Weierstrass Institute for Applied Analysis and Stochastics Coagulation equations and particle systems

Transcription

1 Weierstrass Institute for Applied Analysis and Stochastics Coagulation equations and particle systems Wolfgang Wagner Mohrenstrasse Berlin Germany Tel WIAS workshop, February 2012

2 Plan of the talk 1. Introduction - Smoluchowski s coagulation equation - systems of Brownian particles 2. Spatially homogeneous particle models - direct simulation model - general interactions - asymptotic behavior - mass flow model 3. Phase transition - gelation - explosion Coagulation models WIAS workshop, February 2012 Page 2 (24)

3 Smoluchowski s coagulation equation d dt c(t, x) = 1 x 1 K(x y, y) c(t, x y) c(t, y) 2 y=1 K(x, y) c(t, x) c(t, y) y=1 - average number concentration c(t, x) t 0 x = 1, 2,... - coagulation kernel K(x, y) = ( ) ( ) D(x) + D(y) R(x) + R(y) (x 1/3 + y 1/3) ( x 1/3 + y 1/3) [M. v. Smoluchowski. Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen. Phys. Z., 17: , , 1916.] Coagulation models WIAS workshop, February 2012 Page 3 (24)

4 Convergence results - 1 c(t, r, x) = D(x) r c(t, r, x)+ t x 1 1 K(x y, y) c(t, r, x y) c(t, r, y) 2 y=1 K(x, y) c(t, r, x) c(t, r, y) y=1 n particles move in R 3 according to Brownian motion at distance ε (sphere of influence) they stick together consider N(t, B, x), t 0, B R 3, x = 1, 2,... let n so that n ε = const if 1 N(0, B, x) c(0, r, x) dr B n B R3, x = 1, 2,... then 1 N(t, B, x) c(t, r, x) dr t > 0, B R 3, x = 1, 2,... n B Coagulation models WIAS workshop, February 2012 Page 4 (24)

5 Convergence results - 2 c(t, r, x) = D(x) r c(t, r, x)+ t x 1 1 K(x y, y) c(t, r, x y) c(t, r, y) 2 y=1 K(x, y) c(t, r, x) c(t, r, y) y=1 [R. Lang and. Nguyen. Smoluchowski s theory of coagulation in colloids holds rigorously in the Boltzmann Grad limit. Z. Wahrsch. Verw. Gebiete, 54: , 1980.] D(x) = 1 R(x) = 1 K(x, y) = 1 special case (unlabeled particles), m 0(t, r) = x=1 c(t, r, x) t m0(t, r) = r m0(t, r) 1 m0(t, r)2 2 Coagulation models WIAS workshop, February 2012 Page 5 (24)

6 Convergence results - 3 c(t, r, x) = D(x) r c(t, r, x)+ t x 1 1 K(x y, y) c(t, r, x y) c(t, r, y) 2 y=1 K(x, y) c(t, r, x) c(t, r, y) y=1 K(x, y) = ( ) ( ) D(x) + D(y) R(x) + R(y) [J. R. Norris. Brownian coagulation. Commun. Math. Sci., 2(1):93 101, 2004.] D(x) = x 1/3 R(x) = x 1/3 [F. Rezakhanlou. The coagulating Brownian particles and Smoluchowski s equation. Markov Proc. Rel. Fields, 12: , 2006.] R(x) = x α α [0, 1) + spatial smoothing Coagulation models WIAS workshop, February 2012 Page 6 (24)

7 Spatially homogeneous case - direct simulation process Part 2 ( ) 1(t),..., N(t) (t) t 0 N(0) = n time evolution 1. distribution of the waiting time Prob(τ t) = exp( λ(z) t) where λ(z) = 1 n 1 i<j N K(x i, x j) z = (x 1,..., x N ) 2. distribution of the indices i, j K(x i, x j) 3. replace x i, x j by x i + x j and go to 1. [Marcus (1968), Gillespie (1972), Lushnikov (1978)] Coagulation models WIAS workshop, February 2012 Page 7 (24)

8 Spatially homogeneous case - general interactions ( ) 1(t),..., N(t) (t) t 0 N(0) = n state space of a single particle state space of the system Z = { } (x 1,..., x N ) N : N 0 interactions x i1,..., x il (z 1,..., z k ) = z Z intensities q l (x 1,..., x l, dz) l = 0, 1,..., L Coagulation models WIAS workshop, February 2012 Page 8 (24)

9 Asymptotic behavior empirical measures µ (n) (t, dx) = 1 N(t) δ i (t)(dx) n i=1 t 0 scaling of intensities q 0, q 1,..., q L probability distributions P (n) P(D([0, ), M + b ( ))) Theorem Assumptions on q 0,..., q L and µ (n) (0, dx) f 0(dx) for some f 0 M + b ( )). Then (P (n) ) is relatively compact accumulation points are concentrated on solutions of ( ) Coagulation models WIAS workshop, February 2012 Page 9 (24)

10 Limiting equation ϕ(x) f(t, dx) = ϕ(x) f 0(dx) + t 0 G(ϕ, f(s)) ds t 0 ( ) test functions ϕ C c( ) nonlinear operator G(ϕ, ν) = Z Z [ ] ϕ(z 1) ϕ(z k ) q 0(dz) + L l=1 [ ϕ(z 1) ϕ(z k ) ϕ(x 1)... ϕ(x l ) ν(dx 1)... ν(dx l ) ] q l (x 1,..., x l, dz) Coagulation models WIAS workshop, February 2012 Page 10 (24)

11 Analytical implications Corollary 1 (existence) For each P = lim k P (nk), P ({f : ( ) holds}) = 1. Coagulation models WIAS workshop, February 2012 Page 11 (24)

12 Analytical implications Corollary 1 (existence) For each P = lim k P (nk), P ({f : ( ) holds}) = 1. Corollary 2 (convergence) Assume uniqueness. P ({f}) = 1 P = lim k P (n k) lim n P (n) = δ f lim n µ (n) (t, dx) = f(t, dx) t 0 Coagulation models WIAS workshop, February 2012 Page 11 (24)

13 Example 1 - bimolecular reactions ϕ(x) f(t, dx) = ϕ(x) f 0(dx) + t 0 G(ϕ, f(s)) ds ( ) G(ϕ, ν) = ν(dx 1) [ ] ν(dx 2) ϕ(z 1) ϕ(z k ) ϕ(x 1) ϕ(x 2) q 2(x 1, x 2, dz) Z state space = {1} q 2 = 1 1, 1 1 ( unlabeled coagulation) 1, 1 d dt c(t) = c(t)2 c(t) = t ( annihilating coagulation) d c(t) = 2 c(t)2 dt Coagulation models WIAS workshop, February 2012 Page 12 (24)

14 Example 2 - direct simulation process state space = {1, 2,...} or = (0, ) coagulation jumps x i1, x i2 z 1 intensity q 2(x 1, x 2, dz) = 1 2 K(x1, x2) δx 1+x 2 (dz) nonlinear operator G(ϕ, ν) = 1 [ ] ν(dx 1) ν(dx 2) ϕ(x 1 + x 2) ϕ(x 1) ϕ(x 2) K(x 1, x 2) 2 Smoluchowski s coagulation equation (weak form) d ϕ(x) f(t, dx) = G(ϕ, f(t)) dt Coagulation models WIAS workshop, February 2012 Page 13 (24)

15 Another stochastic coagulation model ( 1(t),..., n(t)) t 0 Ñ(t) = Ñ(0) = n time evolution 1. distribution of the waiting time Prob( τ t) = exp( λ(z) t) where λ(z) = 1 n 1 i, j n K(x i, x j) x j z = (x 1,..., x n) 2. distribution of the indices i, j K(xi, xj) x j 3. replace x i by x i + x j and go to 1. Coagulation models WIAS workshop, February 2012 Page 14 (24)

16 Another stochastic coagulation model - 2 state space = {1, 2,...} or = (0, ) coagulation jumps intensity nonlinear operator G(ϕ, ν) = q 2(x 1, x 2, dz) = x i1, x i2 z 1, z 2 K(x1, x2) x 2 δ x1 +x 2 (dz 1) δ x2 (dz 2) [ ] K(x1, x 2) ν(dx 1) ν(dx 2) ϕ(x 1 + x 2) ϕ(x 1) x 2 limiting equation d ϕ(x) dt f(t, dx) = G(ϕ, f(t)) Coagulation models WIAS workshop, February 2012 Page 15 (24)

17 Equivalence of limiting equations first model second d ϕ(x) f(t, dx) = G(ϕ, f(t)) dt G(ψ, ν) = = = = 1 2 ν(dx 1) ν(dx 1) x 1 ν(dx 1) x 1 ν(dx 1) x 1 model d ψ(x) f(t, dx) = G(ψ, f(t)) dt [ ] K(x1, x 2) ν(dx 2) ψ(x 1 + x 2) ψ(x 1) x 2 ν(dx [ ] 2) ψ(x 1 + x 2) ψ(x 1) x 1 K(x 1, x 2) x 2 ν(dx [ ] 2) ψ(x 1 + x 2) ψ(x 2) x 2 K(x 1, x 2) x 2 ν(dx 2) x 2 K(x 1, x 2) [ (x 1 + x 2) ψ(x 1 + x 2) x 1 ψ(x 1) x 2 ψ(x 2) ] = G(x ψ, ν/x) Coagulation models WIAS workshop, February 2012 Page 16 (24)

18 Equivalence of limiting equations first model second d ϕ(x) f(t, dx) = G(ϕ, f(t)) dt G(ψ, ν) = = = = 1 2 ν(dx 1) ν(dx 1) x 1 ν(dx 1) x 1 so that ϕ(x) = x ψ(x) and ν(dx 1) x 1 model d ψ(x) f(t, dx) = G(ψ, f(t)) dt [ ] K(x1, x 2) ν(dx 2) ψ(x 1 + x 2) ψ(x 1) x 2 ν(dx [ ] 2) ψ(x 1 + x 2) ψ(x 1) x 1 K(x 1, x 2) x 2 ν(dx [ ] 2) ψ(x 1 + x 2) ψ(x 2) x 2 K(x 1, x 2) x 2 ν(dx 2) x 2 K(x 1, x 2) [ (x 1 + x 2) ψ(x 1 + x 2) x 1 ψ(x 1) x 2 ψ(x 2) f(t, dx) = x f(t, dx) mass flow model ] = G(x ψ, ν/x) Coagulation models WIAS workshop, February 2012 Page 16 (24)

19 Explicit solutions for special kernels Part 3 Smoluchowski s coagulation equation d dt c(t, x) = 1 x 1 K(x y, y) c(t, x y) c(t, y) 2 y=1 K(x, y) c(t, x) c(t, y) y=1 K(x, y) m 0(t) m 1(t) m 2(t) t t x + y exp( t) 1 exp(2 t) m k (t) = x k c(t, x) x=1 t 0 k = 0, 1, 2,... c(0, x) = δ 1,x Coagulation models WIAS workshop, February 2012 Page 17 (24)

20 Explicit solutions for special kernels Part 3 Smoluchowski s coagulation equation d dt c(t, x) = 1 x 1 K(x y, y) c(t, x y) c(t, y) 2 y=1 K(x, y) c(t, x) c(t, y) y=1 K(x, y) m 0(t) m 1(t) m 2(t) t t x + y exp( t) 1 exp(2 t) m k (t) = x k c(t, x) x=1 t 0 k = 0, 1, 2,... c(0, x) = δ 1,x K(x, y) const (x + y) unique solution + mass conservation Coagulation models WIAS workshop, February 2012 Page 17 (24)

21 Gelation K(x, y) = x y t [0, 1) [1, ) m 0(t) 1 t 2 m 1(t) 1 m 2(t) 1 1 t 1 2t 1 t m k (t) = x k c(t, x) x=1 c(0, x) = δ 1,x Coagulation models WIAS workshop, February 2012 Page 18 (24)

22 Gelation K(x, y) = x y t [0, 1) [1, ) m 0(t) 1 t 2 m 1(t) 1 m 2(t) 1 1 t 1 2t 1 t m k (t) = x k c(t, x) x=1 c(0, x) = δ 1,x gelation time { } t gel = inf t 0 : m 1(t) < m 1(0) Coagulation models WIAS workshop, February 2012 Page 18 (24)

23 Gelling coagulation kernels - heuristics a homogeneous kernel K(α x, α y) = α ε K(x, y) α > 0 is gelling, if ε > 1 (homogeneity exponent) - rigorous results if K(x, y) C (x y) a for some a > 0.5, C > 0, and K(x, y) = o(x y), if then there exist gelling solutions [Jeon (1998)] ( K(x, y) = C x a y b + x b y a) for some a, b [0, 1], a + b > 1, C > 0, then every weak solution is gelling [Escobedo/Mischler/Perthame (2002)] Coagulation models WIAS workshop, February 2012 Page 19 (24)

24 Gelation in stochastic models Y - locally compact separable metric q(y, dỹ) - compactly bounded kernel Y 0, Y 1, Y 2,... - Markov chain λ(y 0), λ(y 1), λ(y 2),... - waiting time parameters τ 1 < τ 2 < τ 3 <... - jump times 1 q(y, dỹ), λ(y) = q(y, Y) λ(y) explosion time τ = lim l τ l minimal jump process process on Y { } (one-point compactification) { Y Y l : τ l t < τ l+1 (t) = : t τ Coagulation models WIAS workshop, February 2012 Page 20 (24)

25 Regularity and explosion - recall kernel q(y, dỹ) λ(y) = q(y, Y) τ = lim l τ l Markov chain (Y k ) - regular: Prob(τ = ) = 1 explosive: Prob(τ < ) > 0 criterion regularity k=0 1 λ(y k ) = a.s. λ bounded regularity explosion λ unbounded & Y k with appropriate speed Coagulation models WIAS workshop, February 2012 Page 21 (24)

26 Direct simulation process ( ) 1(t),..., N(t) (t) t 0 N(0) = n Y = n N=1 N = {1, 2,...} or (0, ) y = (x 1,..., x N ) λ(y) = 1 2n 1 i j N K(xi, xj) xi, xj xi + xj N(t) and i(t) bounded no explosion Coagulation models WIAS workshop, February 2012 Page 22 (24)

27 Direct simulation process ( ) 1(t),..., N(t) (t) t 0 N(0) = n Y = n N=1 N = {1, 2,...} or (0, ) y = (x 1,..., x N ) λ(y) = 1 2n 1 i j N K(xi, xj) xi, xj xi + xj N(t) and i(t) bounded no explosion largest component for K(x, y) = x y M (n) 1 (t) log n, if t < 1 n 2 3, if t = 1 n, if t > 1 random graph theory [Erdös/Rényi (1960), Bollobás (1985, 2001)] Coagulation models WIAS workshop, February 2012 Page 22 (24)

28 Mass flow process ( 1(t),..., n(t)) t 0 Ñ(t) = Ñ(0) = n Y = n = {1, 2,...} or (0, ) y = (x 1,..., x n) λ(y) = 1 n K(x i,x j ) n i,j=1 x j x i, x j x i + x j, x j Ñ(t) but i(t) unbounded Coagulation models WIAS workshop, February 2012 Page 23 (24)

29 Mass flow process ( 1(t),..., n(t)) t 0 Ñ(t) = Ñ(0) = n Y = n = {1, 2,...} or (0, ) y = (x 1,..., x n) λ(y) = 1 n K(x i,x j ) n i,j=1 x j x i, x j x i + x j, x j Ñ(t) but i(t) unbounded Theorem Assume K(x, y) K(x, y) where K(1, 1) > 0 and for some ε > 1. K(α x, α y) = α ε K(x, y) α > 0 Then the mass flow process explodes almost surely, for any initial distribution. Coagulation models WIAS workshop, February 2012 Page 23 (24)

30 Comments analytical aspects general tool for proving existence numerical aspects original motivation for MF applications in chemical engineering stochastic aspects explosion in finite particle systems Coagulation models WIAS workshop, February 2012 Page 24 (24)

31 Comments analytical aspects general tool for proving existence numerical aspects original motivation for MF applications in chemical engineering stochastic aspects explosion in finite particle systems conjecture t gel = lim n τ (n) Coagulation models WIAS workshop, February 2012 Page 24 (24)

32 Comments analytical aspects general tool for proving existence numerical aspects original motivation for MF applications in chemical engineering stochastic aspects explosion in finite particle systems challenges conjecture t gel = lim n τ (n) - include transport terms (boundary conditions) talk Patterson - gelation in the spatial setting - rigorous derivation of coagulation kernels Coagulation models WIAS workshop, February 2012 Page 24 (24)