Mathematical modeling of self-organized dynamics: from...

Introduction PTW model Vicsek model Numerical schemes

Mathematical modeling of self-organized dynamics:from microscopic to macroscopic description

Sebastien Motsch

CSCAMM, University of Maryland

joint work with : P. Degond, L. Navoret (IMT, Toulouse)

G. Theraulaz, J. Gautrais (CRCA, Toulouse)

UC Davis, Applied Math PDE Seminar

Sebastien Motsch (CSCAMM) Math. model. of self-organized dynamics 6 March 2012 1/ 37

Outline

1 PTW modelExperiments and modelDerivation of a diffusion equation

2 Vicsek modelThe modelDerivation of a hyperbolic system

3 Numerical schemesSplitting methodParticle simulationsMicro vs macro

Motivation

Individuals have only local interactions

There is no leader inside the group (⇒ self-organization)

The global organization of the group is at a much larger scalethan the individual size

How can we connect individual and global dynamics ?

Motivation

Methodology

Individual Based

Model (IBM)

Statistical

analysis

Kinetic

equation

Experiments,

data recording

Macroscopic

equation

Identify the rules Capture the globalfor individual behavior behavior of the system

Methodology

Individual Based

Model (IBM)

Statistical

analysis

Kinetic

equation

Experiments,

data recording

Macroscopic

equation

Methodology

Individual Based

Model (IBM)

Statistical

analysis

Kinetic

equation

Experiments,

data recording

Macroscopic

equation

Outline

Experiments for fish

The diameter of the basin is 4 meters

Species studied: Kuhlia mugil (20-25 cm)

Video , data recorded

An example of trajectory :

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−1.5

−0.5

The norm of the velocity is constant

The trajectory is smooth, the fish seems to turn constantly

An example of trajectory :

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−1.5

−0.5

κ1κ2

The norm of the velocity is constant

The trajectory is smooth, the fish seems to turn constantly

PTW model

The model proposed is the following:

dt= c~τ(θ)

dt= cκ

dκ = −aκ dt + b dBt

where c is the speed, a the inverse of a relaxation time, b theintensity of diffusion.

We call this model “Persistent Turning Walker” (PTW)1.

1Gautrais et al., J. Math. Biol. (2009)Sebastien Motsch (CSCAMM) Math. model. of self-organized dynamics 6 March 2012 8/ 37

PTW model

dt= c~τ(θ)

dt= cκ

PTW model

dt= c~τ(θ)

dt= cκ

PTW model

dt= c~τ(θ)

dt= cκ

PTW model

dt= c~τ(θ)

dt= cκ

Comparison Data and Model

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−1.5

−0.5

Experiment

−2 −1 0 1 2

Simulation (PTW model)

Derivation of a diffusion equation

To analyze the large scale dynamics of the PTW model, it is moreconvenient to manipulate the density distribution of particlesf(t, x, θ, κ).

PTW model Kinetic equation

dt= ~τ(θ)

dt= κ

dκ = −κ dt +√

2α dBt

(in scaled variables)

∂t f + ~τ · ∇~x f

Lf = −κ∂θf +∂κ(κf )+α2∂κ2f

dt= ~τ(θ)

dt= κ

dκ = −κ dt +√

2α dBt

∂t f + ~τ · ∇~x f

dt= ~τ(θ)

dt= κ

dκ = −κ dt +√

2α dBt

∂t f +~τ ·∇~x f +κ∂θf−∂κ(κf ) = α2∂κ2f

dt= ~τ(θ)

dt= κ

dκ = −κ dt +√

2α dBt

∂t f + ~τ · ∇~x f = Lf

Derivation of a macroscopic model

Step 1. Diffusive scaling: t ′ = ε2t, x ′ = εx .In these macroscopic variables, f ε satisfies:

ε∂t fε + ~τ · ∇x f ε =

εL(f ε). (1)

Step 2. Hilbert expansion: f ε = f 0 + εf 1 + ...

Lf 0 = 0 ⇒ f 0 = ρ0(x)N (κ)2π (equilibrium)

with N a Gaussian with zero mean and variance α2.

Step 3. Integrate in (θ, κ):∫θ,κ

(ε∂t f

ε + ~τ · ∇x f ε =1

εL(f ε)

)dθdκ.

ε∂t fε + ~τ · ∇x f ε =

εL(f ε). (1)

(ε∂t f

ε + ~τ · ∇x f ε =1

εL(f ε)

)dθdκ.

ε∂t fε + ~τ · ∇x f ε =

εL(f ε). (1)

(ε∂t f

ε + ~τ · ∇x f ε =1

εL(f ε)

)dθdκ.

ε∂t fε + ~τ · ∇x f ε =

εL(f ε). (1)

(ε∂t f

ε + ~τ · ∇x f ε =1

εL(f ε)

)dθdκ.

ε∂t fε + ~τ · ∇x f ε =

εL(f ε). (1)

(ε∂t f

ε + ~τ · ∇x f ε =1

εL(f ε)

)dθdκ.

Diffusion equation

Thm.2 The distribution f ε solution of (1) satisfies:

f εε→0 ρ0 N (κ)

∂tρ0 +∇~x · J0 = 0,

J0 = −D∇~xρ0,

where D =∫∞

0 exp−α2(−1+s+e−s) ds.

Probabilistic point of view.

= x0 +

∫0cos(θs) ds

ε→0−→ 0 + D Bt′

2Degond, M., J. Stat. Phys. ,Sebastien Motsch (CSCAMM) Math. model. of self-organized dynamics 6 March 2012 12/ 37

Diffusion equation

∂tρ0 +∇~x · J0 = 0,

J0 = −D∇~xρ0,

where D =∫∞

= x0 +

∫0cos(θs) ds

ε→0−→ 0 + D Bt′

Diffusion equation

∂tρ0 +∇~x · J0 = 0,

J0 = −D∇~xρ0,

where D =∫∞

= x0 +

∫0cos(θs) ds

ε→0−→ 0 + D Bt′

Diffusion equation

∂tρ0 +∇~x · J0 = 0,

J0 = −D∇~xρ0,

where D =∫∞

x(t) = x0 +

0cos(θs) ds

ε→0−→ 0 + D Bt′

2Degond, M., J. Stat. Phys. , Chafaı, Cattiaux, M., Asymp. An.Sebastien Motsch (CSCAMM) Math. model. of self-organized dynamics 6 March 2012 12/ 37

Diffusion equation

∂tρ0 +∇~x · J0 = 0,

J0 = −D∇~xρ0,

where D =∫∞

xε(t ′) = εx0 + ε

∫ t′/ε2

0cos(θs) ds

ε→0−→ 0 + D Bt′

Diffusion equation

∂tρ0 +∇~x · J0 = 0,

J0 = −D∇~xρ0,

where D =∫∞

xε(t ′) = εx0 + ε

∫ t′/ε2

0cos(θs) ds

ε→0−→ 0 + D Bt′

Illustration: one trajectory ~x(t)

-10 -5 0 5 10

T=20 (epsilon=1)

-15 -10 -5 0 5 10 15

T=40 (epsilon=0.5)

-30 -20 -10 0 10 20 30

T=100 (epsilon=0.2)

-40 -30 -20 -10 0 10 20 30 40

T=200 (epsilon=0.1)

-60 -40 -20 0 20 40 60

T=400 (epsilon=0.05)

-100 -50 0 50 100

T=2000 (epsilon=0.01)

-150 -100 -50 0 50 100 150

T=4000 (epsilon=0.005)

Fish in interaction

In group, fish are usually aligned

To measure this effect, we observe the velocity of theneighbors in the frame of reference of one fish :

Fish in interaction

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Experimental data

Fish in interaction

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Experimental data

Fish in interaction

Classical model with 3 zones

: attraction

: alignment

: repulsive

Fish in interaction

: attraction

: alignment

: repulsive

Ref.: Aoki (1982), Reynolds (1986),

Huth-Wissel (1992), Couzin et al. (2002),...

Fish in interaction

: alignment

Ref.: Vicsek (1995), Gregoire-Chate (2004),

Cucker-Smale (2007), Ha-Tadmor (2008),...

Outline

Vicsek model (’95)

Discrete dynamics:

xn+1i = xn

i +∆t ωni

ωn+1i = Ω

ni + ε

with Ωni =

∑|xj−xi |<R ωn

j∣∣∣∑|xj−xi |<R ωnj

∣∣∣ , ε noise.

Continuous dynamics:

dxidt = ωi

dωi = (Id− ωi ⊗ ωi )( ν Ωi dt +√

2D dBt)(3)

Remark. eq. (3) + “ν∆t = 1” ⇒ eq. (2)

Discrete dynamics:

xn+1i = xn

i +∆t ωni

ωn+1i = Ω

ni + ε

with Ωni =

∑|xj−xi |<R ωn

dxidt = ωi

2D dBt)(3)

Remark. eq. (3) + “ν∆t = 1” ⇒ eq. (2)

Discrete dynamics:

xn+1i = xn

i +∆t ωni

ωn+1i = Ω

ni + ε

with Ωni =

∑|xj−xi |<R ωn

dxidt = ωi

2D dBt)(3)

Remark. eq. (3) + “ν∆t = 1” ⇒ eq. (2)

Kinetic equation

Under the hypothesis of propagation of chaos3, the density ofparticles f (t, x , ω) satisfies:

∂t f + ω · ∇x f +∇ω · (F f ) = D∆ωf ,

with :

F (x , ω) = (Id− ω ⊗ ω) νΩ(x) , Ω(x) =J(x)

|J(x)|

J(x) =

∫|y−x |<R, ω∗∈S1

ω∗ f (y , ω∗) dy dω∗

The alignment is expressed by the operator ∇ω · Ff ,

The randomness is expressed by D∆ωf .

3Sznitman, Saint-Flour (89)Sebastien Motsch (CSCAMM) Math. model. of self-organized dynamics 6 March 2012 19/ 37

Kinetic equation

∂t f + ω · ∇x f +∇ω · (F f ) = D∆ωf ,

with :

F (x , ω) = (Id− ω ⊗ ω) νΩ(x) , Ω(x) =J(x)

|J(x)|

J(x) =

∫|y−x |<R, ω∗∈S1

Kinetic equation

∂t f + ω · ∇x f +∇ω · (F f ) = D∆ωf ,

with :

F (x , ω) = (Id− ω ⊗ ω) νΩ(x) , Ω(x) =J(x)

|J(x)|

J(x) =

∫|y−x |<R, ω∗∈S1

Kinetic equation

Finally, f satisfies:

∂t f + ω · ∇x f = Q(f ) (4)

with: Q(f ) = −∇ω · (Ff ) + D∆ωf .

The equilibrium of Q(f ) (i.e. Qf = 0) are the Von Misesdistributions:

MΩ(ω) = C exp

(ω · Ω

)where T = D/ν and Ω is an arbitrary direction.

The total momentum is not preserved by the operator:∫ω

Q(f )ω dω 6= 0.

Kinetic equation

∂t f + ω · ∇x f = Q(f ) (4)

with: Q(f ) = −∇ω · (Ff ) + D∆ωf .

MΩ(ω) = C exp

(ω · Ω

Q(f )ω dω 6= 0.

Kinetic equation

∂t f + ω · ∇x f = Q(f ) (4)

with: Q(f ) = −∇ω · (Ff ) + D∆ωf .

MΩ(ω) = C exp

(ω · Ω

Q(f )ω dω 6= 0.

−3 −2 −1 0 1 2 3

Theoretic

Particles

Figure: Local distribution of velocity f (Left) for a simulation in a smalldomain (Right).

Derivation of a hyperbolic system

Step 1. Hydrodynamic scaling: t ′ = εt, x ′ = εx .In these macroscopic variables, f ε satisfies:

∂t fε + ω · ∇x f ε =

εQ(f ε). (5)

⇒ f 0 is an equilibrium: f 0(x , ω) = ρ0(x)MΩ0(x)(ω).

Step 3. Integrate (5) against the collisional invariants∫ω

[∂t fε + ω · ∇x f ε =

εQ(f ε) ]ψ dω

with ψ such that∫ω Q(f )ψ dω = 0.

∂t fε + ω · ∇x f ε =

εQ(f ε). (5)

[∂t fε + ω · ∇x f ε =

εQ(f ε) ]ψ dω

∂t fε + ω · ∇x f ε =

εQ(f ε). (5)

[∂t fε + ω · ∇x f ε =

εQ(f ε) ]ψ dω

∂t fε + ω · ∇x f ε =

εQ(f ε). (5)

[∂t fε + ω · ∇x f ε =

εQ(f ε) ]ψ dω

∂t fε + ω · ∇x f ε =

εQ(f ε). (5)

[∂t fε + ω · ∇x f ε =

εQ(f ε) ]ψ dω

Problem: Only one quantity is preserved by Q.

Momentum is not preserved by the dynamics

Def. ψ is a if for every f satisfying∫ω ωf dω // Ω∫

ωQ(f )ψ dω = 0 ⇒

f Q∗Ωf(ψ) dω = 0 ⇒ ψ =

1ϕΩ(ω)

with ϕΩ a solution of: Q∗(ϕΩ) = ω × Ω.

Then, we can integrate the kinetic equation:∫ω

[∂t fε + ω · ∇x f ε =

εQ(f ε) ]

ϕΩε(ω)

Def. ψ is a collisional invariant if for every f satisfying∫ω ωf dω // Ω∫ω

Q(f )ψ dω = 0 ⇒∫ω

f Q∗Ωf(ψ) dω = 0 ⇒ ψ =

1ϕΩ(ω)

[∂t fε + ω · ∇x f ε =

εQ(f ε) ]

ϕΩε(ω)

Q(f )ψ dω = 0 ⇒∫ω

f Q∗Ωf(ψ) dω = 0 ⇒ ψ =

1ϕΩ(ω)

[∂t fε + ω · ∇x f ε =

εQ(f ε) ]

ϕΩε(ω)

Q(f )ψ dω = 0 ⇒∫ω

f Q∗Ωf(ψ) dω = 0 ⇒ ψ = 1

1ϕΩ(ω)

[∂t fε + ω · ∇x f ε =

εQ(f ε) ]

ϕΩε(ω)

Q(f )ψ dω = 0 ⇒∫ω

f Q∗Ωf(ψ) dω = 0 ⇒ ψ = 1

1ϕΩ(ω)

[∂t fε + ω · ∇x f ε =

εQ(f ε) ]

ϕΩε(ω)

Q(f )ψ dω = 0 ⇒∫ω

f Q∗Ωf(ψ) dω = 0 ⇒ ψ =

1ϕΩ(ω)

[∂t fε + ω · ∇x f ε =

εQ(f ε) ]

ϕΩε(ω)

Q(f )ψ dω = 0 ⇒∫ω

f Q∗Ωf(ψ) dω = 0 ⇒ ψ =

1ϕΩ(ω)

[∂t fε + ω · ∇x f ε =

εQ(f ε) ]

ϕΩε(ω)

Hyperbolic system

f εε→0 ρMΩ(ω)

with:Hyperbolic system

∂tρ+ c1∇x · (ρΩ) = 0,ρ(∂tΩ + c2(Ω · ∇x)Ω) + λ (Id− Ω⊗ Ω)∇xρ = 0,|Ω| = 1

where c1, c2 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 (c1 6= c2).4Degond, M., Math. Models Methods Appli. Sci. (2008)

Hyperbolic system

Remarks:

Hyperbolic system

Remarks:

Hyperbolic system

Remarks:

Hyperbolic system

Remarks:

Applications

Combine PTW and Vicsek model

Degond, M., A Macroscopic Model for a System of Swarming AgentsUsing Curvature Control, J. Stat. Phys. (2011).

Extend the method for attraction-alignment-repulsion model

Liu, Degond, M., Panferov, Hydrodynamic models of self-organizeddynamics, submitted (2011).

Perspectives

Study numerically the kinetic equation

joint work with I. Gamba, J. Haack

Corroborate the macroscopic model with experimental data

project with I. Couzin, S. Garnier

Applications

Combine PTW and Vicsek model

Degond, M., A Macroscopic Model for a System of Swarming AgentsUsing Curvature Control, J. Stat. Phys. (2011).

Extend the method for attraction-alignment-repulsion model

Liu, Degond, M., Panferov, Hydrodynamic models of self-organizeddynamics, submitted (2011).

Perspectives

Study numerically the kinetic equation

joint work with I. Gamba, J. Haack

Corroborate the macroscopic model with experimental data

project with I. Couzin, S. Garnier

Outline

Numerical simulation

We want to numerically solve the macroscopic Vicsek (MV) model:

Two difficulties:

The model is non-conservative...

...and has a geometric constraint

⇒ No available theory to deal with this system.

Two difficulties:

Splitting method

The main idea of this method is to replace the geometricconstraint (|Ω| = 1) by a relaxation operator:

∂tρ+ c1∇x · (ρΩ) = 0,

In the limit η → 0, we recover the original MV model.

To solve numerically this system, we proceed in two steps(splitting):

First, we solve the conservative part (left-hand-side)...

...and then the relaxation part.

Splitting method

∂tρ+ c1∇x · (ρΩ) = 0,

∂t(ρΩ) + c2∇x · (ρΩ⊗ Ω) + λ∇xρ =ρ

η(1− |Ω|2)Ω,

|Ω| = 1

Splitting method

∂tρ+ c1∇x · (ρΩ) = 0,

∂t(ρΩ) + c2∇x · (ρΩ⊗ Ω) + λ∇xρ =ρ

η(1− |Ω|2)Ω,

|Ω| = 1

Splitting method

∂tρ+ c1∇x · (ρΩ) = 0,

∂t(ρΩ) + c2∇x · (ρΩ⊗ Ω) + λ∇xρ =ρ

η(1− |Ω|2)Ω,

|Ω| = 1

Splitting method

∂tρ+ c1∇x · (ρΩ) = 0,

∂t(ρΩ) + c2∇x · (ρΩ⊗ Ω) + λ∇xρ =ρ

η(1− |Ω|2)Ω,

|Ω| = 1

Other numerical methods

In one direction, the system is written:

∂tρ+ c1∂x (ρ cos θ) = 0

∂tθ + c2 cos θ ∂xθ − λsin θ

ρ∂xρ = 0. (6)

Multiplying (6) by 1/ sin θ and integrating in θ, we find aconservative formulation of the MV model.

Solving the conservative formulation gives another method⇒ Conservative method

Remark. Other methods can be developed using the“non-conservative” form of the MV model (e.g. upwind scheme).

∂tρ+ c1∂x (ρ cos θ) = 0

ρ∂xρ = 0. (6)

∂tρ+ c1∂x (ρ cos θ) = 0

ρ∂xρ = 0. (6)

∂tρ+ c1∂x (ρ cos θ) = 0

ρ∂xρ = 0. (6)

Simulations 1

The numerical schemes agree with each other on rarefaction waves(smooth solutions)

Simulations 2

However, the numerical schemes disagree when the solution is ashock wave (non-smooth solutions)

Question : What is the correct solution ? Do we have it ?

⇒ Go back to the microscopic model...

Simulations 2

Particle simulations

Since there is no theoretical solution to test our numericalsimulations, we use the microscopic Vicsek model as a benchmark:

dxεkdt

= ωεk ,

dωεk =1

ε(Id− ωεk ⊗ ωεk)(νΩ

εk dt +

√2D dBt),

Ωεk =

Jεk|Jεk |

, Jεk =∑

j , |xεj −xεk |≤εR

ωεj .

Particle simulations

Since there is no theoretical solution to test our numericalsimulations, we use the microscopic Vicsek model as a benchmark:

dxεkdt

= ωεk ,

dωεk =1

ε(Id− ωεk ⊗ ωεk)(νΩ

εk dt +

√2D dBt),

Ωεk =

Jεk|Jεk |

, Jεk =∑

j , |xεj −xεk |≤εR

ωεj .

We use Riemann problem as initial condition.

Figure: Density ρ: Micro (left) and Macro (right)

We take a cross section of the distribution in the x-direction:

Figure: macro. equation (line) and micro. equation (dot) at time t = 2.

We take a cross section of the distribution in the x-direction:

Figure: macro. equation (line) and micro. equation (dot) at time t = 4.

Micro vs macro

We compare the solutions of the MV model with the particles forthe shock-wave solution:

The splitting method has the “correct speed”.

Micro vs macro

We compare the solutions of the MV model with the particles forthe shock-wave solution:

The splitting method has the “correct speed”.

Contact discontinuity

For an initial condition given as a contact discontinuity, a weaksolution is given by a traveling wave.We observe numerically another type of solution5:

5M., Navoret, SIAM Multiscale Modeling & Simulation (2011)Sebastien Motsch (CSCAMM) Math. model. of self-organized dynamics 6 March 2012 36/ 37

Contact discontinuity

For an initial condition given as a contact discontinuity, a weaksolution is given by a traveling wave.We observe numerically another type of solution5:

5M., Navoret, SIAM Multiscale Modeling & Simulation (2011)Sebastien Motsch (CSCAMM) Math. model. of self-organized dynamics 6 March 2012 36/ 37

General case

How about non-standard initial condition?

General case

How about non-standard initial condition?

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