1 Mathematical models of self-organization Pierre Degond Imperial College London pdegond@imperial.ac.uk (see http://sites.google.com/site/degond/) ↑ ↓ Pierre Degond - Mathematical models of self-organization - CIRM 05/06/2019
Summary 2 1. Introduction 2. Directional coordination: the Vicsek model 3. Body attitude coordination 4. Reflection: network formation models 5. Conclusion ↑ ↓ Pierre Degond - Mathematical models of self-organization - CIRM 05/06/2019
3 1. Introduction ↑ ↓ Pierre Degond - Mathematical models of self-organization - CIRM 05/06/2019
Emergence 4 Emergence is the phenomenon by which: interacting many-particle (or agent) systems exhibit large-scale self-organized structures not explicitly encoded in the agents’ interaction rules Typical emergent phenomena are pattern formation ex: a biological tissue coordination ex: a bird flock self-organization ex: pedestrian lanes Emergence is a key process of life and social systems by which they self-organize into functional systems ↑ ↓ Pierre Degond - Mathematical models of self-organization - CIRM 05/06/2019
Questions 5 Understand link between: individual behavior (micro model: ODE or SDE) & large-scale structure (macro model: PDE) Requires rigorous passage “micro → macro” Why macro models ? Computational time Analysis: stability, bifurcations, . . . Data (images) inform on the macro scale What is special about emergent systems ? “micro → macro” Boltzmann, Hilbert, . . . Lions (94), Villani (10), Hairer (14), Figalli (18) . . . Unusual features Lack of propagation of chaos Lack of conservations: particles are “active” Coexistence of � = phases Complex underlying geometrical structures ⇒ revisit classical concepts ↑ ↓ Pierre Degond - Mathematical models of self-organization - CIRM 05/06/2019
6 2. Directional coordination: the Vicsek model 2.1 Presentation 2.2 Space-homogeneous case: phase transitions 2.3 Space-inhomogeneous case: macroscopic limit ↑ ↓ Pierre Degond - Mathematical models of self-organization - CIRM 05/06/2019
7 Directional coordination: the Vicsek model 2.1 Presentation Tam` as Vicsek (Budapest) ↑ ↓ Pierre Degond - Mathematical models of self-organization - CIRM 05/06/2019
Vicsek model [Vicsek, Czirok, Ben-Jacob, Cohen, Shochet, PRL 95] 8 Individual-Based (i.e. particle) model self-propelled ⇒ all particles have same constant speed = 1 align with their neighbors up to some noise Particle q : position X q ( t ) ∈ R n , velocity direction V q ( t ) ∈ S n − 1 ˙ X q ( t ) = V q ( t ) √ 2 dB q dV q ( t ) = P V ⊥ q ◦ ( kU q dt + t ) U q = J q � | J q | , J q = V j j, | X j − X q |≤ R R = interaction range R k = k ( | J q | ) = alignment frequency V k J q = local particle flux in interaction disk X k U q = neighbors’ average direction ⊥ P V ⊥ q = Id − V q ⊗ V q = orth. proj. on V q ◦ = Stratonovitch: guarantees | V q ( t ) | = 1 , ∀ t “Minimal model” for collective dynamics ↑ ↓ Pierre Degond - Mathematical models of self-organization - CIRM 05/06/2019
Phase transition in Vicsek model 9 Phase transition symmetry breaking disordered → aligned small k large k Simulations by A. Frouvelle Order parameter measures alignment Vk Vk � � � N − 1 � c 1 = q V q � , 0 ≤ c 1 ≤ 1 � � c 1 ∼ 1 c 1 ≪ 1 c 1 vs 1 /k c 1 vs density band formation after [Chat´ e et al, PRE 2008] ↑ ↓ Pierre Degond - Mathematical models of self-organization - CIRM 05/06/2019
Mean-field model 10 f ( x, v, t ) = particle probability density with ( x, v ) ∈ R n × S n − 1 satisfies a Fokker-Planck equation ∂ t f + v · ∇ x f + ∇ v · ( F f f ) = ∆ v f F f ( x, v, t ) = P v ⊥ ( ku f ( x, t )) , P v ⊥ = Id − v ⊗ v u f ( x, t ) = J f ( x, t ) � � | J f ( x, t ) | , J f ( x, t ) = S n − 1 f ( y, w, t ) w dw dy | y − x | <R J f ( x, t ) = particle flux in a neighborhood of x u f ( x, t ) = direction of this flux ku f ( x, t ) = alignment force (with k = k ( | J f | ) ) F f ( x, v, t )) = projection of alignment force on { v } ⊥ P v ⊥ = Id − v ⊗ v = projection on { v } ⊥ ∇ v · , ∇ v : div and grad on S n − 1 ; ∆ v = Laplace-Beltrami on S n − 1 ↑ ↓ Pierre Degond - Mathematical models of self-organization - CIRM 05/06/2019
Remarks 11 From particle to mean-field Requires number of particles N → ∞ Define empirical measure: f N ( x, v, t ) = N − 1 � N q =1 δ ( X q ( t ) ,V q ( t )) ( x, v ) f N → f where f satisfies Fokker-Planck Formal derivation in [D., Motsch: M3AS 18 (2008) 1193] Rigorous convergence proof: Classical: particle models with smooth interaction e.g. [Spohn] Difficulty here is handling constraint | v | = 1 Done for k ( | J f | ) = | J f | in [Bolley Canizo Carrillo: AML 25 (2012) 339] Open for k ( | J f | ) = 1 (difficulty: controling singularity at J f = 0 ) Existence and uniqueness of solutions to Fokker-Planck [Gamba, Kang: ARMA 222 (2016) 317] Other collective dynamics models do not normalize velocities e.g. Cucker-Smale, Motsch-Tadmor → huge literature ↑ ↓ Pierre Degond - Mathematical models of self-organization - CIRM 05/06/2019
12 Directional coordination: the Vicsek model 2.2 Space-homogeneous case: phase transitions [A. Frouvelle, Jian-Guo Liu, SIMA 44 (2012) 791] [PD., A. Frouvelle, Jian-Guo Liu, JNLS 23 (2013), 427] [PD., A. Frouvelle, Jian-Guo Liu, ARMA 216 (2015) 63-115] Amic Frouvelle (Dauphine) Jian-Guo Liu (Duke) ↑ ↓ Pierre Degond - Mathematical models of self-organization - CIRM 05/06/2019
Spatially homogeneous case 13 Forget the space-variable: ∇ x ≡ 0 : f ( v, t ) , v ∈ S n − 1 ∂ t f = −∇ v · ( F f f ) + ∆ v f := Q ( f ) = collision operator u f = J f � S n − 1 f ( v ′ , t ) v ′ dv ′ F f = k ( | J f | ) P v ⊥ u f , | J f | , J f = � Set: ρ ( t ) = f ( v, t ) dv . Then ∂ t ρ = 0 . So, ρ ( t ) = ρ = Constant Global existence results for k ( | J f | ) / | J f | smooth: [Frouvelle Liu: SIMA 44 (2012) 791] & [D. Frouvelle Liu: JNLS 23 (2013) 427 & ARMA 216 (2015) 63] for k ( | J f | ) = 1 : [Figalli Kang Morales: ARMA 227 (2018) 869] Equilibria: solutions of Q ( f ) = 0 ↑ ↓ Pierre Degond - Mathematical models of self-organization - CIRM 05/06/2019
Simulation of convergence to equilibrium 14 Histogram of positions and velocity velocity directions vectors of particles in ( − π, π ) in periodic box Simulation by S. Motsch ↑ ↓ Pierre Degond - Mathematical models of self-organization - CIRM 05/06/2019
Equilibria are VMF distributions 15 (VMF = Von Mises-Fisher) given by e κ u · v f ( v ) = ρ M κu ( v ) , M κu ( v ) = e κ u · v dv � where orientation u ∈ S n − 1 is arbitrary and concentration parameter κ = k ( | J f | ) Order parameter: c 1 ( κ ) = � M κu ( v ) u · v dv ∈ [0 , 1] , c 1 ( κ ) ր Compatibility equation: | J f | = ρc 1 ( κ ) = ρc 1 ( k ( | J f | )) introducing j ( κ ) = inverse function of k ( | J f | ) , can be recast in ρ = j ( κ ) κ = 0 or c 1 ( κ ) j ( κ ) Number of roots and local monotony of c 1 ( κ ) determine number of equilibria and their stability ↑ ↓ Pierre Degond - Mathematical models of self-organization - CIRM 05/06/2019
Examples 16 | J | Ex. 1: k ( | J | ) = 1+ | J | : continuous phase transition Ex. 2: k ( | J | ) = | J | + | J | 2 : discontinuous phase transition Ex. 1 Ex.2 ↑ ↓ Pierre Degond - Mathematical models of self-organization - CIRM 05/06/2019
Free energy 17 � f ln f dv − Φ( | J f | ) with Φ ′ = k Free energy: F ( f ) = d Free energy dissipation: dt F ( f ) = −D ( f ) ≤ 0 � � 2 dv � � D ( f ) = τ ( | J f | ) f � ∇ v f − k ( | J f | )( v · u f ) f is an equilibrium iff D ( f ) = 0 Free energy decays with time towards an equilibrium Unstable VMF are local max or saddle-points of F Stable VMF are local min of F F estimates L 2 -distance to local equilibrium: � f ( t ) − ρM κu f ( t ) � 2 L 2 ∼ F ( f ( t )) − F ( ρM κu f ( t ) ) ց Convergence to equilibrium with explicit rate relies on entropy-entropy dissipation estimates: cf Villani, . . . D ( f ) ≥ 2 λ κ ( F ( f ) − F ( M κu ))+ “small” ↑ ↓ Pierre Degond - Mathematical models of self-organization - CIRM 05/06/2019
18 Directional coordination: the Vicsek model 2.3 Space-inhomogeneous case: macroscopic limit [PD, S. Motsch: M3AS 18 Suppl. (2008) 1193] [PD., A. Frouvelle, Jian-Guo Liu, JNLS 23 (2013), 427] [PD., A. Frouvelle, Jian-Guo Liu, ARMA 216 (2015) 63-115] Sebastien Motsch (Arizona State) ↑ ↓ Pierre Degond - Mathematical models of self-organization - CIRM 05/06/2019
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