek Model The flocking transition seems to exhibit power law scalings scalings: U c U c U L

20 Vicsek Model The flocking transition seems to exhibit power law scalings scalings: U c U c U L scaling function and exponent relation U c (L) f(l/ ) f(x) ( O(1) x 1 x x. 1 correlation length

21 Vicsek Model The flocking transition seems to exhibit power law scalings scalings: U c U c U L scaling function and exponent relation U c (L) f(l/ ) f(x) ( O(1) x 1 x x. 1 correlation length U L c (L) L c (L) exponent identity =

22 Vicsek Model The Vicsek Model is like an active version of the celebrated XY model of equilibrium statistical mechanics it is continuous to a state with long range order, exhibits critical exponents, finite size effects, and scaling behaviour hv(r) v(0)i const except, the XY model famously does not have long range order hv(r) v(0)i r 2x it is a novel type of non-equilibrium phase transition

23 Vicsek Model It took over 10 years to discover that this paradigm is not quite correct The general consensus now is that the transition is discontinuous (a) order parameter (b) Binder cumulant G(,L)=1 hu 4 (t)i t 3hU 2 (t)i 2 t (c) variance (d) probability distribution black circles: angular noise red triangles: vectorial noise Chaté et al., Phys Rev E 77, (2008)

24 Nature of the Ordered Phase There are travelling bands of high density and order separated by disordered low density regions Chaté et al., Phys Rev E 77, (2008)

25 Nature of the Ordered Phase There are travelling bands of high density and order separated by disordered low density regions Chaté et al., Phys Rev E 77, (2008)

Nature of the Ordered Phase There are travelling bands of high density and order separated by disordered low density regions sharp leading edge diffuse trailing edge (exponential tail)

26 Nature of the Ordered Phase There are travelling bands of high density and order separated by disordered low density regions sharp leading edge diffuse trailing edge (exponential tail) characteristic width speed close to individual propulsion speed v0 So the transition in the Vicsek model is discontinuous and exhibits phase separation Chaté et al., Phys Rev E 77, (2008)

27 Variations on the Vicsek Model Given its importance and simplicity, many variants of the Vicsek model have been constructed and studied to see what is robust and what can be altered angular versus vectorial noise metric-free interactions Delaunay triangulation An issue with the Vicsek model is that unbounded flocks disperse cohesion visual information projection term

28 Metric-free Models The neighbours each bird interacts with are chosen topologically, rather than by distance, using a Delaunay triangulation (equivalently Voronoi tessellation) Ginelli & Chaté, Phys Rev Lett 105, (2010) Pearce & Turner, New J Phys 16, (2014)

29 Metric-free Models The neighbours each bird interacts with are chosen topologically, rather than by distance, using a Delaunay triangulation (equivalently Voronoi tessellation) Ginelli & Chaté, Phys Rev Lett 105, (2010) Pearce & Turner, New J Phys 16, (2014)

30 Metric-free Models The neighbours each bird interacts with are chosen topologically, rather than by distance, using a Delaunay triangulation (equivalently Voronoi tessellation) This rule agrees with observations of starling flocks in Rome (Starflag project) A boundary term provides cohesion (effective surface tension) Ginelli & Chaté, Phys Rev Lett 105, (2010) Pearce & Turner, New J Phys 16, (2014)

Metric-free Models The neighbours each bird interacts with are chosen topologically, rather than by distance, using a Delaunay triangulation (equivalently Voronoi tessellation) This rule agrees with

31 Metric-free Models The neighbours each bird interacts with are chosen topologically, rather than by distance, using a Delaunay triangulation (equivalently Voronoi tessellation) This rule agrees with observations of starling flocks in Rome (Starflag project) A boundary term provides cohesion (effective surface tension) order parameter Binder cumulant (noise) There are no bands and the transition is now continuous Ginelli & Chaté, Phys Rev Lett 105, (2010) Pearce & Turner, New J Phys 16, (2014)

32 The Type of Order in Active Matter Flocks are polar they have a macroscopic direction Many other systems are apolar, or nematic there is alignment but it is not a vector polar nematic

33 Liquid Crystals 1980s

34 Liquid Crystals 1980s 2010s

35 Liquid Crystals 1980s 2010s

36 Liquid Crystals nematic blue phase cholesteric photo: Michi Nakata twist grain boundary focal conic helical nanofilament (B4)

Liquid Crystals Chaikin-Lubensky broken symmetry

molecules nematic broken rotational symmetry

smectic A 1d broken translational symmetry

37 Liquid Crystals Chaikin-Lubensky broken symmetry ordered mesophases composed of long, thin, rod-like molecules nematic broken rotational symmetry molecules align along a common axis (director) smectic A 1d broken translational symmetry molecules form (fluid) layers smectic C + tilt relative to the layer normal

38 Polar or Nematic Equilibrium concepts of symmetry apply also to active systems to characterise the nature of their order first moment pi (r) = h i i = Z i P (r, ) d orientational probability distribution individual orientation second moment Qij (r) = i j 1 d ij = Z i j P (r, ) d 1 d ij result for isotropic distribution

39 Dry Active Matter Can we understand collective motion? Is every system different, or are there typical classes? Are there model systems that capture some of the richness and diversity? This involves a blend of the familiar with the unfamiliar familiar unfamiliar flocks are collections of individuals that move around and interact through collisions the collision rule does not conserve momentum Active matter may be viewed as a material that possesses a new type of mechanics; it represents a new paradigm for non-equilibrium phenomena We would like to identify its key organising principles and develop a unified description across many length scales

Active Matter Lectures for the 2011 ICTP School on Mathematics and Physics of Soft and Biological Matter Lecture 3: Hydrodynamics of SP Hard Rods

Active Matter Lectures for the 2011 ICTP School on Mathematics and Physics of Soft and Biological Matter Lecture 3: of SP Hard Rods M. Cristina Marchetti Syracuse University Baskaran & MCM, PRE 77 (2008);