Asymptotics of the Trapping Reaction

Transcription

1 Asymptotics of the Trapping Reaction Alan Bray Isaac Newton Institute 20 April 2006 Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

2 COWORKERS Richard Blythe Satya Majumdar Lucian Anton Richard Smith Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

3 Tout le malheur des hommes vient d une seule chose, qui est de ne savoir pas demeurer en repos, dans une chambre. (Pascal, Pensées, fragment 139 (1670)). Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

4 Tout le malheur des hommes vient d une seule chose, qui est de ne savoir pas demeurer en repos, dans une chambre. (Pascal, Pensées, fragment 139 (1670)). All the misfortune of man comes from the fact that he does not stay peacefully in his room. Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

5 INTRODUCTION The trapping reaction: A + B B (Ovchinnikov and Zeldovich (1978)) with diffusion constants D A, D B. Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

6 INTRODUCTION The trapping reaction: A + B B (Ovchinnikov and Zeldovich (1978)) with diffusion constants D A, D B. The two-species annihilation reaction: A + B 0 (Toussaint and Wilczek (1983)) with initial concentrations ρ A (0) < ρ B (0). Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

7 INTRODUCTION The trapping reaction: A + B B (Ovchinnikov and Zeldovich (1978)) with diffusion constants D A, D B. The two-species annihilation reaction: A + B 0 (Toussaint and Wilczek (1983)) with initial concentrations ρ A (0) < ρ B (0). Rate equation approach (trapping reaction): dρ A dt = λρ A ρ B Q(t) ρ A(t) ρ A (0) = exp( λρ Bt) Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

8 However, spatial fluctuations are important for d 2 (reactions are diffusion-limited). Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

9 However, spatial fluctuations are important for d 2 (reactions are diffusion-limited). Exact asymptotic forms (Bramson and Lebowitz 1988): exp( λ d t d/2 ), d < 2 Q(t) exp( λ 2 t/ ln t), d = 2 exp( λ d t), d > 2 Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

10 However, spatial fluctuations are important for d 2 (reactions are diffusion-limited). Exact asymptotic forms (Bramson and Lebowitz 1988): exp( λ d t d/2 ), d < 2 Q(t) exp( λ 2 t/ ln t), d = 2 exp( λ d t), d > 2 Our primary goals are 1 To determine exactly the coefficients λ d for d 2. 2 To determine the leading corrections to the asymptotic forms. 3 To investigate the spatial fluctuations of surviving A-particles. Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

11 UPPER and LOWER BOUNDS ON Q(t) (AB and Blythe (2002,2003)) Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

12 UPPER and LOWER BOUNDS ON Q(t) (AB and Blythe (2002,2003)) The Upper Bound (d=1) ( Pascal Principle ) ( Q(t) Q U (t) = Q TAP (t) = exp 4 ) ρ B (D B t) 1/2 π where Q TAP (t) is the result for the Target Annihilation Problem, corresponding to D A = 0 (static A-particle with mobile traps). Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

13 UPPER and LOWER BOUNDS ON Q(t) (AB and Blythe (2002,2003)) The Upper Bound (d=1) ( Pascal Principle ) ( Q(t) Q U (t) = Q TAP (t) = exp 4 ) ρ B (D B t) 1/2 π where Q TAP (t) is the result for the Target Annihilation Problem, corresponding to D A = 0 (static A-particle with mobile traps). For general dimensions d 2 the results have the Bramson-Lebowitz forms { exp( λ d t d/2 ), d < 2 Q U (t) = exp( λ 2 t/ ln t), d = 2 with λ d = { 2 πd sin ( ) πd 2 (4πDB ) d/2 ρ B, d < 2 4πD B ρ B, d = 2 Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

14 The Target Annihilation Problem (d=1) Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

15 The Target Annihilation Problem (d=1) A B-particle (trap), starting at x, has not yet reached the target (located at x = 0) at time t with probability ( ) x q(t) = erf 4DB t Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

16 The Target Annihilation Problem (d=1) A B-particle (trap), starting at x, has not yet reached the target (located at x = 0) at time t with probability ( ) x q(t) = erf 4DB t Averaging over the initial position x, uniformly in the interval( L, L), gives q = 1 1 L ( ) x dx erfc = 1 1 2L L 4DB t L 4 π DB t Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

17 The Target Annihilation Problem (d=1) A B-particle (trap), starting at x, has not yet reached the target (located at x = 0) at time t with probability ( ) x q(t) = erf 4DB t Averaging over the initial position x, uniformly in the interval( L, L), gives q = 1 1 L ( ) x dx erfc = 1 1 2L L 4DB t L 4 π DB t The probability that none of N traps has reached the target is ( Q TAP (t) = N ( DB t) exp 4 ) ρ B DB t L π π (in the limit N, L with ρ B = N/L fixed). Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

18 The Lower Bound (d=1) Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

19 The Lower Bound (d=1) Create a fictitious box with edges at x = ±l/2, and consider the subset of trajectories in which 1 There are no B-particles initially inside the box. 2 The A-particle stays in the box up to time t. 3 No B-particles enter the box up to time t. Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

20 The Lower Bound (d=1) Create a fictitious box with edges at x = ±l/2, and consider the subset of trajectories in which 1 There are no B-particles initially inside the box. 2 The A-particle stays in the box up to time t. 3 No B-particles enter the box up to time t. These trajectories are a subset of all trajectories for which the A-particle survives till time t, so Q(t) Q L (t) max l exp( ρ B l π 2 D A t/l 2 ) Q TAP (t) exp[ const.(ρ 2 B D At) 1/3 ] Q TAP (t) The prefactor is subdominant for t : ( Q L (t) exp 4 ) ρ B (D B t) 1/2 const.(ρ 2 π B D At) 1/3 Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

21 Compare with data from Mehra and Grassberger (2002): 6 Numerical data Bounds ln P(t) / (ρ 2 Dt) 1/ ln(ρ 2 Dt) Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

22 In two dimensions (Mehra and Grassberger (2002)): 16 -ln t ln Q(t) / (ρdt) Numerical data Asymptote t Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

23 ELIMINATING THE B-PARTICLES (AB, Majumdar, Blythe (2003)) Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

24 ELIMINATING THE B-PARTICLES (AB, Majumdar, Blythe (2003)) Consider the A and B particles to be non-interacting and consider events in which different B-particles hit the A-particle for the first time. These events are described by a Poisson distribution the probability that the A-particle, with a given trajectory z(t), has been hit by n different B-particles up to time t is p n = µn n! exp( µ) where µ = µ[z(τ)], 0 τ t, is a functional of the A-particle trajectory z(t). Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

25 ELIMINATING THE B-PARTICLES (AB, Majumdar, Blythe (2003)) Consider the A and B particles to be non-interacting and consider events in which different B-particles hit the A-particle for the first time. These events are described by a Poisson distribution the probability that the A-particle, with a given trajectory z(t), has been hit by n different B-particles up to time t is p n = µn n! exp( µ) where µ = µ[z(τ)], 0 τ t, is a functional of the A-particle trajectory z(t). The probability of no hits up to time t is Q(t) = p 0 (t) z = exp( µ[z]) z where the average... z is taken over all A-particle trajectories weighted with the usual Wiener measure. Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

26 To determine µ[z] we calculate, in two ways, the probability density for a B-particle to reach z(t) at time t (in a noninteracting system): ρ B = t 0 dt µ(t ) G B ( z(t), t z(t ), t ) where ( G B z(t), t z(t ), t ) 1 = [4πD B (t t exp ( [z(t) z(t )] 2 ) )] 1/2 4D B (t t ) is the B-particle diffusion propagator. For the target problem (z(t) = 0 for all t) this gives µ(t) = µ 0 (t) = 4 ρ B (D B t) 1/2 π as before. Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

27 PROOF OF THE PASCAL PRINCIPLE (AB, Majumdar, Blythe (2003); Moreau et al. (2003)) In general we can write µ[z] = µ 0 + µ 1 [z]. Then µ 1 [z] satisfies the equation: where µ 1 [z] = 1 π t 0 dt 1 t t1 t1 0 dt 2 t1 t 2 µ(t 2 ) K(t 1, t 2 ) ( [z(t1 ) z(t 2 )] 2 ) K(t 1, t 2 ) = 1 exp 4D B (t 1 t 2 ) The obvious inequalities K(t 1, t 2 ) 0 and µ[z] 0 prove that i.e. the Pascal principle is proved. µ 1 [z] 0 µ[z] µ 0 Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

28 PRE-ASYMPTOTIC CORRECTIONS (Anton, AB (2004)) Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

29 PRE-ASYMPTOTIC CORRECTIONS (Anton, AB (2004)) We attempt a perturbative treatment valid for D A D B. Expand µ 1 [z] to O(z 2 ): ρ t B dt t1 1 dt 2 µ 1 [z] = 2π 3/2 D 1/2 0 (t t 1 ) 1/2 0 B t 1/2 2 (t 1 t 2 ) [z(t 1) z(t 2 )] 2 3/2 Then Q(t) = exp[ µ 0 (t)] exp( µ 1 [z]) z where the average is evaluated using the Wiener measure ( P[z] exp 1 t ) ż 2 (τ) dτ 4D A This is merely (!) the ratio of two Gaussian integrals, but is still nontrivial. We can simplify by restricting the path integrals to paths that begin and end at the origin: z(0) = 0 = z(t), and make the Fourier decomposition z(τ) = 2 a ( n nπτ ) DA t π n sin t n=1 Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19 0

30 This gives exp( µ 1 [z]) z = where n=1 ( ) ( dan exp 1 2π 2 an g ) A mn a n a m n m,n g = (4/π 7/2 )ρ B D A t/db A mn = 1 mn 1 0 dx x 1 x φ m (x, y) = [sin(mπx) sin(mπy)] 0 dy y(x y) 3/2 φ m(x, y)φ n (x, y) Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

31 This gives exp( µ 1 [z]) z = where n=1 ( ) ( dan exp 1 2π 2 an g ) A mn a n a m n m,n g = (4/π 7/2 )ρ B D A t/db A mn = 1 mn 1 0 dx x 1 x φ m (x, y) = [sin(mπx) sin(mπy)] 0 dy y(x y) 3/2 φ m(x, y)φ n (x, y) The convexity inequality, exp x exp x, gives [ ] exp( µ 1 [z]) z exp 1 ln(1 + ga nn ) 2 Using A nn 2π 2 n 3/2 for n gives finally the improved lower bound, valid for D A D B, [ Q(t) exp 4 ρ B DB t 1 (32ρ 2 D 2 ) 1/3 ] A π B t 3 D B Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19 n=1

32 6 5 Numerical data Bounds -ln Q(t)/(ρ 2 Dt) 1/ ln(ρ 2 Dt) Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

33 SPATIAL FLUCTUATIONS OF SURVIVING TRAJECTORIES (AB, Majumdar, Blythe (2003); Anton, AB (2005)) Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

34 SPATIAL FLUCTUATIONS OF SURVIVING TRAJECTORIES (AB, Majumdar, Blythe (2003); Anton, AB (2005)) Using once more the quadratic action functional gives the probability to find z(t) = x as p(x, t) = N exp( S[z cl ]; x, t) where z cl is the classical path that minimises the action functional S[z] = 1 t ż 2 (τ) dτ + µ[z] 4D A 0 under the boundary conditions z(0) = 0, z(t) = x. Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

35 SPATIAL FLUCTUATIONS OF SURVIVING TRAJECTORIES (AB, Majumdar, Blythe (2003); Anton, AB (2005)) Using once more the quadratic action functional gives the probability to find z(t) = x as p(x, t) = N exp( S[z cl ]; x, t) where z cl is the classical path that minimises the action functional S[z] = 1 t ż 2 (τ) dτ + µ[z] 4D A 0 under the boundary conditions z(0) = 0, z(t) = x. The calculation is lengthy and leads to the surprising result that p(x, t) is Gaussian with variance x 2 DB t = const., ρ B independent of D A (!), for large t. Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

36 To see how this is possible, consider the simpler problem of B-particles to the right of a fixed target located at x = 0. The probability distribution for the location of the left-most particle at time t, given that none of the particles has reached the target, is easily calculated: p(x, t) = x ( σ 2 (t) exp x 2 ) 2σ 2 (t) where ( πdb t σ(t) = ρ 2 B ) 1/4 Now suppose there are B-particles to left and right. The typical gap size between the rightmost of the particles starting on the left and the leftmost of the particles starting on the right, given no interactions between the two groups, is of order (D B t/ρ 2 B )1/4. An intervening A-particle can explore this space with negligible probability cost provided D A D B. Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19

37 SUMMARY 1 Exact asymptotic forms have been obtained for the A-particle survival probability in the trapping reaction A + B B, for d 2. 2 Leading corrections to asymptopia have been obtained in d = 1 (extensions to all d 2 are straightforward in principle, but have not yet been done). 3 Surviving A-particles have subdiffusive fluctuations, x 2 (D B t) 1/2 /ρ B for small D A, this result requiring t before D A 0. Extensions to all d 2 are again possible in principle. Alan Bray (Isaac Newton Institute) Asymptotics of the Trapping Reaction 20 April / 19