Different types of phase transitions for a simple model of alignment of oriented particles

Transcription

1 Different types of phase transitions for a simple model of alignment of oriented particles Amic Frouvelle Université Paris Dauphine Joint work with Jian-Guo Liu (Duke University, USA) and Pierre Degond (Institut de Mathématiques de Toulouse, France) KI-Net workshop on Kinetic description of social dynamics: From consensus to flocking CSCAMM, November 8th, 2012 Amic Frouvelle Alignment interaction with different types of phase transitions November /23

2 Goal: macroscopic description of some animal societies Local interactions without leader Emergence of macroscopic structures, phase transitions Images Benson Kua (flickr) and Amic Frouvelle Alignment interaction with different types of phase transitions November /23

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 (and the intensity of the noise) a function of the (norm of the) local mean momentum. Amic Frouvelle Alignment interaction with different types of phase transitions November /23

4 Outline 1 2 Second order (or continuous) phase transition First order (discontinuous) phase transition Hysteresis Amic Frouvelle Alignment interaction with different types of phase transitions November /23

5 Individual dynamics Particles at positions: X 1,..., X N in R n. Orientations ω 1,..., ω N in S (unit sphere). dx k = ω k dt dω k = νp ω ω k dt + 2τ P k ω db k 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 = ν( J k ) and τ = τ( J k ), no singularity if ν( J ) J Lipschitz. is Amic Frouvelle Alignment interaction with different types of phase transitions November /23

6 Kinetic description Assumptions : K with finite second moment, and K, J ν( J ) J and τ bounded Lipschitz. Theorem (following Bolley, Cañizo, Carrillo, 2012) Probability density function f (x, ω, t), as N : t f + ω x f + ν( J f ) ω (P ω ω f f ) = τ( J f ) ω f ω f = J f J f, J f = K J f, J f = ωf (x, ω, t) dω. ω S Tool : coupling process + estimations. d X k = ω k dt d ω k = ν( J f N t )P ω ω k f N t dt + 2τ( J f N t )P ω db k t k ft N = law( X 1, ω 1 ) = law( X k, ω k ) Amic Frouvelle Alignment interaction with different types of phase transitions November /23

7 Space-homogeneous version Reduced equation, for a function f (ω, t): t f = Q(f ), Q(f ) = ν( J f ) ω (P ω Ω f f ) + τ( J f ) ω f, Ω f = J f J f, J f (t) = f (ω, t) ω dω. Key parameter: the conserved quantity ρ = S f. Writing h( J ) = ν( J ) τ( J ), we get Q(f ) = τ( J f ) ω (e h( J f )ω Ω f ω (e h( J f )ω Ω f f )). Main assumption: J h( J ) is increasing. Its inverse: σ. S Amic Frouvelle Alignment interaction with different types of phase transitions November /23

8 Equilibria 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 κ f = h( J f ), we can write Q under the form: [ ( f Q(f ) = τ( J f ) ω M κf Ω f ω M κf Ω f )] Equilibria: f eq = ρm κω, for some Ω S. Then J feq = ρ J MκΩ = ρc(κ), and κ = κ feq = h( J feq ). Compatibility condition: κ = h(ρc(κ)), i.e. σ(κ) = ρc(κ). Amic Frouvelle Alignment interaction with different types of phase transitions November /23..

9 Solutions to the compatibility condition Uniform distribution f = ρ: always equilibrium. Possible shapes for function c σ (= 1 ρ )? 2 informations: 1 ρ c as κ 0, and 0 as κ κ max (the maximum of h( J )). c σ 0 0 κ Amic Frouvelle Alignment interaction with different types of phase transitions November /23

10 Existence, uniqueness, regularity, positivity, bounds Theorem 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 different types of phase transitions November /23

11 Main tool: Onsager free energy Free energy: F(f ) = S f ln f Φ( J f ), with dφ d J = h( J ). Dissipation: D(f ) = τ( J f ) S f ω(ln f h( J f )ω Ω f ) 2 0. d dt F + D = 0 F(f ) 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 Refined: if the roots of σ(κ) = ρc(κ) are isolated, there exists a solution κ such that: lim J f (t) = ρc(κ ) and s R, lim f (t) ρm t t κ Ωf (t) H s = 0. Amic Frouvelle Alignment interaction with different types of phase transitions November /23

12 Local analysis near uniform equilibria σ(κ) Critical value: ρ c = lim (0, + ]. κ 0 c(κ) Theorem: Strong unstability Exponential stability ρ > ρ c : if J f0 0, then f cannot converge to the uniform equilibrium. ρ < ρ c : There exists a universal constant δ such that if f 0 ρ H s < δ, then we have for all t 0 f (t) ρ H s f 0 ρ H s e λt 1 1 δ f, with λ = (n 1)τ 0 (1 ρ ). 0 ρ H s ρ c Tools: linearization for the evolution of J f, and then energy estimates for the whole equation. Amic Frouvelle Alignment interaction with different types of phase transitions November /23

13 Around a nonisotropic equilibria ρm κω Proposition: weak stability/unstability We denote by F κ the value of F(ρM κω ) (independent of Ω S). ( σ c ) (κ) < 0 (unstable): in any neighborhood of ρm κω, there exists f 0 such that F(f 0 ) < F κ. ( σ c ) (κ) > 0 (stable): If f 0 H s K, with s > n 1 2, then there exists δ > 0 and C (depending only on K and s), such that if f 0 ρm κω L 2 δ for some Ω S, then for all t 0, we have F(f ) F κ, and f ρm κωf L 2 C f 0 ρm κωf0 L 2. Tools: expansion of F F κ, and Sobolev interpolation. Amic Frouvelle Alignment interaction with different types of phase transitions November /23

14 Stronger stability: exponential convergence to equilibrium Theorem: exponential stability in case ( σ c ) (κ) > 0 For all s > n 1 2, there exist universal constants δ > 0 and C > 0, such that if f 0 ρm κω H s < δ for some Ω S, there exists Ω S such that with f ρm κω H s C f 0 ρm κωf0 H s e λt, λ = cτ(σ) σ Λ κ ( σ c ), where Λ κ is the best constant for the following weighted Poincaré inequality: g 2 MκΩ Λ κ (g g MκΩ ) 2 MκΩ Amic Frouvelle Alignment interaction with different types of phase transitions November /23

15 Outline Second order (or continuous) phase transition First order (discontinuous) phase transition Hysteresis 1 2 Second order (or continuous) phase transition First order (discontinuous) phase transition Hysteresis Amic Frouvelle Alignment interaction with different types of phase transitions November /23

16 Second order (or continuous) phase transition First order (discontinuous) phase transition Hysteresis General statements in the case ( σ c ) > 0 for all κ If ρ < ρ c, then the solution converges exponentially fast towards the uniform distribution f = ρ. If ρ = ρ c, the solution converges to the uniform distribution. If ρ > ρ c and J f0 0, then there exists Ω such that f converges exponentialy fast to the von Mises distribution f = ρm κω, where κ > 0 is the unique positive solution to the equation ρc(κ) = σ(κ). We can then define c (order parameter) as a function of ρ, and this function is continuous. Critical exponent β: when c(ρ) (ρ ρ c ) β. Can be any number in (0, 1], as one can artificially choose σ(κ) = c(κ)(1 + κ 1 β ). Amic Frouvelle Alignment interaction with different types of phase transitions November /23

17 Second order (or continuous) phase transition First order (discontinuous) phase transition Hysteresis Practical criteria for continuous phase transition Lemma is a nonincreasing function of J, then we are in the previous case of a continuous phase transition. In that case, the critical exponent β, if it exists, can only take values in [ 1 2, 1]. If h( J ) J In that case, we also have a special cancellation (related to the so-called conformal Laplacian), which gives global exponential decay when ρ < ρ c : Proposition If ρ < ρ c, there exists a universal constant C such that we have for all t 0 f (t) ρ H s C f 0 ρ H s e λt, with λ = (n 1)τ min (1 ρ ρ c ). Amic Frouvelle Alignment interaction with different types of phase transitions November /23

18 A special example Second order (or continuous) phase transition First order (discontinuous) phase transition Hysteresis 1 Focus: ν( J ) = J and τ( J ) = (related to the so-called 1 + J extrinsic noise ). In that case, we get h( J ) = J + J 2, so we are not indeed in the previous case. We obtain σ(κ) = 1 2 ( 1 + 4κ 1) c σ n=2 n= κ Amic Frouvelle Alignment interaction with different types of phase transitions November /23

19 The phase diagram 1 Second order (or continuous) phase transition First order (discontinuous) phase transition Hysteresis 0.8 Order parameter c n=2 n= Density ρ Amic Frouvelle Alignment interaction with different types of phase transitions November /23

20 Comparison of Free energy levels 0.1 Second order (or continuous) phase transition First order (discontinuous) phase transition Hysteresis Fκ F(ρ) n=2 n= Density ρ Amic Frouvelle Alignment interaction with different types of phase transitions November /23

21 Rates of convergence Second order (or continuous) phase transition First order (discontinuous) phase transition Hysteresis 3 Rate of convergence λ 2 1 uniform von Mises von Mises 0 1 uniform 2 3 Density ρ n=2 n=3 4 Amic Frouvelle Alignment interaction with different types of phase transitions November /23

22 Second order (or continuous) phase transition First order (discontinuous) phase transition Hysteresis Numerical illustration of the hysteresis phenomena Change of scale f = f ρ. The parameter ρ can now be considered as a free parameter that we let evolve in time Density ρ Amic Frouvelle Alignment interaction with different types of phase transitions November /23