A model of alignment interaction for oriented particles with phase transition

Transcription

1 A model of alignment interaction for oriented particles with phase transition Amic Frouvelle ACMAC Joint work with Jian-Guo Liu (Duke University, USA) and Pierre Degond (Institut de Mathématiques de Toulouse, France) Amic Frouvelle Alignment interaction with phase transition October /21

2 Goal: macroscopic description of some animal societies Local interactions without leader Emergence of macroscopic structures Images Amic Frouvelle Benson Kua (flickr) and Alignment interaction with phase transition October /21

3 Modeling of interacting self-propelled particles Vicsek et al. (1995). Discrete in time (interval t), alignment only, synchronous reorientation. New direction = Mean direction of neighboring particles at previous step + Noise Simulations: phase transition phenomenon, emergence of coherent structures. Degond-Motsch (2008). Time-continuous version: relaxation (with constant rate ν) towards the local mean direction. Hydrodynamic limit without phase transition phenomenon. Model presented here: making ν proportional to the local mean momentum. Amic Frouvelle Alignment interaction with phase transition October /21

4 Outline 1 Presentation of the model Kinetic model Hydrodynamic scaling The phase transition 2 General results Convergence to equilibrium Amic Frouvelle Alignment interaction with phase transition October /21

5 Individual dynamics Presentation of the model Kinetic model Hydrodynamic scaling The phase transition Particles at positions: X 1,..., X N in R n. Orientations ω 1,..., ω N in S (unit sphere). { dxk = ω k dt dω k = ν(id ω k ω k ) ω k dt + 2d(Id ω k ω k ) db k t Target direction: ω k = J k J k, J k = 1 N N K( X j X k ) ω j. j=1 Setting ν = J k ν 0, no more singularity (binary interactions): { dxk = ω k dt dω k = ν 0 (Id ω k ω k ) J k dt + 2d(Id ω k ω k ) db k t Amic Frouvelle Alignment interaction with phase transition October /21

6 Kinetic description Presentation of the model Kinetic model Hydrodynamic scaling The phase transition Theorem (Bolley, Cañizo, Carrillo, 2011) Probability density function f (x, ω, t), as N : t f + ω x f + ν 0 ω ((Id ω ω)j f f ) = d ω f J f (x, ω, t) = K( y x ) υ f (y, υ, t) dy dυ. y R n, υ S Tool : coupling process + estimations. d X k = ω k dt d ω k = ν 0 (Id ω k ω k ) J f N t dt + 2d(Id ω k ω k ) dbt k ft N = law( X 1, ω 1 ) = law( X k, ω k ) Amic Frouvelle Alignment interaction with phase transition October /21

7 Hydrodynamic scaling Presentation of the model Kinetic model Hydrodynamic scaling The phase transition Scaling, with ε 1 (and K 0 = R n K(x)dx): f ε (x, ω, t) = ν 0 K 0 f ( 1 1 dεx, ω, dε t). Mean-field reduced and rescaled equation: ε( t f ε + ω x f ε ) = Q(f ε ) + O(ε 2 ), with an effect of localization in space: Q(f ) = ω ((Id ω ω)j f f ) + ω f, J f (x, t) = f (x, ω, t) ω dω. Since (Id ω ω)j = ω (J ω), we get S Q(f ) = ω (e ω J f ω (e ω J f f )). Amic Frouvelle Alignment interaction with phase transition October /21

8 Local equilibria Presentation of the model Kinetic model Hydrodynamic scaling The phase transition Definitions: Fisher von Mises distribution M κω (ω) = eκ ω Ω S eκ υ Ω dυ. Orientation Ω S, concentration κ 0. Order parameter: c(κ) = J MκΩ = π cos θ 0 eκ cos θ sin n 2 θ dθ π 0 eκ cos θ sin n 2 θ dθ For J f = κ f Ω f, we can write Q under the form: [ ( )] f Q(f ) = ω M κf Ω f ω. M κf Ω f Local equilibria: f eq = ρm κω, for some Ω S. Compatibility condition: κ = κ feq = J feq = ρ J κω = ρc(κ).. Amic Frouvelle Alignment interaction with phase transition October /21

9 Presentation of the model Kinetic model Hydrodynamic scaling The phase transition Solutions to the compatibility condition, equilibria Proposition The function κ c(κ) κ is decreasing, its limit is 1 n when κ 0. Rate of convergence r(ρ) 6 ε 4 ε 2 ε n = 2 n = 3 n = Density ρ ρ n, only one solution: κ = 0. Uniform equilibrium ( stable ). ρ > n, uniform equilibrium ( unstable ) for κ = 0. Unique solution κ(ρ) > 0. Manifold of equilibria ( stable ): ρm κ(ρ)ω, Ω S. Amic Frouvelle Alignment interaction with phase transition October /21

10 Presentation of the model Kinetic model Hydrodynamic scaling The phase transition Formal derivation of macroscopic models Theorem (P. Degond, AF, J.-G. Liu) Region where ρ ε (x, t) n ε: hydrodynamic model. As ε 0, the limit of f ε is f 0 = ρ(x, t)m κ(ρ)ω(x,t) (formally), and the functions ρ, Ω satisfy the system: { t ρ + x (ρ c Ω) = 0, ρ ( t Ω + c (Ω x )Ω) + λ (Id Ω Ω) x ρ = 0. Region where ρ ε (x, t) n ε: diffusion model. First order correction in ε for ρ ε : t ρ ε = ε n 1 x ( ) 1 n ρ ε xρ ε. Amic Frouvelle Alignment interaction with phase transition October /21

11 General results Convergence to equilibrium Stochastic model and its mean-field limit Orientations ω k only. { dωk = (Id ω k ω k )J k dt + 2τ (Id ω k ω k ) db k t, J k = 1 N Nj=1 ω j. Limit as N (Bolley-Cañizo-Carrillo): t f = ω ((Id ω ω)j[f ]f ) + τ ω f, with J[f ] = Doi equation: { t f = (f Ψ) + τ f, S ω f dω. Ψ(ω) = S K(ω, ω) f ( ω) d ω (= J[f ] ω). Here K = ω ω (dipolar potential). Polymers: ω ω (Onsager) or (ω ω) 2 (Maier Saupe). Amic Frouvelle Alignment interaction with phase transition October /21

12 General results Convergence to equilibrium Existence, uniqueness, regularity, positivity, bounds Theorem (AF, J.-G. Liu, 2011) For an initial probability measure f 0 H s (S), (for an arbitrary s): Existence and uniqueness of a weak solution f. Global solution, in C (R + S), and f > 0 for t > 0. Instantaneous regularity estimates and uniform bounds: ( f (t) 2 H s+m C ) t m f 0 2 H s. Tool: spherical harmonics decomposition. Nonlinearity: finite number of coefficients. Amic Frouvelle Alignment interaction with phase transition October /21

13 Onsager free energy General results Convergence to equilibrium Free energy (non convex): F(f ) = τ S f ln f 1 2 J[f ] 2. Dissipation term: D(f ) = S f ω(τ ln f ω J[f ]) 2 0. d dt F + D = 0. The free energy F(f ) is decreasing towards F. LaSalle s principle Limit set: E = {f C (S) D(f ) = 0 and F(f ) = F }. lim t inf f (t) g H s = 0. g E Amic Frouvelle Alignment interaction with phase transition October /21

14 Equilibria General results Convergence to equilibrium D(f ) = 0 (no dissipation) Q(f ) = 0 (stationary solution) Equivalent conditions: f critical point of F τ ln f J[f ] ω = Cte Amounts to f = M κω, with the compatibility condition τκ = c(κ). Proposition The function κ c(κ) κ is decreasing, its limit is 1 n when κ 0. τ 1 n. Only one solution: κ = 0. Uniform equilibrium (unique minimizer of F). τ < 1 n : uniform equilibrium for κ = 0, not minimizing F. Only positive solution: κ(τ). Manifold of equilibria: M κ(τ)ω, Ω S. Amic Frouvelle Alignment interaction with phase transition October /21

15 Additional conservation relation General results Convergence to equilibrium Conformal Laplacian: n 1 Y l = l(l + 1)... (l + n 2)Y l, for a spherical harmonic Y l of degree l. Proposition For g H n+1 2 (S) with mean zero, S g n 1 g = 0. Norms g n 1 2 H 2 Conservation relation 1 d 2 dt f 1 2 H n 1 2 = 1 S g n 1g, n 3 g 2 H 2 = τ f 1 n 3 2 H 2 = 1 S g n 1 g: + 1 (n 2)! J[f ] 2. For τ > 1 n, it is an entropy dissipation! Global exponential convergence with rate (n 1)(τ 1 n ) towards uniform equilibrium (in any H s ). Amic Frouvelle Alignment interaction with phase transition October /21

16 General results Convergence to equilibrium Case τ < 1 n : the moving Fisher von Mises distribution ODE for J[f (t)]: J[f 0 ] 0 J[f (t)] 0 for all t. J[f (t)] We define Ω(t) = J[f (t)]. Expansion around the moving equilibrium M κ(τ)ω(t) : f = (1 + h)m κ(τ)ω(t), Notation MκΩ for the mean against this measure. Unique decomposition of the form f = (1 + α [ ω Ω(t) c(κ(τ))] + g)m κ(τ)ω(t), with g MκΩ = 0 and gω MκΩ = 0. By LaSalle s principle an unstability of the uniform distribution: h, α and g converge to 0. Amic Frouvelle Alignment interaction with phase transition October /21

17 General results Convergence to equilibrium Exponential convergence to a fixed equilibrium Poincaré inequality: h 2 MκΩ Λ κ (h h MκΩ ) 2 MκΩ Expansion of D(f ) and F(f ) F, in α 2 and g 2 MκΩ, with h small: exponential decay of f M κω(t) with rate r for all r < r (τ) = (c(κ) 2 + nτ 1)Λ κ. Dynamics of Ω(t): dω dt = (Id Ω Ω) g ω Ω ω M κω So we get dω dt C g 2 MκΩ Ce rt : exponential convergence of Ω to some Ω S. Interpolation + uniform bounds: exponential convergence to M κω in any H s. Amic Frouvelle Alignment interaction with phase transition October /21

18 Special cases General results Convergence to equilibrium Critical case τ = 1 n. Same type of method, with the decomposition f = 1 + α cos θ α2 (cos 2 θ 1 n ) α3 (cos 3 θ 3 n+2 cos θ) + g, with α = n f cos θ and cos θ = ω Ω. Expansion of D(f ) in terms of g 2 and α 6, and of F(f ) in terms of g 2 and α 4 : algebraic convergence in 1 t towards the uniform distribution. Case of no noise: τ = 0. The norm of J[f ] increases and the H n 1 2 norm of f explodes... Amic Frouvelle Alignment interaction with phase transition October /21