Different types of phase transitions for a simple model of alignment of oriented particles Amic Frouvelle CEREMADE Université Paris Dauphine Joint work with Jian-Guo Liu (Duke University, USA) and Pierre Degond (Institut de Mathématiques de Toulouse, France) Kinetic Description of Multiscale Phenomena ACMAC, June 21st, 2013 Amic Frouvelle Alignment interaction with different types of phase transitions June 2013 1/15
Goal: macroscopic description of some animal societies Local interactions without leader Emergence of macroscopic structures, phase transitions Example : Vicsek model [1995], following more realistic similar models (Aoki [1982], Reynolds [1987], Huth-Wissel [1992]). Images Benson Kua (flickr) and Amic Frouvelle Alignment interaction with different types of phase transitions June 2013 2/15
Kinetic mean-field model The kinetic model Spatially homogeneous framework Stability issues Ingredients : moving at unit speed, alignment with neighbors, orientational noise. Theorem (following Bolley, Cañizo, Carrillo, 2012) Probability density function f (x, v, t) of finding an individual at x R n, with speed v S (unit sphere of R n ): t f + v x f }{{} transport + ν( J f ) v ( v (v Ω f ) f ) }{{} alignment = τ( J f ) v f }{{} diffusion Local momentum: J f = v S v f (x, v, t) dv. Target orientation: Ω f = J f J f, with J f = K x J f. Assumptions : K with finite second moment, and K, J ν( J ) J and τ bounded Lipschitz. Amic Frouvelle Alignment interaction with different types of phase transitions June 2013 4/15
Space-homogeneous version The kinetic model Spatially homogeneous framework Stability issues Smoluchowski equation on the sphere Reduced equation, for a function f (v, t): t f = Q(f ), Q(f ) = τ( J f ) v [ v f k( J f ) v (v Ω f ) f ] Ω f = J f J f, J f (t) = f (v, t) v dv. Key parameter: the conserved quantity ρ = S f. Key function: k( J ) = ν( J ) τ( J ) (competition between alignment and noise). More precisely, its inverse κ j(κ): Main assumption: J k( J ) increasing. κ = k( J ) J = j(κ) Amic Frouvelle Alignment interaction with different types of phase transitions June 2013 5/15 S
Equilibria The kinetic model Spatially homogeneous framework Stability issues Definitions: von Mises Fisher distribution M κω (v) = e κ v Ω S eκ w Ω dw. Orientation Ω S, concentration κ 0. Order parameter: c(κ) = J MκΩ = We rewrite Q: Q(f ) = τ( J f ) ω π 0 cos θ eκ cos θ sin n 2 θ dθ π 0 eκ cos θ sin n 2 θ dθ [M k( Jf )Ωf ω ( Compatibility relation for steady-states f M k( Jf )Ω f. )] Q(f ) = 0 f = ρm κω, with Ω S and j(κ) = ρc(κ).. Amic Frouvelle Alignment interaction with different types of phase transitions June 2013 6/15
Local analysis near uniform equilibria The kinetic model Spatially homogeneous framework Stability issues j(κ) Critical value: ρ c = lim (0, + ]. κ 0 c(κ) Theorem: Strong instability Exponential stability ρ > ρ c : if J f0 0, then f cannot converge to the uniform equilibrium. ρ < ρ c : There exists a 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 June 2013 7/15
The kinetic model Spatially homogeneous framework Stability issues Close to a nonisotropic equilibria ρm κω Theorem: Exponential stability in case ( j c ) (κ) > 0 For all s > n 1 2, there exist constants δ > 0 and C > 0, such that if f 0 ρm κω0 H s < δ for some Ω 0 S, there exists Ω S such that f ρm κω H s C f 0 ρm κω0 H s e λt, with λ = cτ(j) j Λ κ ( j c ), where Λ κ is the best constant for the following weighted Poincaré inequality: g 2 MκΩ Λ κ (g g MκΩ ) 2 MκΩ Tools: Free energy F(f ) = S f ln f Φ( J f ), with dφ d J = k( J ), and its dissipation D(f ) = τ( J f ) S f ω(ln f h( J f )ω Ω f ) 2. Instability of ρm κω if ( j c ) (κ) < 0. Amic Frouvelle Alignment interaction with different types of phase transitions June 2013 8/15
Second order (or continuous) phase transition First order (discontinuous) phase transition Hysteresis General statements in the case ( j 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) = j(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 j(κ) = c(κ)(1 + κ 1 β ). Amic Frouvelle Alignment interaction with different types of phase transitions June 2013 9/15
A phase diagram: k( J ) = J + J 2 1 Second order (or continuous) phase transition First order (discontinuous) phase transition Hysteresis Order parameter c c c 0.8 0.6 c 0.4 0.2 n=2 n=3 0 1 ρ ρ ρ c ρ c 4 Density ρ c c c Amic Frouvelle Alignment interaction with different types of phase transitions June 2013 10/15
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. 1 1 Order parameter c (kinetic) 0.8 0.6 0.4 0.2 Order parameter c (particles) 0.8 0.6 0.4 0.2 0 0.5 1 1.5 2 Density ρ 2.5 3 0 0.5 1 1.5 2 Density ρ 2.5 3 Amic Frouvelle Alignment interaction with different types of phase transitions June 2013 11/15
Scalings for the kinetic equation Scalings Ordered phase, hydrodynamic model Disordered phase, diffusion 2 scaling parameters : ε (hydrodynamic scaling) and η (characteristic length for the observation kernel K). Reduced kinetic equation ε( t f + v x f ) + K 2 η 2[ v (P v l f f ) m f v f ] = Q(f ) + O(η 4 ), with l f = ν( J f ) P J f Ω x J f + (Ω f x J f ) ν ( J f )Ω f, f m f = (Ω f x J f ) τ ( J f ), Limit as ε 0, in the cases where η = O(ε), or η = O( ε)? Amic Frouvelle Alignment interaction with different types of phase transitions June 2013 12/15
Scalings Ordered phase, hydrodynamic model Disordered phase, diffusion Branch of a stable nonisotropic equilibrium Stable branch of von Mises distributions given by ρ κ(ρ). Theorem: formal hydrodynamic limit When ε 0, in a region where f ε f 0 = ρ(x, t)m κ(ρ)ω(x,t), the functions ρ, Ω satisfy the following system: { t ρ + x (ρ c Ω) = 0, ρ ( t Ω + c(ω x )Ω) + Θ P Ω x ρ = K 2 δp Ω x (ρcω). c = cos θ Mκ, Θ = 1 κ + ρ κ dκ dρ ( c c), δ = ν(σ) c ( n 1 κ + c). Scaling parameter K 2 = lim ε 0 K 2 η 2 ε. Hyperbolicity linked to the critical exponent β (second order), or non hyperbolicity in the neighborhood of ρ (first order). Amic Frouvelle Alignment interaction with different types of phase transitions June 2013 13/15
Region where ρ c ρ ε (x, t) ε Scalings Ordered phase, hydrodynamic model Disordered phase, diffusion Chapman Enskog expansion. Theorem: formal diffusion correction As ε 0, at first order, in a region where f ε f 0 = ρ(x, t), f ε is (formally) given by f ε (x, v, t) = ρ ε (x, t) ε n ρ c v x ρ ε (x, t) (n 1)nτ 0 (ρ c ρ ε (x, t)), And the density ρ ε satisfies the following diffusion equation: t ρ ε = ε ρ ( c 1 x (n 1)nτ 0 ρ c ρ ε xρ ε). Second order phase transition : boundary region where ρ ε (x, t) ρ c = O(ε)? How to connect the two models? Amic Frouvelle Alignment interaction with different types of phase transitions June 2013 14/15