1 Complex systems: Self-organization vs chaos assumption P. Degond Institut de Mathématiques de Toulouse CNRS and Université Paul Sabatier pierre.degond@math.univ-toulouse.fr (see http://sites.google.com/site/degond/) Joint work with E. Carlen and B. Wennberg ; numerical simulations by R. Chatelin
Summary 2 1. Examples 2. Chaos property in particle systems 3. Binary particle dynamics on S 1 : the CLD & BDG dynamics 4. Chaos property for CLD & BDG 5. Conclusion
3 1. Examples
Complex system 4 System with locally interacting agents emergence of spatio-temporal coordination patterns, structures, correlations, synchronization No leader / only local interactions
Vicsek model [Vicsek et al, PRL 95] 5 Time-discrete model: t n = n t k-th individual R X k ω k Xk n : position at tn ωk n: velocity with ωn k = 1 X n+1 k ω n+1 k = X n k + ω n k t = ω k n + noise (uniform in small angle interval) ω n k = Jn k J n k, Jn k = j, X n j Xn k R ω n j Alignment to neighbours mean velocity plus noise
Phase transition [Vicsek et al, PRL 95] 6 Phase transition to disorder Order parameter α = N 1 j ω j 2 N = particle number 0 α 1 Measures alignment S 1 ω k S 1 ω k α 1 α 1: ω aligned α 1 α 1: ω random
Phase transition to aligned state 7 As noise decreases [Vicsek et al, PRL 95] As density increases [Vicsek et al, PRL 95] α(n) α(noise level) Band formation [Chaté et al] Particle positions ρ (black) and α (red) (cross section)
Self-organization 8 Vicsek dynamics exhibits self-organization & emergence of coherent structures supposes the build-up of correlations between particles Kinetic and Hydrodynamic models rely on the chaos assumption When N is large, particles are statistically independent Question: are kinetic and hydrodynamic models relevant for Complex Systems? Goal: provide illustrative examples
9 2. Chaos property in particle systems
Method 10 Construct the Master equation Tells us the passage F N (t n ) F N (t n+1 ) where F N (v 1,...,v N ) = N-particle probability distribution Note: F N invariant under permutations of {v 1,...,v N } Compute the marginals F (j) N (v 1,...,v j ) = Master eq. eq. for the marginals F N dv j+1...dv N Eqs. for the marginals not closed (BBGKY hierarchy) Marginals: fixed number of variables when N
Binary interactions 11 Hierarchy: F (j) N (tn+1 ) = J (j) (F (j+1) N (t n )) Taking the limit N simplifies the problem If N large, system is not influenced by the state of one given particle Particles become independent Chaos assumption F (j) (v 1,...,v j ) = j k=1 F (1) (v k )
Binary interactions (cont) 12 Suppose at t = 0: particles are independent F (j) (v 1,...,v j ) t=0 = F (1) (v k ) t=0 If N finite: Dynamics builds up correlations instantaneously If N, correlations tend to 0 for Hard-Sphere Dynamics [Lanford], T s.t. t [0, T] F (j) (v 1,...,v j ) t F (1) (v k ) t as N BBGKY hierarchy converges to the Boltzmann eq.
Related questions 13 As N : Dynamics becomes irreversible entropy functional H which ց in time Dissipation Equilibria = states of maximal disorder For classical systems (e.g. rarefied gases) strong relation between these concepts Is this still true for self-organization processes? will some of these concepts survive while others won t?
14 3. Binary particle dynamics on S 1 : the CLD & BDG dynamics
Dynamics on S 1 15 Setting N particles with velocities v k S 1 i.e. v k R 2 with v k = 1 Space homogeneous problem kein x!!! All particles can interact State of the system at the n-th iterate Z N (t n ) = (v 1,...,v N )(t n ) (S 1 ) N t n = n t Discrete stochastic dynamics Z N (t n ) Z N (t n+1 )
Ex. 1: Space-homogeneous Vicsek dynamics16 Compute average direction v = k v k / k v k Add independent noise v k = v w k g(z) proba on S 1, symmetric g(z) = g(z ) w k : N independent random var. drawn according to g Note: Multiplicative group structure of S 1 Also use phases θ s.t. v = e iθ v k S 1 v k v All particles interact no reduction using marginals
Ex 2. A binary Vicsek dynamics: BDG 17 After [Bertin, Droz, Gregoire] Pick a pair {i, j} at random probability P ij = 2/N(N 1) average direction: v ij = (v i + v j )/ v i + v j Add independent noise drawn according to g: v i = v ij w i v j = v ij w j All particles but {i,j} unchanged Variant (acception-rejection) S 1 v i v i v v j v j Collision performed with probability h(v i v j) s.t. 0 h 1
Ex 3. Choose the Leader (CLD) 18 Pick an ordered pair (i, j) at random Probability P ij = 1/N(N 1) Then, i joins j plus noise w drawn according to g v i = v j w All particles but i unchanged S 1 v i v i v j
19 4. Chaos property in BDG and CLD dynamics
Noise scaling 20 Outline Compute the masters eq. and the marginals Let N while scaling noise variance appropriately Assumptions on noise distribution as N : g N δ(v) Var(g N ) = σ2 N i.e. MSD(g N) = O( 1 N ) Goal: find eqs. for the marginals as N and t = O( 1 N ) (continuous time limit) 2
Master eq: methodology 21 Take any observable φ(v 1,..., v N ) Denote Z N (t n ) = (v 1,...,v N )(t n ) the state of the system at time t n Markov transition operator Q φ(v 1,...,v N ) = E{φ(Z N (t n+1 )) Z N (t n ) = (v 1,...,v N )} Denote F N (v 1,..., v N ) = N-particle proba: E{φ(Z N (t n+1 ))} = φ F N (t n+1 )dz = (Q φ)f N (t n )dz F N (t n+1 ) = QF N (t n ) where Q = adjoint of Q
Example: CLD 22 Q φ(v 1,...,v N ) = 1 φ(v 1,...,wv j,...,v j,...,v N )g(w)dw N(N 1) S 1 i j QF N (v 1,...,v N ) = 1 g(v j vi N(N 1) i) i j S 1 F N (v 1,...,w i,...,v N )dw i
Example: BDG 23 Q φ(v 1,...,v N ) = 2 N(N 1) i<j { h( v i vj ) S 1 φ(v 1,...,v i,...,v j,...,v N )g(vijv i)g(v ijv j)dv i dv j } + (1 h( v i vj ))φ(v 1,...,v N ) with mid-direction v ij defined by v ij = (v i + v j )/ v i + v j
N in CLD 24 Small noise limit g N δ Var(g N ) = σ 2 /N t = O(1/N 2 ) First marginal: t f (1) (σ 2 /2) 2 θ 1 f (1) = 0 Second marginal: t f (2) (σ 2 /2) θ1,θ 2 f (2) +2f (2) = (f (1) (θ 1 )+f (1) (θ 2 ))δ(θ 2 θ 1 )
Stationary states as t 25 f (1) f (1) eq = 1: uniform distribution on S 1 f (2) f (2) eq the unique solution of (σ 2 /2) θ1,θ 2 f + 2f = 2δ(θ 2 θ 1 ) f (2) eq (θ 1,θ 2 ) f (1) eq (θ 1 )f (1) eq (θ 2 ) Chaos assumption violated f (2) eq peaked at θ 1 = θ 2 coherent motion but no preferred mean direction
Numerical simulations 26 Experimental protocol simulations with N = 10 2, 10 3, 10 4 & 10 5 particles wait until stationary state Pick one i and a pair (i,j) at random Redo the simulation M times to avoid correlations Plot histograms of θ 1 and (θ 1,θ 2 ) of these M samples Compare with theoretical f eq (1) and f eq (2)
(1) (2) feq & feq : experiments N = 103 27 σ=π σ = π/10 σ = π/100
(2) feq : experiments vs theory N = 103 28 σ=π σ = π/10 σ = π/100
N in BDG 29 Small noise limit and continuous time limit g N δ Var(g N ) = σ 2 /N t = O(1/N 2 ) Strong bias ( grazing collisions ) h N / h N δ Var(h N / h N ) = τ 2 /N Goal: in the limit N : Compare the relative influence of the noise σ and the grazing bias τ
Explicit hierarchy 30 t f (1) = (σ 2 τ 2 ) 2 θf (2) (θ,θ) θ=θ1 t f (2) = (σ 2 τ 2 )( 2 θf (3) (θ,θ 2,θ) θ=θ1 + 2 θf (3) (θ 1,θ,θ) θ=θ2 ). t f (j) = (σ 2 τ 2 ) j θf 2 (j+1) (θ 1,...,θ k 1,θ,θ k+1,...,θ j,θ) θ=θk k=1
Interpretation 31 If chaos assumption holds, f (1) (θ) satisfies t f = (σ 2 τ 2 ) (f 2 ) = 2(σ 2 τ 2 ) (f f ) nonlinear heat equation σ > τ: well-posed ; noise added wider than initial spread σ < τ: ill-posed ; noise added narrower: concentration? BUT: Chaos assumption does not hold Existence for hierarchy? infinitely many stationary states
32 5. Conclusion
Observations & Future work 33 Simple dynamics of aggregation do not satisfy chaos assumption How can kinetic theory survive this situation? Requires rethinking of classical concepts (entroypy, dissipation, irreversibility, equilibria,... ) Spatialization Kinetic & fluid models application to practical systems (swarming, trail formation, construction,... )