Collective dynamics in life sciences Lecture 2: the Vicsek model - PowerPoint PPT Presentation

1 Collective dynamics in life sciences Lecture 2: the Vicsek model Pierre Degond Imperial College London pdegond@imperial.ac.uk (see http://sites.google.com/site/degond/) Joint works with: Amic Frouvelle (Dauphine), Jian-Guo Liu (Duke), S´ ebastien Motsch (ASU) ↑ ↓ Pierre Degond - Collective dynamics in life sciences, Lect. # 2 - CIRM, April 2017

Summary 2 1. The Vicsek model 2. Mean-Field model 3. Self-Organized Hydrodynamics (SOH) 4. Properties of the SOH model and extensions 5. Conclusion ↑ ↓ Pierre Degond - Collective dynamics in life sciences, Lect. # 2 - CIRM, April 2017

3 1. The Vicsek model ↑ ↓ Pierre Degond - Collective dynamics in life sciences, Lect. # 2 - CIRM, April 2017

Vicsek model [Vicsek, Czirok, Ben-Jacob, Cohen, Shochet, PRL 95] 4 Individual-Based (aka particle) model self-propelled ⇒ all particles have same constant velocity a align with their neighbours up to a certain noise Time-discrete model k , at t n = n ∆ t k -th particle position X n k , velocity direction V n X n +1 = X n k + aV n | V n k ∆ t, k | = 1 R k V k k = J n � ¯ J n V n V n k X k k = j , |J n k | j, | X n j − X n k |≤ R ) = arg ( ¯ arg ( V n +1 V n k + τ n k ) k τ n k drawn uniformly in [ − τ, τ ] ; R = interaction range J n k = local particle flux in interaction disk ¯ V n k = neighbors’ average direction ↑ ↓ Pierre Degond - Collective dynamics in life sciences, Lect. # 2 - CIRM, April 2017

5 2. Mean-Field model ↑ ↓ Pierre Degond - Collective dynamics in life sciences, Lect. # 2 - CIRM, April 2017

Time continuous Vicsek model 6 Passage to time continuous dynamics: requires introduction of new parameter: interaction frequency ν ˙ X k ( t ) = aV k ( t ) √ k ◦ ( ν ¯ 2 τ dB k dV k ( t ) = P V ⊥ V k dt + t ) , P V ⊥ k = Id − V k ⊗ V k V k = J k � ¯ J k = V j , √ 2 τ dB k ν ¯ |J k | V k dt t j, | X j − X k |≤ R V k dV k ¯ V k Recover original Vicsek by: S 1 Time discretization ∆ t s.t. ν ∆ t = 1 Gaussian noise → uniform here ( X k , V k ) ∈ R n × R n , n ≥ 2 Dimension n = 2 ; ↑ ↓ Pierre Degond - Collective dynamics in life sciences, Lect. # 2 - CIRM, April 2017

Mean-field model 7 f ( x, v, t ) = particle probability density ν ¯ v f v satisfies a Fokker-Planck equation F f v f ¯ S 1 ∂ t f + av · ∇ x f + ∇ v · ( F f f ) = τ ∆ v f F f ( x, v, t ) = P v ⊥ ( ν ¯ v f ( x, t )) , P v ⊥ = Id − v ⊗ v v 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 v f ( x, t ) = direction of this flux ¯ F f ( x, v, t )) = projection of the flux direction on v ⊥ ( x, v ) ∈ R n × S n − 1 ; ∇ v · , ∇ v : div and grad on S n − 1 ∆ v Laplace-Beltrami operator on the sphere ↑ ↓ Pierre Degond - Collective dynamics in life sciences, Lect. # 2 - CIRM, April 2017

Passage to dimensionless units 8 Highlights important physical scales & small parameters Choose time scale t 0 , space scale x 0 = at 0 Set f scale f 0 = 1 /x n 0 , F scale F 0 = 1 /t 0 τ = τt 0 , ¯ R = R Introduce dimensionless parameters ¯ ν = νt 0 , ¯ x 0 x = x 0 x ′ , t = t 0 t ′ , f = f 0 f ′ , F = F 0 F ′ Change variables Get the scaled Fokker-Planck system (omitting the primes): ∂ t f + v · ∇ x f + ∇ v · ( F f f ) = ¯ τ ∆ v f F f ( x, v, t ) = P v ⊥ (¯ ν ¯ v f ( x, t )) , P v ⊥ = Id − v ⊗ v v 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 ↑ ↓ Pierre Degond - Collective dynamics in life sciences, Lect. # 2 - CIRM, April 2017

Macroscoping scaling 9 τ = 1 Choice of t 0 such that ¯ ε , ε ≪ 1 Macroscopic scale: there are many velocity diffusion events within one time unit k := ¯ ν Assumption 1: τ = O (1) ¯ Social interaction and diffusion act at the same scale ν − 1 = O ( ε ) , i.e. mean-free path is microscopic Implies ¯ ¯ Assumption 2: R = ε Interaction range is microscopic ν − 1 and of the same order as mean-free path ¯ R = O ( √ ε ) : interaction range still small Possible variant: ¯ but large compared to mean-free path. To be investigated later ↑ ↓ Pierre Degond - Collective dynamics in life sciences, Lect. # 2 - CIRM, April 2017

Fokker-Planck under macroscopic scaling 10 With Assumption 2 ( ¯ R = O ( ε ) ) Interaction is local at leading order: by Taylor expansion: � J f = J f + O ( ε 2 ) , J f ( x, t ) = S n − 1 f ( x, w, t ) w dw J f ( x, t ) = local particle flux. From now on, neglect O ( ε 2 ) term Fokker-Planck eq. in scaled variables ε ( ∂ t f ε + v · ∇ x f ε ) + ∇ v · ( F ε f ε ) = ∆ v f ε F ε ( x, v, t ) = kP v ⊥ u f ε ( x, t ) u f ε ( x, t ) = J f ε � S n − 1 f ε ( x, w, t ) w dw | J f ε | , J f ε ( x, t ) = Hydrodynamic model is obtained in the limit ε → 0 ↑ ↓ Pierre Degond - Collective dynamics in life sciences, Lect. # 2 - CIRM, April 2017

11 3. Self-Organized Hydrodynamics (SOH) ↑ ↓ Pierre Degond - Collective dynamics in life sciences, Lect. # 2 - CIRM, April 2017

Collision operator 12 Model can be written ∂ t f ε + v · ∇ x f ε = 1 εQ ( f ε ) with collision operator Q ( f ) = −∇ v · ( F f f ) + ∆ v f F f = kP v ⊥ u f u f = J f � | J f | , J f = S n − 1 f ( x, w, t ) w dw When ε → 0 , f ε → f (formally) such that Q ( f ) = 0 ⇒ importance of the solutions of Q ( f ) = 0 (equilibria) Q acts on v -variable only ( ( x, t ) are just parameters) ↑ ↓ Pierre Degond - Collective dynamics in life sciences, Lect. # 2 - CIRM, April 2017

Algebraic preliminaries 13 Force F f can be written: F f ( v ) = k ∇ v ( u f · v ) Note u f independent of v ( ( x, t ) are fixed) Rewrite: � � Q ( f )( v ) = ∇ v · − f k ∇ v ( u f · v ) + ∇ v f � � = ∇ v · f ∇ v ( − k u f · v + ln f ) Let u ∈ S n − 1 be given: Solutions of ∇ v ( − k u · v + ln f ) = 0 are proportional to : e ku · v f ( v ) = M ku ( v ) := � S n − 1 e ku · v dv von Mises-Fisher (VMF) distribution ↑ ↓ Pierre Degond - Collective dynamics in life sciences, Lect. # 2 - CIRM, April 2017

VMF distribution 14 Again: e ku · v M ku ( v ) := � S n − 1 e ku · v dv u ∈ S n − 1 : orientation k > 0 : concentration parameter; � Order parameter: c 1 ( k ) = S n − 1 M ku ( v ) u · v dv ր k → c 1 ( k ) , 0 ≤ c 1 ( k ) ≤ 1 � Flux: S n − 1 M ku ( v ) v dv = c 1 ( k ) u Here: concentration parameter k and order parameter c 1 ( k ) are constant ↑ ↓ Pierre Degond - Collective dynamics in life sciences, Lect. # 2 - CIRM, April 2017

Equilibria 15 Definition: equilibrium manifold E = { f ( v ) | Q ( f ) = 0 } Theorem: E = { ρM ku for arbitrary ρ ∈ R + and u ∈ S n − 1 } Note: dim mediumblue E = n Proof: follows from entropy inequality: 2 � �� f f � � � H ( f ) = Q ( f ) M kuf dv = − M ku f � ∇ v ≤ 0 � � M kuf � f � � �� follows from Q ( f ) = ∇ v · M ku f ∇ v M kuf f Then, Q ( f ) = 0 implies H ( f ) = 0 and M kuf = Constant and f is of the form ρM ku Reciprocally, if f = ρM ku , then, u f = u and Q ( f ) = 0 ↑ ↓ Pierre Degond - Collective dynamics in life sciences, Lect. # 2 - CIRM, April 2017

Use of equilibria 16 f ε → f as ε → 0 with v → f ( x, v, t ) ∈ E for all ( x, t ) Implies that f ( x, v, t ) = ρ ( x, t ) M ku ( x,t ) Need to specify the dependence of ρ and u on ( x, t ) Requires n equations since ( ρ, u ) ∈ R + × S n − 1 are determined by n independent real quantities f satisfies ∂ t f + v · ∇ x f = lim ε → 0 1 ε Q ( f ε ) Problem: lim ε → 0 1 ε Q ( f ε ) is not known Trick: Collision invariant ↑ ↓ Pierre Degond - Collective dynamics in life sciences, Lect. # 2 - CIRM, April 2017

Collision invariant 17 � is a function ψ ( v ) such that Q ( f ) ψ dv = 0 , ∀ f Form a linear vector space C Multiply eq. by ψ : ε − 1 term disappears Find a conservation law: � � � � � � ∂ t S n − 1 f ( x, v, t ) ψ ( v ) dv + ∇ x · S n − 1 f ( x, v, t ) ψ ( v ) v dv = 0 � Have used that ∂ t or ∇ x and . . . dv can be interchanged Limit fully determined if dim C = dim E = n C = Span { 1 } . Interaction preserves mass but no other quantity Due to self-propulsion, no momentum conservation dim C = 1 < dim E = n . Is the limit problem ill-posed ? ↑ ↓ Pierre Degond - Collective dynamics in life sciences, Lect. # 2 - CIRM, April 2017

Use of CI: mass conservation eq. 18 Proof that ψ ( v ) = 1 is a CI ? � � Obvious. Q ( f ) = ∇ v · . . . is a divergence � By Stokes theorem on the sphere, Q ( f ) dv = 0 Use of the CI ψ ( v ) = 1 : Get the conservation law � � � � � � ∂ t S n − 1 f ( x, v, t ) dv + ∇ x · S n − 1 f ( x, v, t ) v dv = 0 With f = ρM ku we have � � f ( x, v, t ) dv = ρ ( x, t ) , f ( x, v, t ) v dv = ρc 1 u We end up with the mass conservation eq. ∂ t ρ + c 1 ∇ x · ( ρu ) = 0 ↑ ↓ Pierre Degond - Collective dynamics in life sciences, Lect. # 2 - CIRM, April 2017