Alignment processes on the sphere

Transcription

1 Alignment processes on the sphere Amic Frouvelle CEREMADE Université Paris Dauphine Joint works with : Pierre Degond (Imperial College London) and Gaël Raoul (École Polytechnique) Jian-Guo Liu (Duke University) Young Researchers Workshop: Stochastic and deterministic methods in kinetic theory Duke University, November 8th December nd, 016 Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

2 Context: alignment of self-propelled particles Unit speed, local interactions without leader Emergence of patterns Purpose of this talk : alignment mechanisms of (kinetic) Vicsek and BDG models. Focus on the alignment process only : no noise, no space! Images Benson Kua (ickr) and Amic Frouvelle Alignment processes on the sphere Nov.-Dec. 016 /4

3 Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

4 Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

5 Toy model in R n Velocities v i (t) R n, for 1 i N. Each velocity is attracted by the others, with strengths m j such that N j=1 m j = 1. N dv i dt = m j (v j v i ) = J v i where J = j=1 N m i v i. i=1 Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

6 Toy model in R n Velocities v i (t) R n, for 1 i N. Each velocity is attracted by the others, with strengths m j such that N j=1 m j = 1. N dv i dt = m j (v j v i ) = J v i where J = j=1 Conservation of momentum : J is constant. We get N m i v i. v i (t) = J + e t a i, with a i = v i (0) J R n and N i=1 m i a i = 0. i=1 Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

7 Toy model in R n Velocities v i (t) R n, for 1 i N. Each velocity is attracted by the others, with strengths m j such that N j=1 m j = 1. N dv i dt = m j (v j v i ) = J v i where J = j=1 Conservation of momentum : J is constant. We get N m i v i. v i (t) = J + e t a i, with a i = v i (0) J R n and N i=1 m i a i = 0. Gradient ow structure If E = 1 i,j m i m j v i v j, then E = i m i v i J. dv Then vi E = m i de i, and so = dt dt i m i dv i dt = E. i=1 Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

8 Coupled nonlinear ordinary dierential equations Velocities v i (t) S, the unit sphere of R n, for 1 i N. Each velocity is attracted by the others, with strengths m j such that N j=1 m j = 1, under the constraint that it stays on the sphere. If v S, we denote P v the orthogonal projection on the tangent space of S at v. So P v u = u (v u)v, for any u R n. dv i dt = P v i N j=1 m j (v j v i ) = P v J where J = i N m i v i. i=1 We indeed get d v i dt = v i P v J = 0. i No more conservation here! Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

9 Same gradient ow structure Dene as in the toy model : E = 1 N N m i m j v i v j = m i m j (1 v i v j ) = 1 J 0. i,j=1 i,j=1 Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

10 Same gradient ow structure Dene as in the toy model : E = 1 N m i m j v i v j = i,j=1 N m i m j (1 v i v j ) = 1 J 0. i,j=1 Notice that we have v (v u) = P v u, (for v S, u R n, and where v is the gradient on the sphere). So vi E = N j=1 dv i m i m j P v v j = m i i dt. Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

11 Same gradient ow structure Dene as in the toy model : E = 1 N m i m j v i v j = i,j=1 N m i m j (1 v i v j ) = 1 J 0. i,j=1 Notice that we have v (v u) = P v u, (for v S, u R n, and where v is the gradient on the sphere). So de dt ( = d J dt vi E = ) = N i=1 N j=1 dv i m i m j P v v j = m i i dt. m i dv i dt = N m i ( J (v i J) ) 0. Then J is increasing, so if J(0) 0, then J(t) 0 for all t, and Ω(t) = J(t) J(t) is well-dened. Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4 i=1

12 Convergence, relatively to Ω 1 d J dt N = J m i (1 (v i Ω) ) 0. i=1 Hence J is increasing, bounded and with bounded second derivative (we compute it and get that everything is continuous on S). Therefore its derivative must converge to 0 as t +. Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

13 Convergence, relatively to Ω 1 d J dt N = J m i (1 (v i Ω) ) 0. i=1 Hence J is increasing, bounded and with bounded second derivative (we compute it and get that everything is continuous on S). Therefore its derivative must converge to 0 as t +. Front or Back particles v i (t) Ω(t) ±1 as t +, for 1 i N. Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

14 Convergence, relatively to Ω 1 d J N = J m i (1 (v i Ω) ) 0. dt i=1 Hence J is increasing, bounded and with bounded second derivative (we compute it and get that everything is continuous on S). Therefore its derivative must converge to 0 as t +. Front or Back particles v i (t) Ω(t) ±1 as t +, for 1 i N. Convergence of Ω(t)? dω dt = P Ω MΩ, with M(t) = N i=1 m i P v i (a n n matrix). Easy to show that dω dt is L in time, but L 1?... Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

15 Case with all v i at the front If v i Ω 1 for all i, then J = i m i v i Ω 1. We have 1 d dt v i v j = J Ω vi + v j v i v j. Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

16 Case with all v i at the front If v i Ω 1 for all i, then J = i m i v i Ω 1. We have 1 d dt v i v j = J Ω vi + v j v i v j. Exponential estimates The quantities (1 v i v j ) = 1 v i v j, 1 J, 1 v i Ω and P v Ω = 1 (v i Ω) are all bounded by Ce t. i Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

17 Case with all v i at the front If v i Ω 1 for all i, then J = i m i v i Ω 1. We have 1 d dt v i v j = J Ω vi + v j v i v j. Exponential estimates The quantities (1 v i v j ) = 1 v i v j, 1 J, 1 v i Ω and P v Ω = 1 (v i Ω) are all bounded by Ce t. i We then obtain dω N dt = m i (1 v i Ω)P Ω v i = O(e 3t ), i=1 which gives that Ω(t) Ω S. Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

18 Asymptotic behaviour of each v i After some computations, using the previous estimates, we get d v i dt = dv i dt dv i dt v i + O(e 3t ). This gives that dv i = a dt i e t + O(e t ) for some a i R n. Using it with the expression above, we obtain a better estimate for v i. Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

19 Asymptotic behaviour of each v i After some computations, using the previous estimates, we get d v i dt = dv i dt dv i dt v i + O(e 3t ). This gives that dv i = a dt i e t + O(e t ) for some a i R n. Using it with the expression above, we obtain a better estimate for v i. Theorem (if all particles are front): There exists Ω S, and a i {Ω } R n, for 1 i N such that N i=1 m i a i = 0 and that, as t +, v i (t) = (1 a i e t ) Ω + e t a i + O(e 3t ) for 1 i N, Ω(t) = Ω + O(e 3t ). We see that Ω(t) acts as a nearly conserved quantity, and we recover results asymptotically similar to the case of the toy model. Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

20 Only one can nish at the back We denote λ > 0 the limit of J (increasing). Recall that 1 d dt v i v j = J vi + v j v i v j. If v i (0) v j (0), we cannot have v i Ω 1 and v j Ω 1 (repulsion). Up to renumbering, only v N is going to the back. We then get J N 1 i=1 m i m N = 1 m N = λ (so m N < 1 ). Same as before: v i v j = O(e λt ) (for i, j N). Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

21 Only one can nish at the back We denote λ > 0 the limit of J (increasing). Recall that 1 d dt v i v j = J vi + v j v i v j. If v i (0) v j (0), we cannot have v i Ω 1 and v j Ω 1 (repulsion). Up to renumbering, only v N is going to the back. We then get J N 1 i=1 m i m N = 1 m N = λ (so m N < 1 ). Same as before: v i v j = O(e λt ) (for i, j N). Convexity argument If all v i are in the same hemisphere, they stay in it. Therefore v N Cone((v i ) 1 i<n ). We then get the same estimates (with e t replaced by e λt, except the one in O(e 3t ) for dω dt, since 1 v N Ω does not converge to 0). Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

22 Faster convergence of v N After some computations, we get d v N dt = dv N dt dv N dt v N + O(e 3λt ). Therefore we get d dt dv N dt = dv N dt + O(e 3λt ). Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

23 Faster convergence of v N After some computations, we get d v N dt = dv N dt dv N dt v N + O(e 3λt ). Therefore we get d dt dv N dt = dv N dt + O(e 3λt ). Lemma: Tail of a perturbed ODE (exponential) If x C 1 (R), such that dx dt = x + O(e αt ), with α > 0. If x is uniformly bounded, then x(t) = O(e αt ). Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

24 Faster convergence of v N After some computations, we get d v N dt = dv N dt dv N dt v N + O(e 3λt ). Therefore we get d dt dv N dt = dv N dt + O(e 3λt ). Lemma: Tail of a perturbed ODE (exponential) If x C 1 (R), such that dx dt = x + O(e αt ), with α > 0. If x is uniformly bounded, then x(t) = O(e αt ). Therefore dv N dt = O(e 3λt ). This gives that P Ω v N = 1 J dv N dt = O(e 3λt ). Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

25 Asymptotic behaviour of each v i Same method to get dω = dt O(e 3λt ), except for the term in (1 v N Ω)P Ω v N for which we use the previous analysis. We then get the same kind of asymptotic expansions. Theorem (if v N is the only back particle): There exists Ω S, and a i {Ω } R n, for 1 i < N such that N 1 i=1 m i a i = 0 and that, as t +, v i (t) = (1 a i e λt ) Ω + e λt a i + O(e 3λt ) for i N, v N (t) = Ω + O(e 3λt ), Ω(t) = Ω + O(e 3λt ). Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

26 Aggregation equation on the sphere PDE for the empirical distribution Dene f (t) = N i=1 δ vi (t) P(S) (a probability measure on the sphere), then f is a weak solution of the following PDE: t f + v (f P v J f ) = 0, where J f = S v df (v). (1) Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

27 Aggregation equation on the sphere PDE for the empirical distribution Dene f (t) = N i=1 δ vi (t) P(S) (a probability measure on the sphere), then f is a weak solution of the following PDE: t f + v (f P v J f ) = 0, where J f = S v df (v). (1) Theorem Given a probability measure f 0, there exists a unique global (weak) solution to the aggregation equation given by (1). Tools: optimal transport, or for this case harmonic analysis (Fourier for n = ), which gives well-posedness in Sobolev spaces. Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

28 Properties of the model Characteristics Suppose that the function J(t) is given, we dene Φ t (v) the solution of the following ODE: dφ t (v) dt = P Φt(v) J(t) with Φ 0(v) = v. Then the solution to the (linear) equation t f + v (f P v J) = 0 is given by f (t) = Φ t #f 0 (push-forward): if ψ C 0 (S), then S ψ(v)d(φ t#f 0 )(v) = S ψ(φ t(v))df 0 (v). Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

29 Properties of the model Characteristics Suppose that the function J(t) is given, we dene Φ t (v) the solution of the following ODE: dφ t (v) dt = P Φt(v) J(t) with Φ 0(v) = v. Then the solution to the (linear) equation t f + v (f P v J) = 0 is given by f (t) = Φ t #f 0 (push-forward): if ψ C 0 (S), then S ψ(v)d(φ t#f 0 )(v) = S ψ(φ t(v))df 0 (v). We can perform computations exactly as for the particles: d dt J f = P v J f df (v) = P v f J f = M f J f. S Increase of J f (t), integrability of J f M f J f. All moments are C. Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

30 Convergence of Ω(t) We have Ω = dω dt = P Ω M f Ω, which is L in time, but L 1? After a few computations, we obtain where u = d dt Ω = Ω ( 1 Ω M f Ω (u P Ω v) ) f + J (1 (v Ω) )u P Ω v f, Ω Ω (when it is well-dened, 0 otherwise). Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

31 Convergence of Ω(t) We have Ω = dω dt = P Ω M f Ω, which is L in time, but L 1? After a few computations, we obtain where u = d dt Ω = Ω ( 1 Ω M f Ω (u P Ω v) ) f + J (1 (v Ω) )u P Ω v f, Ω Ω (when it is well-dened, 0 otherwise). Lemma: Tail of a perturbed ODE (integrability) If dx dt = x + g where x is bounded, g L1 (R + ), then x(t) L 1 (R + ). Therefore we get that Ω is integrable, and then Ω(t) Ω S. Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

32 Convergence of f Proposition: unique back Suppose that J(t) R n \ {0} is given (and continuous), with Ω(t) = J(t) J(t) converging to Ω S. Then there exists a unique v back S such that the solution v(t) of dv = P dt v J(t) with v(0) = v back satises v(t) Ω as t +. Conversely, if v(0) v back, then v(t) Ω as t. Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

33 Convergence of f Proposition: unique back Suppose that J(t) R n \ {0} is given (and continuous), with Ω(t) = J(t) J(t) converging to Ω S. Then there exists a unique v back S such that the solution v(t) of dv = P dt v J(t) with v(0) = v back satises v(t) Ω as t +. Conversely, if v(0) v back, then v(t) Ω as t. Theorem Convergence in Wasserstein distance to mδ Ω + (1 m)δ Ω, where m is the mass of {v b } with respect to the measure f 0. In particular, if f 0 has no atoms, then f δ Ω. No rate... Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

34 Collisional kinetic model Stability of Dirac masses Toy model in R n : midpoint collision (sticky particles) Speeds v i R n (1 i N), Poisson clocks: v i, v j v i +v j. Kinetic version (large N) : evolution of f t (v) P (R n ) t f t (v) = f t (v + w)f t (v w)dw f t (v). R n Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

35 Collisional kinetic model Stability of Dirac masses Toy model in R n : midpoint collision (sticky particles) Speeds v i R n (1 i N), Poisson clocks: v i, v j v i +v j. Kinetic version (large N) : evolution of f t (v) P (R n ) t f t (v) = (v) df t (v )df t (v ) f (t, v). R n R n δ v +v Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

36 Collisional kinetic model Stability of Dirac masses Toy model in R n : midpoint collision (sticky particles) Speeds v i R n (1 i N), Poisson clocks: v i, v j v i +v j. Kinetic version (large N) : evolution of f t (v) P (R n ) t f t (v) = (v) df t (v )df t (v ) f (t, v). R n R n δ v +v Conservation of center of mass v = Rn v df (v). Second moment m = R n v v d df (v): m dt = m. Exponential convergence towards a Dirac mass W (f t, δ v ) = W (f 0, δ v )e t 4. Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

37 Collisional kinetic model Stability of Dirac masses Toy model in R n : midpoint collision (sticky particles) Speeds v i R n (1 i N), Poisson clocks: v i, v j v i +v j. Kinetic version (large N) : evolution of f t (v) P (R n ) t f t (v) = (v) df t (v )df t (v ) f (t, v). R n R n δ v +v Conservation of center of mass v = Rn v df (v). Second moment m = R n v v d df (v): m dt = m. Exponential convergence towards a Dirac mass W (f t, δ v ) = W (f 0, δ v )e t 4. Decreasing energy: E(f ) = R n R n v u df (v) df (u) (equal to m ), no need for v in this denition. Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

38 Midpoint model on the sphere Collisional kinetic model Stability of Dirac masses Kernel K(v, v, v ): probability density that a particle at position v interacting with another one at v is found at v after collision. t f t (v) = S S K(v, v, v ) df t (v )df t (v ) f t (v). v v m When v v, K(, v, v ) = δ vm, where v m = v +v v +v. v Supp K(, v, v ) v Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

39 Midpoint model on the sphere Collisional kinetic model Stability of Dirac masses Kernel K(v, v, v ): probability density that a particle at position v interacting with another one at v is found at v after collision. t f t (v) = S S K(v, v, v ) df t (v )df t (v ) f t (v). v v m When v v, K(, v, v ) = δ vm, where v m = v +v v +v. v Supp K(, v, v ) v Energy E(f ) = S S d(v, u) df (v) df (u). Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

40 Link Energy Wasserstein Collisional kinetic model Stability of Dirac masses Useful Lemma Markov inequalities For f P(S), there exists v S such that for all v S: W (f, δ v ) E(f ) 4 W (f, δ v ), For such a v and for all κ > 0, we have df (v) 1 E(f ), κ {v S; d(v, v) κ} {v S; d(v, v) κ} d(v, v) df (v) 1 E(f ). κ Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

41 Evolution of the energy Collisional kinetic model Stability of Dirac masses 1 d dt S S S E(f ) = α(v, v, u) df (v ) df (v ) df (u). Local contribution to the variation of energy α(v, v, u) = d(v, u) K(v, v, v ) dv d(v, u) + d(v, u). S Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

42 Evolution of the energy Collisional kinetic model Stability of Dirac masses 1 d dt S S S E(f ) = α(v, v, u) df (v ) df (v ) df (u). Local contribution to the variation of energy α(v, v, u) = d(v, u) K(v, v, v ) dv d(v, u) + d(v, u). S v a v m a Conguration of Apollonius: b m b v α(v, v, u) = m b + b. u Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

43 Evolution of the energy Collisional kinetic model Stability of Dirac masses 1 d dt S S S E(f ) = α(v, v, u) df (v ) df (v ) df (u). Local contribution to the variation of energy α(v, v, u) = d(v, u) K(v, v, v ) dv d(v, u) + d(v, u). S v a v m a Conguration of Apollonius: b m b v α(v, v, u) = m b + b. u Flat case (we get a ): α(v, v, u) = 1 4 d(v, v ). Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

44 Collisional kinetic model Stability of Dirac masses Error estimates in Apollonius' formula Lemma: global estimate (only triangular inequalities) For all v, v, u S, we have α(v, v, u) 1 4 d(v, v ) + d(v, v ) min ( d(v, u), d(v, u) ). Amic Frouvelle Alignment processes on the sphere Nov.-Dec. 016 /4

45 Collisional kinetic model Stability of Dirac masses Error estimates in Apollonius' formula Lemma: global estimate (only triangular inequalities) For all v, v, u S, we have α(v, v, u) 1 4 d(v, v ) + d(v, v ) min ( d(v, u), d(v, u) ). Lemma: local estimate, more precise For all κ 1 < π 3, there exists C 1 > 0 such that for all κ κ 1, and all v, v, u S with max(d(v, u), d(v, u), d(v, v )) κ, we have α(v, v, u) 1 4 d(v, v ) + C 1 κ d(v, v ). Spherical Apollonius: 1 (cos b + cos b ) = cos a cos m. Amic Frouvelle Alignment processes on the sphere Nov.-Dec. 016 /4

46 Collisional kinetic model Stability of Dirac masses Decreasing energy Control on displacement We set ω := {v S; d(v, v) 1 κ}, and we cut the triple integral in four parts following if v, v, u is in ω or not. 1 d dt E(f ) E(f ) C κ E(f ) + 1 E(f ) }{{} κ Local lemma E(f ) κ }{{} Global lemma + Markov (and Cauchy-Schwarz). Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

47 Collisional kinetic model Stability of Dirac masses Decreasing energy Control on displacement We set ω := {v S; d(v, v) 1 κ}, and we cut the triple integral in four parts following if v, v, u is in ω or not. 1 d dt E(f ) E(f ) C κ E(f ) + 1 E(f ) }{{} κ Local lemma E(f ) κ }{{} Global lemma + Markov (and Cauchy-Schwarz) Same kind of cut to show that v(t) satises Cauchy's criteria (for t ) and converges with the same rate as E(f ).. Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

48 Collisional kinetic model Stability of Dirac masses Decreasing energy Control on displacement We set ω := {v S; d(v, v) 1 κ}, and we cut the triple integral in four parts following if v, v, u is in ω or not. 1 d dt E(f ) E(f ) C κ E(f ) + 1 E(f ) }{{} κ Local lemma E(f ) κ }{{} Global lemma + Markov (and Cauchy-Schwarz) Same kind of cut to show that v(t) satises Cauchy's criteria (for t ) and converges with the same rate as E(f ). Theorem: local stability of Dirac masses There exists C 1 > 0 and η > 0 such that for all solution f C(R +, P(S)) with initial condition f 0 satisfying W (f 0, δ v0 ) < η for a v 0 S, there exists v S such that W (f t, δ v ) C 1 W (f 0, δ v0 ) e 1 4 t.. Amic Frouvelle Alignment processes on the sphere Nov.-Dec /4

49 Thanks!