A new flocking model through body attitude coordination

A new flocking model through body attitude coordination Sara Merino Aceituno (Imperial College London) Pierre Degond (Imperial College London) Amic Frouvelle (Paris Dauphine) ETH Zürich, November 2016 Transport phenomena in collective dynamics: from micro to social hydrodynamics

Section 1 Introduction

Emergent phenomena: auto-organisation Many agents Local interactions

Figure: Scale bar, 10 µm. Source: A study of synchronisation between the flagella of bull spermatozoa, with related observations, David M. Woolley, Rachel F. Crockett, William D. I. Groom, Stuart G. Revell Journal of Experimental Biology 2009 212: 2215-2223.

Description of the system to model Vicsek model

Description of the system to model Vicsek model Body attitude coordination

Previous results for the Vicsek model Equations at different scales Vicsek model [Degond, Motsch, 08] (X i (t), ω i (t)) R n S n, i = 1,..., N dx i = cω i dt, ( c > 0 dω i = P ω ν ω i + ) 2DB i i t ν, D > 0 ω i := J i J i, J i := N K( X i (t) X j (t) )ω j Fokker-Planck type equation f = f (t, x, ω) ( t f ε + cω x f ε = 1 ε f ω M Ω[f ] ε ω M )+O(ε) Ω[f ] M Ω (ω) exp ( ν D ω Ω) (von-mises equilibria) Ω ε [f ] S ω f ε (t, x, ω ) dω n Macroscopic eq. SOH Relations ft N = 1 N N i=1 δ x i (t),ω i (t) (empirical distribution) N [Bolley, Carrillo, Cañizo] f ε = f ε (t, x, v) [Figalli, Kang, Morales] NON conserved quant. ρ ε = f ε dv ρ ω f ε dω Ω ε Ω S n ε 0

Previous results for the Vicsek model Equations at different scales Vicsek model [Degond, Motsch, 08] (X i (t), ω i (t)) R n S n, i = 1,..., N dx i = cω i dt, ( c > 0 dω i = P ω ν ω i + ) 2DB i i t ν, D > 0 ω i := J i J i, J i := N K( X i (t) X j (t) )ω j Fokker-Planck type equation f = f (t, x, ω) ( t f ε + cω x f ε = 1 ε f ω M Ω[f ] ε ω M )+O(ε) Ω[f ] M Ω (ω) exp ( ν D ω Ω) (von-mises equilibria) Ω ε [f ] S ω f ε (t, x, ω ) dω n Macroscopic eq. SOH ρ = ρ(t, x), Ω = Ω(t, x) { t ρ + x (c 1 ρω) = 0, ρ ( t Ω + c 2 (Ω x )Ω) + c 3 P Ω ( x ρ) = 0 Relations ft N = 1 N N i=1 δ x i (t),ω i (t) (empirical distribution) N [Bolley, Carrillo, Cañizo] f ε = f ε (t, x, v) [Figalli, Kang, Morales] NON conserved quant. ρ ε = f ε dv ρ ω f ε dω Ω ε Ω S n ε 0 f ε f = ρ(t, x)m Ω (ω) (local equilibria) [Jiang, Xiong, Zhang] Generalised Collision Invariant Simulation by Sébastien Motsch (ASU)

Section 2 New results: body attitude coordination

Derivation of the Individual Based Model (X i, A i ) i {1,...,N}, X i R 3, A i. Averaged body orientation: M k (t) := 1 N N K( X i (t) X k (t) )A i (t). (1) i=1

Derivation of the Individual Based Model (X i, A i ) i {1,...,N}, X i R 3, A i. Averaged body orientation: M k (t) := 1 N N K( X i (t) X k (t) )A i (t). (1) i=1 Lemma (Polar decomposition of a square matrix, Golub 12) Given a matrix M M, if det(m) 0 then there exists a unique orthogonal matrix A (given by A = M( M T M) 1 ) and a unique symmetric positive definite matrix S such that M = AS. The matrix A minimizes the quantity 1 N N i=1 K( X i(t) X k (t) ) A i (t) A 2 among the elements of.

Mean-field limit Proposition (Formal) When the number of agents in (2)-(3) N, its corresponding empirical distribution f N (x, A, t) = 1 N N k=1 δ (Xk (t),a k (t)) converges weakly to f = f (x, A, t), (x, A, t) R 3 [0, ) satisfying t f + v 0 Ae 1 x f = D A f A (ff [f ]) (4) F [f ] := νp TA ( M[f ]) M[f ] = PD(M[f ]), M[f ](x, t) := R 3 K ( x x ) f (x, A, t)a da dx where PD(M[f ]) corresponds to the orthogonal matrix obtained on the Polar Decomposition of M[f ].

Macroscopic limit Adimensional analysis and rescaling: t f ε + Ae 1 x f ε = 1 ε Q(f ε ) + O(ε) (5) F 0 [f ] := ν P TA (Λ[f ]) Λ[f ] = PD(λ[f ]), λ[f ](x, t) := Q(f ) := D A f A (ff 0 [f ]) f (x, A, t)a da

Macroscopic limit Adimensional analysis and rescaling: t f ε + Ae 1 x f ε = 1 ε Q(f ε ) + O(ε) (5) F 0 [f ] := ν P TA (Λ[f ]) Λ[f ] = PD(λ[f ]), λ[f ](x, t) := Q(f ) := D A f A (ff 0 [f ]) = D A with equilibria given by ρ(t, x)m Λ (A): M Λ (A) = 1 ( ) ν(a Λ) Z exp, D f (x, A, t)a da [ M Λ[f ] A ( Consistency relation: λ[m Λ0 ] = c 1 Λ 0 where c 1 (0, 1). f M Λ[f ] )]. M Λ (A) da = 1, Λ, (6)

Key steps of the proof: the continuity equation and the non-conservation of momentum t f ε + Ae 1 x f ε = 1 ε Q(f ε ) + O(ε) By conservation of mass: t ρ ε + x Ae 1 f ε da = O(ε), ρ ε (t, x) := } {{ } =:j[f ε] in the limit (formally) ρ ε ρ (remember that f ε ρm Λ ) j[f ε ] ρ Ae 1 M Λ (A) da =: j by the consistency relation, we have that Conclude: j = ρc 1 Λe 1. t ρ(t, x) + x (ρ(t, x) c 1 Λ(t, x)e 1 ) = 0. f ε (x, A, t) da

Key steps of the proof: the continuity equation and the non-conservation of momentum t f ε + Ae 1 x f ε = 1 ε Q(f ε ) + O(ε) Equation for Λ? By the consistency relation: AM Λ (A) da = c 1 Λ but the momentum is not conserved! 1 AQ(f ε )(A) da 0 ε

The Generalised Collision Invariant Define the operator notice in particular that Q(f, Λ 0 ) := A ( ( )) f M Λ0 A M Λ0 Q(f ) = Q(f, Λ[f ]).

The Generalised Collision Invariant Define the operator notice in particular that Q(f, Λ 0 ) := A Using this operator we define: ( ( )) f M Λ0 A M Λ0 Q(f ) = Q(f, Λ[f ]). Definition (Generalised Collision Invariant) For a given Λ 0 we say that a real-valued function ψ : R is a Generalised Collision Invariant associated to Λ 0, ψ GCI (Λ 0 ) for short, if Q(f, Λ 0 )ψ da = 0 for all f s.t P TΛ0 (λ[f ]) = 0.

Why this helps? Consider ψ f GCI (Λ f ), then [ t f ε + Ae 1 x f ε] ψ f da = 1 ε = 1 ε = O(ε) Q(f ε )ψ f da + O(ε) Q(f ε, Λ f )ψ f da + O(ε)

Why this helps? Consider ψ f GCI (Λ f ), then [ t f ε + Ae 1 x f ε] ψ f da = 1 ε = 1 ε = O(ε) Q(f ε )ψ f da + O(ε) Q(f ε, Λ f )ψ f da + O(ε) Reference: Ning Jiang, Linjie Xiong, Teng-Fei Zhang, Hydrodynamic limits of the kinetic self-organized models; Existence of the GCI: Lax-Milgram; Computation of the GCI: differential geometry in

Computation of the GCI Proposition (Equation for the GCI) We have that ψ GCI (Λ 0 ) if and only if exists C T Λ0 such that A (M Λ0 A ψ) = C AM Λ0. (7)

Computation of the GCI Proposition (Equation for the GCI) We have that ψ GCI (Λ 0 ) if and only if exists C T Λ0 such that A (M Λ0 A ψ) = C AM Λ0. (7) Proposition (Existence of the GCI) For a given B T Λ fixed, there exists a unique ψ B H 1 () up to a constant, satisfying the relation (7).

Proposition (Non-constant GCI) Let P A, then the unique solution ψ H0 1 () of (8) is given by Going back to the GCI ψ(a), we can write it as ψ(b) = P B ψ 0 ( 1 2tr(B)). (9) ψ(a) = P (Λ T A) ψ 0 (Λ A) (10) where ψ 0 is constructed as follows: Let ψ 0 : R R be the unique solution to 1 )) sin 2 (θ/2) θ (sin 2 m(θ) sin θ (θ/2)m(θ) θ (sin θ ψ 0 ψ 2 sin 2 0 = sin θm(θ) θ/2 (11) where m(θ) = M Id (B) = exp(d 1 σ( 1 2 + cos θ))/z. Then ψ 0 (θ) = ψ 0 ( 1 2 tr(b)) by the relation 1 2 tr(b) = 1 2 + cos θ. ψ 0 is 2π-periodic, even and negative.

Back to the limit t f ε + Ae 1 x f ε = 1 ε Q(f ε ) + O(ε). With GCI ψλ P(A) = (P ΛT A)ψ 0 (Λ A) for any P antisymmetric: [ ] [ t (ρm Λ ) + Ae 1 x (ρm Λ )] P Λ T A ψ 0 (Λ A) da = 0 for all P A therefore X := since [ t (ρm Λ ) + Ae 1 x (ρm Λ )] (Λ T A A T Λ)ψ 0 (Λ A) da = 0. M = P S P: antisymmetric matrices, S, symmetric matrices. Compute the value of each term.

The macroscopic equations Theorem ((Formal) macroscopic limit) When ε 0 in the kinetic equation it holds that f ε f = f (x, A, t) = ρm Λ (A), Λ = Λ(t, x), ρ = ρ(t, x) 0. Consider the orthogonal basis {Λe 1 =: Ω, Λe 2 =: u, Λe 3 =: v}, where {e 1, e 2, e 3 } is the canonical basis. The equation for Λ expressed as the evolution of this basis is ρd t Ω + P Ω (c 3 x ρ + c 4 ρr) = 0, (12) ρd t u u (c 3 x ρ + c 4 ρr) Ω + c 4 ρδv = 0, (13) ρd t v v (c 3 x ρ + c 4 ρr) Ω c 4 ρδu = 0, (14) where D t := t + c 2 (Ω x ), (15) δ := [(Ω x )u] v + [(u x )v] Ω + [(v x )Ω] u, (16) r := ( x Ω)Ω + ( x u)u + ( x v)v. (17)

Proposition (Properties of δ and r) Let Λ = Λ(x) : R 3 be a function taking values on. It holds that δ Λ = δ ΛA and r Λ = r ΛA for any A. (18) where δ Λ and r Λ correspond to the expressions in (16) and (17), respectively. Suppose now that Λ = Λ(x) = exp ([b(x)] ) Λ(x 0 ) where b = b(x) : R 3 R 3 is a smooth function around a fixed x 0 R 3 where b(x 0 ) = 0, then δ Λ (x 0 ) = ( x b) (x 0 ), r Λ (x 0 ) = ( x b) (x 0 ).

Future work Simulations; new modelling using quaternions: joint with Pierre Degond (Imperial College London), Amic Frouvelle (Université Paris Dauphine) and Ariane Trescases (University of Cambridge). Study emergent structures. Compare them with the ones appearing in the Vicsek model (sperm dynamics). Some preliminary results: Maciek Biskupiak s video. Simulation by Maciek Biskupiak

Special thanks to: THE ORGANISERS Gianluca Crippa (Universität Basel) Siddhartha Mishra (ETH Zrich) Eitan Tadmor (University of Maryland / ETH-ITS Zrich) AND Andrea Bianda and Anil Zenginoglu AND Sponsors: EPSRC (Engineering and Physical Sciences Research Council, UK) & KI-NET & FIM at ETH Zürich

Summary: body attitude coordination Equations at different scales Individual Based Model (X i (t), A i (t)) R 3, i = 1,..., N dx i = v 0 A i e 1 dt, ( v 0 > 0 da i = P TAi νpd(m k )dt + 2 ) DdWt i M i (t) := 1 N N k=1 K( X k(t) X i (t) )A k (t). Fokker-Planck type equation f = f (t, x, A) ( t f ε + cω x f ε = 1 ε f A M Λ[f ] ε A M )+O(ε) Λ[f ] M Λ (A) exp ( ν D A Λ) (von-mises eq. ) Λ ε [f ] A f ε (t, x, A ) da Macroscopic eq. ρ = ρ(t, x), Λ = Λ(t, x) t ρ + x (c 1 ρλe 1 ) = 0, ρd t Ω + P Ω (c 3 x ρ + c 4 ρr) = 0, ρd t u u (c 3 x ρ + c 4 ρr) Ω + c 4 ρδv = 0, ρd t v v (c 3 x ρ + c 4 ρr) Ω c 4 ρδu = 0, Relations ft N = 1 N N i=1 δ X i (t),a i (t) (empirical distribution) N f ε = f ε (t, x, v) ρ ε = f ε dv ρ NON conserved quant. Consistency relation: A f ε da = c 1 Λ ε Λ ε 0 f ε f = ρ(t, x)m Λ (A)