Models of collective displacements: from microscopic to macroscopic description Sébastien Motsch CSCAMM, University of Maryland joint work with : P. Degond, L. Navoret (IMT, Toulouse) SIAM Analysis of PDEs, San-Diego Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 1/ 21
Outline 1 Microscopic description 2 Derivation of macroscopic models 3 Numerical simulations Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 2/ 21
Motivation Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 3/ 21
Motivation Individuals only have local interactions There is no leader inside the group ( self-organization) The global organization of the group is at a much larger scale than the individual size Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 3/ 21
Motivation Individuals only have local interactions There is no leader inside the group ( self-organization) The global organization of the group is at a much larger scale than the individual size How can we connect individual and global dynamics? Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 3/ 21
Methodology Experiments, data recording Macroscopic equation Statistical analysis Kinetic equation Individual Based Model (IBM) Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 4/ 21
Methodology Experiments, data recording Macroscopic equation Statistical analysis Kinetic equation Individual Based Model (IBM) Identify the rules of the individual behavior Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 4/ 21
Methodology Experiments, data recording Macroscopic equation Statistical analysis Kinetic equation Individual Based Model (IBM) Identify the rules of the individual behavior Capture the global behavior of the system Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 4/ 21
Outline 1 Microscopic description 2 Derivation of macroscopic models 3 Numerical simulations Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 5/ 21
Boid models Classical model with 3 zones : attraction : alignment : repulsive Ref.: Aoki (1982), Reynolds (1986), Huth-Wissel (1992), Couzin et al. (2002),... Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 6/ 21
Boid models : alignment Ref.: Vicsek (1995), Gregoire-Chaté (2004), Cucker-Smale (2007), Ha-Tadmor (2008),... Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 6/ 21
Vicsek model ( 95) Discrete dynamics: x ω i i R Ω i x n+1 i = x n i + t ω n i ω n+1 with Ω n i = i = Ω n i + ɛ x j x i <R ωn j x j x i <R ωn j (1), ɛ noise. Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 7/ 21
Vicsek model ( 95) Discrete dynamics: x ω i i R Ω i x n+1 i = x n i + t ω n i ω n+1 with Ω n i = i = Ω n i + ɛ x j x i <R ωn j x j x i <R ωn j (1), ɛ noise. Continuous dynamics: dx i dt = ω i dω i = (Id ω i ω i )( ν Ω i dt + 2D db t ). (2) Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 7/ 21
Vicsek model ( 95) Discrete dynamics: x ω i i R Ω i x n+1 i = x n i + t ω n i ω n+1 with Ω n i = i = Ω n i + ɛ x j x i <R ωn j x j x i <R ωn j (1), ɛ noise. Continuous dynamics: dx i dt = ω i dω i = (Id ω i ω i )( ν Ω i dt + 2D db t ). (2) Remark. eq. (2) + ν t = 1 eq. (1) Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 7/ 21
Figure: Simulation of the Vicsek model: left position of particles, right density and mean velocities. Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 8/ 21
Outline 1 Microscopic description 2 Derivation of macroscopic models 3 Numerical simulations Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 9/ 21
Kinetic equation In the limit N +, the density distribution of particles f (t, x, ω) satisfies: with : t f + ω x f + ω (F f ) = D ω f, Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 10/ 21
Kinetic equation In the limit N +, the density distribution of particles f (t, x, ω) satisfies: with : t f + ω x f + ω (F f ) = D ω f, F (x, ω) = (Id ω ω) νω(x), Ω(x) = J(x) J(x) J(x) = ω f (y, ω ) dy dω y x <R, ω S 1 Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 10/ 21
Kinetic equation Kinetic equation t f + ω x f = Q(f ) (3) with: Q(f ) = ω (Ff ) + D ω f. Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 11/ 21
Kinetic equation Kinetic equation t f + ω x f = Q(f ) (3) with: Q(f ) = ω (Ff ) + D ω f. The equilibrium of Q(f ) (i.e. Qf = 0) are the Von Mises distributions: ( ) ω Ω M Ω (ω) = C exp T where T = D/ν and Ω is an arbitrary direction. Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 11/ 21
Kinetic equation Kinetic equation t f + ω x f = Q(f ) (3) with: Q(f ) = ω (Ff ) + D ω f. The equilibrium of Q(f ) (i.e. Qf = 0) are the Von Mises distributions: ( ) ω Ω M Ω (ω) = C exp T where T = D/ν and Ω is an arbitrary direction. The total momentum is not preserved by the operator: Q(f )ω dω 0. ω Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 11/ 21
1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Density distribution 1.2 1 0.8 0.6 0.4 0.2 0 Numeric Theoretic π π/2 ¼ π/2 π θ Figure: Simulation of the microscopic model (left) and the corresponding distribution of velocity angle (right). Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 12/ 21
Derivation of a macroscopic model Step 1. We use a hydrodynamic scaling: t = εt, x = εx. In these macroscopic variables, f ε satisfies: t f ε + ω x f ε = 1 ε Q(f ε ). (4) Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 13/ 21
Derivation of a macroscopic model Step 1. We use a hydrodynamic scaling: t = εt, x = εx. In these macroscopic variables, f ε satisfies: t f ε + ω x f ε = 1 ε Q(f ε ). (4) Step 2. Expand f ε (Hilbert expansion): f ε = f 0 + εf 1 +... f 0 equilibrium: f 0 = ρ 0 M Ω 0. Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 13/ 21
Derivation of a macroscopic model Step 1. We use a hydrodynamic scaling: t = εt, x = εx. In these macroscopic variables, f ε satisfies: t f ε + ω x f ε = 1 ε Q(f ε ). (4) Step 2. Expand f ε (Hilbert expansion): f ε = f 0 + εf 1 +... f 0 equilibrium: f 0 = ρ 0 M Ω 0. Step 3. Integrate (4) against the collisional invariants [ t f ε + ω x f ε = 1 ε Q(f ε )] ψ dω ω with ψ such that ω Q(f )ψ dω = 0. Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 13/ 21
Collisional invariants Problem: Only one quantity is preserved by Q. Momentum is not preserved by the dynamics Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 14/ 21
Collisional invariants Problem: Only one quantity is preserved by Q. Momentum is not preserved by the dynamics Def. ψ is a collisional invariant if for every f ω Q(f )ψ dω = 0 Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 14/ 21
Collisional invariants Problem: Only one quantity is preserved by Q. Momentum is not preserved by the dynamics Def. ψ is a collisional invariant if for every f ω Q(f )ψ dω = 0 ω f Q Ω f (ψ) dω = 0 Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 14/ 21
Collisional invariants Problem: Only one quantity is preserved by Q. Momentum is not preserved by the dynamics Def. ψ is a collisional invariant if for every f ω Q(f )ψ dω = 0 ω f Q Ω f (ψ) dω = 0 ψ = 1 Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 14/ 21
Collisional invariants Problem: Only one quantity is preserved by Q. Momentum is not preserved by the dynamics Def. ψ is a collisional invariant if for every f ωf dω //Ω ω ω Q(f )ψ dω = 0 ω satisfying f Q Ω f (ψ) dω = 0 ψ = 1 Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 14/ 21
Collisional invariants Problem: Only one quantity is preserved by Q. Momentum is not preserved by the dynamics Def. ψ is a collisional invariant if for every f ωf dω //Ω ω ω Q(f )ψ dω = 0 ω satisfying f Q Ω f (ψ) dω = 0 ψ = { 1 ϕ Ω (ω) with ϕ Ω a solution of: Q (ϕ Ω ) = ω Ω. Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 14/ 21
Collisional invariants Problem: Only one quantity is preserved by Q. Momentum is not preserved by the dynamics Def. ψ is a collisional invariant if for every f ωf dω //Ω ω ω Q(f )ψ dω = 0 ω satisfying f Q Ω f (ψ) dω = 0 ψ = { 1 ϕ Ω (ω) with ϕ Ω a solution of: Q (ϕ Ω ) = ω Ω. Then, we can integrate the kinetic equation: [ t f ε + ω x f ε = 1 ( ) ε Q(f ε 1 )] dω. ϕ Ω ε(ω) ω Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 14/ 21
Macroscopic model Thm. 1 The distribution f ε solution of (4) satisfies: f ε ε 0 ρ M Ω (ω) 1 Degond, M., Math. Models Methods Appli. Sci. (2008) Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 15/ 21
Macroscopic model Thm. 1 The distribution f ε solution of (4) satisfies: with: Macroscopic model f ε ε 0 ρ M Ω (ω) t ρ + c 1 x (ρω) = 0, ρ( t Ω + c 2 (Ω x )Ω) + λ (Id Ω Ω) x ρ = 0, Ω = 1 where c 1, c 2 and λ depend on T = D/ν. 1 Degond, M., Math. Models Methods Appli. Sci. (2008) Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 15/ 21
Macroscopic model Thm. 1 The distribution f ε solution of (4) satisfies: with: Macroscopic model f ε ε 0 ρ M Ω (ω) t ρ + c 1 x (ρω) = 0, ρ( t Ω + c 2 (Ω x )Ω) + λ (Id Ω Ω) x ρ = 0, Ω = 1 where c 1, c 2 and λ depend on T = D/ν. Remarks: the system obtained is hyperbolic......but non-conservative (due to the constraint Ω = 1) ρ and Ω have different convection speeds (c 1 c 2 ). 1 Degond, M., Math. Models Methods Appli. Sci. (2008) Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 15/ 21
Outline 1 Microscopic description 2 Derivation of macroscopic models 3 Numerical simulations Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 16/ 21
Numerical simulations We want to solve numerically the hyperbolic system: t ρ + c 1 x (ρω) = 0, ρ( t Ω + c 2 (Ω x )Ω) + λ (Id Ω Ω) x ρ = 0, Ω = 1 Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 17/ 21
Numerical simulations We want to solve numerically the hyperbolic system: t ρ + c 1 x (ρω) = 0, ρ( t Ω + c 2 (Ω x )Ω) + λ (Id Ω Ω) x ρ = 0, Ω = 1 Problem The model is non-conservative Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 17/ 21
Numerical simulations We want to solve numerically the hyperbolic system: t ρ + c 1 x (ρω) = 0, ρ( t Ω + c 2 (Ω x )Ω) + λ (Id Ω Ω) x ρ = 0, Ω = 1 Problem The model is non-conservative and has a geometric constraint Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 17/ 21
Numerical simulations We want to solve numerically the hyperbolic system: t ρ + c 1 x (ρω) = 0, t (ρω) + c 2 x (ρω Ω) + λ x ρ = ρ η (1 Ω 2 )Ω, Ω = 1 Problem The model is non-conservative and has a geometric constraint Method: Replace the geometric constraint ( Ω = 1) by a relaxation operator. Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 17/ 21
Numerical simulations We want to solve numerically the hyperbolic system: t ρ + c 1 x (ρω) = 0, t (ρω) + c 2 x (ρω Ω) + λ x ρ = ρ η (1 Ω 2 )Ω, Ω = 1 Problem The model is non-conservative and has a geometric constraint Method: Replace the geometric constraint ( Ω = 1) by a relaxation operator. In the limit η 0, we recover the original PDE. Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 17/ 21
Micro Vs Macro We use Riemann problems as initial condition: Figure: Density ρ: Micro (left) and Macro (right) Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 18/ 21
We take a cross section of the distribution in the x-direction 2 : 3 2.5 ρ θ 2 1.5 1 0.5 0 0 2 4 6 8 10 x Figure: macro. equation (line) and micro. equation (dot) at time t = 2. 2 M, Navoret, SIAM Multiscale Modeling & Simulation (2011) Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 19/ 21
We take a cross section of the distribution in the x-direction 2 : 3 2.5 ρ θ 2 1.5 1 0.5 0 0 2 4 6 8 10 x Figure: macro. equation (line) and micro. equation (dot) at time t = 4. 2 M, Navoret, SIAM Multiscale Modeling & Simulation (2011) Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 19/ 21
Introduction Microscopic description Derivation of macroscopic models Numerical simulations Micro Vs Macro Se bastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 20/ 21
Conclusion & Perspectives Summary Generalization of the notion of collisional invariant derivation of a hyperbolic system from the Vicsek model Numerical method to solve the hyperbolic system Good agreement with the microscopic model Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 21/ 21
Conclusion & Perspectives Summary Generalization of the notion of collisional invariant derivation of a hyperbolic system from the Vicsek model Numerical method to solve the hyperbolic system Good agreement with the microscopic model Perspectives Add an attraction/repulsion rule joint work with J-G Liu, P. Degond, V. Panferov Solve numerically the kinetic equation joint work with I. Gamba, J. Haack Corroborate the Vicsek model with real experimental data project with I. Couzin, S. Garnier Sébastien Motsch (CSCAMM) Mathematical modeling of collective displacements 14 November 2011 21/ 21