THREE-BODY INTERACTIONS DRIVE THE TRANSITION TO POLAR ORDER IN A SIMPLE FLOCKING MODEL

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1 THREE-BODY INTERACTIONS DRIVE THE TRANSITION TO POLAR ORDER IN A SIMPLE FLOCKING MODEL Purba Chatterjee and Nigel Goldenfeld Department of Physics University of Illinois at Urbana-Champaign

2 Flocking in Actomyosin motility assays V. Schallar et al, Nature volume 467, (2010) 2

3 Minimal model of flocking Ԧv t = v 0 eiθ t θ t + 1 = θ t R + η T. Vicsek et al, Phys. Rev. Lett. 75, 1226 (1995) 3

4 Minimal model of flocking Ԧv t = v 0 eiθ t θ t + 1 = θ t R + η Low noise/ high density better alignment T. Vicsek et al, Phys. Rev. Lett. 75, 1226 (1995) 4

5 Minimal model of flocking Ԧv t = v 0 eiθ t θ t + 1 = θ t R + η Low noise/ high density better alignment High noise/ low density poor alignment T. Vicsek et al, Phys. Rev. Lett. 75, 1226 (1995) 5

6 Minimal model of flocking Ԧv t = v 0 eiθ t θ t + 1 = θ t R + η Low noise/ high density better alignment High noise/ low density poor alignment Isotropic phase Low density High noise Polar ordered phase High density Low noise T. Vicsek et al, Phys. Rev. Lett. 75, 1226 (1995) 6

7 Minimal model of flocking iθ t Ԧv t = v 0 e θ t + 1 = θ t R + η Low noise/ high density better alignment High noise/ low density poor alignment Phase coexistence Isotropic phase Low density High noise Polar ordered phase High density Low noise T. Vicsek et al, Phys. Rev. Lett. 75, 1226 (1995) 7

8 Minimal model of flocking Mapping between Active fluid and Equilibrium fluid η T density Volume 1 Phase coexistence Gas phase Low density High noise T. Vicsek et al, Phys. Rev. Lett. 75, 1226 (1995) Liquid phase High density Low noise 8

9 Minimal model of flocking Low density High noise High density Low noise T. Vicsek et al, Phys. Rev. Lett. 75, 1226 (1995) 9

10 Minimal model of flocking Low density High density T. Vicsek et al, Phys. Rev. Lett. 75, 1226 (1995) 10

11 Minimal model of flocking A. P. Solon and J. Tailleur, Phys. Rev. E 92, (2015) 11

12 Noise strength Gas Minimal model of flocking density Liquid Coexistence H. Chaté et al, Phys. Rev. Lett. 114, (2015) 12

13 Noise strength Questions 1) What interactions are important for the ordering transition? density H. Chaté et al, Phys. Rev. Lett. 114, (2015) 13

14 Noise strength Questions 1) What interactions are important for the ordering transition? 2) What is the effect of stochasticity? density H. Chaté et al, Phys. Rev. Lett. 114, (2015) 14

15 Noise strength Questions Stochastic Theory 2-Body Interactions 3-body Interactions 1) What interactions are important for the ordering transition? Mean Field Theory 2) What is the effect of stochasticity? density H. Chaté et al, Phys. Rev. Lett. 114, (2015) 15

16 The Active Ising Model (AIM) N particles, each carrying ± spin. 1D lattice of size L, sites labelled by i. n ± i number of + and - spins on site i. ρ i = n + i + n i local density at site i. m i = n + i n i local magnetization at site i. No exclusion: arbitrary number of ± spins on each site. A. P. Solon and J. Tailleur, Phys. Rev. E 92, (2015) 16

17 Interactions 17

18 Interactions Self-Propulsion: + N D(1+ε) + i N i+1 N D(1 ε) i N i+1 + N D(1 ε) + i N i 1 N D(1+ε) i N i 1 where ε 0,1 D(1 + ε) D(1 ε) D(1 ε) D(1 + ε) 18

19 Interactions (contd.) Alignment: Random spin flip: N i + T N i N i T N i + Two-body interaction N i + + N i r 2/ρ i 2N i + N + i + N r 2/ρ i i 2N i Three body interaction 2N + i + N r 2 3/ρ i + i 3N i N i + + 2N i r 3/ρ i 2 3N i 19

20 Stochastic Hydrodynamics Individual level Model n = (n +, n ) Master Equation P(n, t) Fokker-Planck Equation P(ρ, m; x, t) Kramers-Moyal expansion, continuum limit Langevin Equation (Ito) ρ x, t, m(x, t) spde 20

21 Stochastic Hydrodynamics t ρ = xx ρ v x m, t m = xx m v x ρ m 2 T r r 3 m 2 2 ρ β ρ T + β β ρ2 m 2 η v = 2Dε, β = r r 3 4 η x, t η x, t = δ x x δ(t t ) 21

22 Stochastic Hydrodynamics t ρ = xx ρ v x m, t m = xx m v x ρ m 2 T r r 3 m 2 2 ρ β ρ T + β β ρ2 m 2 η v = 2Dε, β = r r 3 4 η x, t η x, t = δ x x δ(t t ) No noise in ρ because the noise comes from the alignment interactions, and the local ρ is only affected by the hopping processes, not alignment. 22

23 Stochastic Hydrodynamics t ρ = xx ρ v x m, t m = xx m v x ρ m 2 T r r 3 m 2 2 ρ β ρ T + β β ρ2 m 2 η v = 2Dε, β = r r 3 4 η x, t η x, t = δ x x δ(t t ) No noise in ρ because the noise comes from the alignment interactions, and the local ρ is only affected by the hopping processes, not alignment. The gradient of m changes ρ, and m is noisy, and this is why ρ has fluctuations as well 23

24 Noise strength 2-Body Interactions 3-body Interactions Stochastic Theory Mean Field Theory? H. Chaté et al, Phys. Rev. Lett. 114, (2015) density 24

25 Mean Field Theory (MFT) t ρ = xx ρ v x m, t m = xx m v x ρ m 2 T r r 3 2 m 2 ρ 2 25

26 Mean Field Theory (MFT) t ρ = xx ρ v x m, t m = xx m v x ρ m 2 T r r 3 2 m 2 ρ 2 Homogeneous steady states: For T > r 3, homogeneous isotropic 4 ρ = ρ 0, m = m 0 = 0 26

27 Mean Field Theory (MFT) t ρ = xx ρ v x m, t m = xx m v x ρ m 2 T r r 3 2 m 2 ρ 2 Homogeneous steady states: For T > r 3, homogeneous isotropic 4 ρ = ρ 0, m = m 0 = 0 For T < r 3 4, homogeneous polar order ρ = ρ 0, m = m 0 = ±ρ 0 r 3 4T r 3 27

28 MFT Phase Diagrams ρ 0 = 2 Phase diagram in mean field approximation exhibits no phase coexistence. 28

29 Noise strength 2-Body Interactions 3-body Interactions Stochastic Theory Mean Field Theory Gas, Liquid, no coexistence. H. Chaté et al, Phys. Rev. Lett. 114, (2015) density 29

30 Noise strength 2-Body Interactions 3-body Interactions Stochastic Theory Mean Field Theory Gas, Liquid, no coexistence.? H. Chaté et al, Phys. Rev. Lett. 114, (2015) density 30

31 Mean Field Theory (MFT) t ρ = xx ρ v x m, t m = xx m v x ρ m 2 T r r 3 2 m 2 ρ 2 For r 3 = 0, only stable homogeneous state: ρ = ρ 0, m = m 0 = 0 No ordered state with just 2-body interactions. 31

32 Noise strength 2-Body Interactions 3-body Interactions Stochastic Theory Mean Field Theory Only Gas phase Gas, Liquid, no coexistence. H. Chaté et al, Phys. Rev. Lett. 114, (2015) density 32

33 Beyond Mean Field Theory MFT : P ρ, m; x, t = δ ρ ρҧ δ m ഥm 33

34 ҧ Beyond Mean Field Theory MFT : P ρ, m; x, t = δ ρ ρҧ δ m ഥm Weak Fluctuation Theory (WFT) : P ρ, m; x, t = N ρ ρ, ҧ a ρ ρ N m ഥm, a m ρҧ 34

35 ҧ Beyond Mean Field Theory MFT : P ρ, m; x, t = δ ρ ρҧ δ m ഥm Weak Fluctuation Theory (WFT) : P ρ, m; x, t = N ρ ρ, ҧ a ρ ρ N m ഥm, a m Assumption 1: Fluctuations are Gaussian Assumption 2: The variances of these Gaussian distributions are proportional to the average density ρ, ҧ i.e. locally the number of fluctuating degrees of freedom is proportional to ρҧ ρҧ 35

36 Testing the WFT approximation WFT 36

37 Beyond Mean Field Theory MFT : P ρ, m; x, t = δ ρ ρҧ δ m ഥm Weak Fluctuation Theory (WFT) : P ρ, m; x, t = N ρ ρ, ҧ a ρ m 2 T r r 3 2 m 2 Where r = 3r 3 a m /4 T c = r 3 4 r ρ = T c MFT r/ρ ρҧ N m ഥm, a m ρҧ ρ 2 m 2 T r r ρ + r 3 2 m 2 ρ 2 37

38 MFT, WFT and spde MFT: t ρ = xx ρ v x m, WFT: t m = xx m v x ρ m 2 T r 3 4 t ρ = xx ρ v x m, + r 3 2 m 2 ρ 2 t m = xx m v x ρ m 2 T r 3 + r + r 3 m 2 4 ρ 2 ρ 2 spde t ρ = xx ρ v x m, t m = xx m v x ρ m 2 T r r 3 m 2 2 ρ β ρ T + β β ρ2 m 2 η 38

39 MFT, WFT and spde MFT: WFT: t ρ = xx ρ v x m, t m = xx m v x ρ m 2 T r 3 4 t ρ = xx ρ v x m, + r 3 2 m 2 ρ 2 No fluctuations or correlations. t m = xx m v x ρ m 2 T r 3 + r + r 3 m 2 4 ρ 2 ρ 2 spde t ρ = xx ρ v x m, t m = xx m v x ρ m 2 T r r 3 m 2 2 ρ β ρ T + β β ρ2 m 2 η 39

40 MFT, WFT and spde MFT: WFT: t ρ = xx ρ v x m, t m = xx m v x ρ m 2 T r 3 4 t ρ = xx ρ v x m, + r 3 2 t m = xx m v x ρ m 2 T r 3 + r + r 3 m 2 4 ρ 2 ρ 2 spde t ρ = xx ρ v x m, m 2 ρ 2 No fluctuations or correlations. Weak fluctuationsrenormalized MFT t m = xx m v x ρ m 2 T r r 3 m 2 2 ρ β ρ T + β β ρ2 m 2 η 40

41 MFT, WFT and spde MFT: WFT: t ρ = xx ρ v x m, t m = xx m v x ρ m 2 T r 3 4 t ρ = xx ρ v x m, + r 3 2 t m = xx m v x ρ m 2 T r 3 + r + r 3 m 2 4 ρ 2 ρ 2 spde t ρ = xx ρ v x m, m 2 ρ 2 No fluctuations or correlations. Weak fluctuationsrenormalized MFT t m = xx m v x ρ m 2 T r r 3 m 2 2 ρ β ρ T + β β ρ2 m 2 η Full stochastic theory 41

42 Noise strength 2-Body Interactions 3-body Interactions Stochastic Theory? Mean Field Theory Only Gas phase Gas, Liquid, no coexistence. H. Chaté et al, Phys. Rev. Lett. 114, (2015) density 42

43 Weak Fluctuation Theory Phase Diagrams T=0.5 The Weak Fluctuation Theory recovers the phase coexistence regime. 43

44 WFT Simulation: Coexistence Phase In the coexistence phase, liquid fraction: φ = ρ 0 ρ g ρ l ρ g T=0.1 Increasing density only increases the liquid fraction. 44

45 Noise strength 2-Body Interactions 3-body Interactions Stochastic Theory Gas, liquid AND coexistence phase. Mean Field Theory Only Gas phase Gas, liquid, no coexistence. H. Chaté et al, Phys. Rev. Lett. 114, (2015) density 45

46 Noise strength 2-Body Interactions 3-body Interactions Stochastic Theory Gas, liquid AND coexistence phase.? Mean Field Theory Only Gas phase Gas, liquid, no coexistence. H. Chaté et al, Phys. Rev. Lett. 114, (2015) density 46

47 Absence of Three-body Interactions WFT: t ρ = xx ρ v x m, t m = xx m v x ρ m 2 T r 3 + r + r 3 m 2 4 ρ 2 ρ 2 47

48 Absence of Three-body Interactions WFT: t ρ = xx ρ v x m, t m = xx m v x ρ m 2 T r 3 + r + r 3 m 2 4 ρ 2 ρ 2 Only Gas phase when r 3 = 0 48

49 Absence of Three-body Interactions WFT: t ρ = xx ρ v x m, t m = xx m v x ρ m 2 T r 3 + r + r 3 m 2 4 ρ 2 ρ 2 spde t ρ = xx ρ v x m, Only Gas phase when r 3 = 0 t m = xx m v x ρ m 2 T r r 3 m 2 2 ρ β ρ T + β β ρ2 m 2 η 49

50 Absence of Three-body Interactions WFT: t ρ = xx ρ v x m, t m = xx m v x ρ m 2 T r 3 + r + r 3 m 2 4 ρ 2 ρ 2 spde t ρ = xx ρ v x m, Only Gas phase when r 3 = 0 t m = xx m v x ρ m 2 T r r 3 m 2 2 ρ β ρ T + β β ρ2 m 2 η 50

51 Absence of Three-body Interactions WFT: t ρ = xx ρ v x m, t m = xx m v x ρ m 2 T r 3 + r + r 3 m 2 4 ρ 2 ρ 2 spde t ρ = xx ρ v x m, Only Gas phase when r 3 = 0 t m = xx m v x ρ m 2 T r r 3 m 2 2 ρ β ρ T + β β ρ2 m 2 η spde exhibits Gas phase and Switching phases. 51

52 spde Simulation: Absence of three-body interactions With r 3 = 0, no homogeneous ordered state For ρ > ρ c (T) = r 2 /2T 2 : only Gas phase, m 0 = 0. 52

53 spde Simulation: Absence of three-body interactions With r 3 = 0, no homogeneous ordered state For ρ > ρ c (T) = r 2 /2T 2 : only Gas phase, m 0 = 0. 53

54 spde Simulation: Absence of three-body interactions With r 3 = 0, no homogeneous ordered state spde: t m = xx m v x ρ m 2 T r r 3 m 2 2 ρ β ρ T + β β ρ2 m 2 η For ρ < ρ c = r 2 /2T 2, local switching of magnetization between ± m max (x) = ρ(x) T+r 2/2 r 2 /2 54

55 spde Simulation: Absence of three-body interactions With r 3 = 0, no homogeneous ordered state spde: t m = xx m v x ρ m 2 T r r 3 m 2 2 ρ β ρ T + β β ρ2 m 2 η For ρ < ρ c = r 2 /2T 2, local switching of magnetization between ± m max (x) = ρ(x) T+r 2/2 r 2 /2 ρ 0 = 10 < ρ c, T = 0.1, r 3 = 0 55

56 spde Simulation: Absence of three-body interactions With r 3 = 0, no homogeneous ordered state spde: t m = xx m v x ρ m 2 T r r 3 m 2 2 ρ β ρ T + β β ρ2 m 2 η For ρ < ρ c = r 2 /2T 2, local switching of magnetization between ± m max (x) = ρ(x) T+r 2/2 r 2 /2 ρ 0 = 10 < ρ c, T = 0.1, r 3 = 0 56

57 spde Simulation: Absence of three-body interactions With r 3 = 0, no homogeneous ordered state spde: No global order. m 2 ρ β Switching due to number ρ fluctuations, which is a collective effect at low density. t m = xx m v x ρ m 2 T r r 3 2 For ρ < ρ c = r 2 /2T 2, local switching of magnetization between ± m max (x) = ρ(x) T+r 2/2 r 2 /2 ρ 0 = 10 < ρ c, T = 0.1, r 3 = 0 T + β β ρ2 m 2 η 57

58 spde Simulation: Absence of three-body interactions With r 3 = 0, no homogeneous ordered state 58

59 Noise strength Stochastic Theory 2-Body Interactions Gas, switching phase at low densities-no GLOBAL ORDER 3-body Interactions Gas, liquid AND coexistence phase. Mean Field Theory Only Gas phase Gas, liquid, no coexistence. H. Chaté et al, Phys. Rev. Lett. 114, (2015) density 59

60 Noise strength Stochastic Theory 2-Body Interactions Gas, switching phase at low densities-no GLOBAL ORDER 3-body Interactions Gas, liquid AND coexistence phase. Gillespie Simulations of Individual Level Model (ILM)? Mean Field Theory Only Gas phase Gas, liquid, no coexistence. H. Chaté et al, Phys. Rev. Lett. 114, (2015) density 60

61 Results from Gillespie Simulations T = 0.1 ρ 0 = 1.0 T = 0.1 ρ 0 = 5.0 T = 0.1 ρ 0 = 10.0 D = 1, ε = 0.9, r 2 = 1, r 3 = 4, L =

62 Weak Fluctuation Theory vs Gillespie Simulation results WFT Simulation T=0.5 T=0.1 WFT and Simulation results agree qualitatively. Density is slaved to the magnetization and its fluctuations 62

63 Noise strength Stochastic Theory 2-Body Interactions Gas, switching phase at low densities-no GLOBAL ORDER 3-body Interactions Gas, liquid AND coexistence phase. Gillespie Simulations of Individual Level Model (ILM) Gas, liquid AND coexistence phase Mean Field Theory Only Gas phase Gas, liquid, no coexistence. H. Chaté et al, Phys. Rev. Lett. 114, (2015) density 63

64 Noise strength Stochastic Theory 2-Body Interactions Gas, switching phase at low densities-no GLOBAL ORDER 3-body Interactions Gas, liquid AND coexistence phase. Gillespie Simulations of Individual Level Model (ILM)? Gas, liquid AND coexistence phase Mean Field Theory Only Gas phase Gas, liquid, no coexistence. H. Chaté et al, Phys. Rev. Lett. 114, (2015) density 64

65 Gillespie Simulations: Absence of Three-body Interactions In the absence of three-body interactions (r 3 =0): no homogeneous ordered phase. Switching states for low densities. 65

66 Gillespie Simulations: Absence of Three-body Interactions In the absence of three-body interactions (r 3 =0): no homogeneous ordered phase. Switching states for low densities. Gillespie simulations of the Individual Level Model exhibit switching states, like in the full spde 66

67 Noise strength Stochastic Theory 2-Body Interactions Gas, switching phase at low densities-no GLOBAL ORDER 3-body Interactions Gas, liquid AND coexistence phase. Gillespie Simulations of Individual Level Model (ILM) Gas, switching phase at low densities-no GLOBAL ORDER Gas, liquid AND coexistence phase Mean Field Theory Only Gas phase Gas, liquid, no coexistence. H. Chaté et al, Phys. Rev. Lett. 114, (2015) density 67

68 Conclusion In the absence of three-body interactions there is no ordering transition. With three-body interactions, correct phase diagram for flocking is recovered. Local switching of magnetization at low densities, due to large relative number fluctuations. Future Directions Generalising results to higher dimensions. Generalising to the case of continuous orientational parameter. 68

69 70

70 Reserve slides 71

71 തX = X 1, X 2, X N, Reactions R μ, μ=1,2,,m Gillespie P τ, μ dτ = a μ exp a 0 τ probability that the next reaction takes place between time t + τ and time t + τ + dτ, and is a μ reaction, given state തX at time t. a μ probability per unit time that a μ reaction takes place. M a 0 = μ a μ Algorithm: 1. Generate τ, μ according to P τ, μ at each time t. 2. Update തX to reflect R μ having taken place. 3. Advance time by τ. 4. Repeat from step 1. Generating τ, μ : 1. Generate two uniformly distributed random numbers r 1, r 2. R 1 : X 1 c 1 X 2, a 1 = c 1 X 1 R 2 : X 1 + X 2 c 2 2X 1, a 2 = c 2 X 1 X 2 R 3 : 2X 1 + X 2 c 3 3X 1, a 3 = c 3X 1 (X 1 1)X Choose τ = 1 ln 1 a 0 r 1 3. Choose μ: σ μ 1 μ ν=1 a ν < a 0 r 2 σ ν=1 a ν 73

72 t ρ = xx ρ v x m, WFT Steady States t m = xx m v x ρ m 2 T r r ρ + r 3 2 Homogeneous steady states: For T > r 3, homogeneous isotropic 4 ρ = ρ 0, m = m 0 = 0 For T < r 3 4 For ρ < ϕ g = 4r/(r 3 4T), homogeneous isotropic, m 0 = 0 For ρ ϕ g, ϕ l, coexistence : homogeneous ordered state exists but unstable ---->travelling fronts. v r_3[v 2 T + D r 4 3 4T 2 ] + 2v 2 T + Dr 3 (r 3 4T) φ l = φ g 4v 2T + Dr_3(r 3 4T) m 2 ρ 2 For ρ > ϕ l, homogeneous polar order, m 0 = ±ρ 0 r 3 4T r 3 74

73 Switching Phase Biancalani et al, Phys. Rev. Lett. 112, (2014) 75

74 Derivation of spde 76

75 Derivation of spde (contd.) 77

76 Derivation of spde (contd.) 78

77 Derivation of spde (contd.) 79

78 Derivation of spde (contd.) 80

79 Derivation of spde (contd.) 81

80 Derivation of spde (contd.) 82

81 Results from Exact Stochastic Simulations (contd.) Giant number fluctuations in the coexistence phase. 83

PHYS 563 term Paper The Flocking Transition : A Review of The Vicsek Model

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