Reaction-Diffusion Problems

Transcription

1 Field Theoretic Approach to Reaction-Diffusion Problems at the Upper Critical Dimension Ben Vollmayr-Lee and Melinda Gildner Bucknell University Göttingen, 30 November 2005

2 Reaction-Diffusion Models Doi Second Quantized Representation Field Theory 3A (, A, 2A) Reactions in One Dimension Renormalization Group Calculation Smoluchowski Theory Simulations Summary

3 Reaction-Diffusion Models classical particles hop randomly on lattice one-species or multi-species react when occupying the same site Α+Α Α Α+Β?

4 Reaction-Diffusion Models classical particles hop randomly on lattice short- and long-range hops (diffusion vs. Levy flights) quenched disorder varying lattices, networks, continuum

5 Reaction-Diffusion Models classical particles hop randomly on lattice one-species or multi-species react when occupying the same site Α+Α Α Α+Β?

6 Reaction-Diffusion Models classical particles hop randomly on lattice one-species or multi-species general one-species decay: ka la reversible: A + B C directed percolation: A + A A and A A + B conserves n A n B slow mode

7 Reaction-Diffusion Models classical particles hop randomly on lattice one-species or multi-species react when occupying the same site Α+Α Α Α+Β?

8 Reaction-Diffusion Models classical particles hop randomly on lattice one-species or multi-species react when occupying the same site reaction rate or probability capture radius (continuum) site occupancy restrictions

9 Reaction-Diffusion Models classical particles hop randomly on lattice one-species or multi-species react when occupying the same site Α+Α Α Α+Β?

10 Master Equation t P (α, t) = β [ ] w β α P (β, t) w α β P (α, t) α, β specified by occupation numbers {n} = (n 1, n 2,...) rates w α β for states connected by hops and reactions Poisson initial conditions

11 Diffusion t P ({n}, t) = D [ h 2 (n i +1) P (n i +1, n j 1, t) n i P ij A + A 0 Reaction ] +(n j +1) P (n i 1, n j +1, t) n j P [ ] t P (n, t) = λ (n+2)(n+1) P (n+2, t) n(n 1) P (n, t)

12 Diffusion t P ({n}, t) = D [ h 2 (n i +1) P (n i +1, n j 1, t) n i P ij A + A 0 Reaction ] +(n j +1) P (n i 1, n j +1, t) n j P [ ] t P (n, t) = λ (n+2)(n+1) P (n+2, t) n(n 1) P (n, t) Yuck!

13 Reaction-Diffusion Models Doi Second Quantized Representation Field Theory 3A (, A, 2A) Reactions in One Dimension Renormalization Group Calculation Smoluchowski Theory Simulations Summary

14 Doi Representation introduce â, â at each site with [â i, â j ] = δ ij occupation state {n} = i (â i )n i 0

15 Doi Representation introduce â, â at each site with [â i, â j ] = δ ij occupation state {n} = i (â i )n i 0 define nonequilibrium state vector φ(t) = {n} P ({n}, t) {n} master equation becomes t φ(t) = Ĥ φ(t)

16 Doi Hamiltonian Ĥ = ĤD + ĤR Diffusion: Ĥ D = D (â h 2 i â j )(â i â j ) ij Reaction: A + A : Ĥ R = λ(â 2 â 2 â 2 )

17 Doi Hamiltonian Ĥ = ĤD + ĤR Diffusion: Ĥ D = D (â h 2 i â j )(â i â j ) ij Reaction: A + A : Ĥ R = λ(â 2 â 2 â 2 ) ka la: Ĥ R = λ(â k â k â l â k )

18 Doi Hamiltonian Ĥ = ĤD + ĤR Diffusion: Ĥ D = D (â h 2 i â j )(â i â j ) ij Reaction: A + A : Ĥ R = λ(â 2 â 2 â 2 ) ka la: Ĥ R = λ(â k â k â l â k ) A+B C: Ĥ R = λ(â ˆb âˆb ĉ âˆb) [compare to master equation]

19 Expectation Values Formal solution φ(t) = e Ĥt φ(0) Nonequilibrium averages Q(t) = Q({n})P ({n}, t) = ˆQ φ(t) {n} Requires projection state = 0 e i a i

20 Expectation Values Formal solution φ(t) = e Ĥt φ(0) Nonequilibrium averages Q(t) = Q({n})P ({n}, t) = ˆQ φ(t) {n} Requires projection state = 0 e i a i Probability conservation: 1 = e Ĥt φ(0) Ĥ = 0.

21 Reaction-Diffusion Models Doi Second Quantized Representation Field Theory 3A (, A, 2A) Reactions in One Dimension Renormalization Group Calculation Smoluchowski Theory Simulations Summary

22 Coherent State Representation [Peliti 85] Coherent states: â z = z z, complex z Identity operator: 1 = d 2 z π z z Divide into N = t/ t time slices: Q = ˆQ(e Ĥ t )... (e Ĥ t ) φ(0) Insert identity at each site between each slice: z i,t φ(x, t)

23 Field Theory for ka la [BPL 94] Complex fields φ and φ 1 + φ Expectation value: Q = 1 N D(φ, φ) Q(φ)e S[φ, φ]

24 Field Theory for ka la [BPL 94] Complex fields φ and φ 1 + φ Expectation value: Q = 1 N D(φ, φ) Q(φ)e S[φ, φ] Action S = S D + S R d d x n 0 φt=0 S D = d d x dt φ( t D 2 )φ S R = d d x dt k i=1 λ 0c i φi φ k c k = 1, c i depend only on k, l

25 Why is this Field Theory Useful? Apply RG to extract universal quantities asymptotic late times, small currents noneq critical points, e.g. directed percolation No ad hoc assumptions, as in Langevin equations Flexible: can handle many generalizations Review Article U.C. Täuber, M. Howard, and BPV-L, J. Phys. A, R79 (2005).

26 Reaction-Diffusion Models Doi Second Quantized Representation Field Theory 3A (, A, 2A) Reactions in One Dimension Renormalization Group Calculation Smoluchowski Theory Simulations Summary

27 Rate Equations Assume particles remain randomly distributed A + B reaction: t a = t b = Γab ka la reaction: t a = Γak a (Γt) 1/(k 1) { 1/t 2A (, A) 1/t 1/2 3A (, A, 2A) Valid for single-species when d > d c = 2/(k 1)

28 One-Species Reactions with d d c : particles become anti-correlated, slower decay: A(Dt) d/2 d < d c { a Ã(ln t/dt) d c/2 2 2A la d = d c = 1 3A la const.t d c/2 d > d c decay amplitudes A and à are universal, as demonstrated by RG calculations [BPL 94]

29 Tests of RG Predictions à = 1/4π for A + A A in d c = 2 matches exact solution [Bramson and Griffeaths 80]

30 Tests of RG Predictions à = 1/4π for A + A A in d c = 2 matches exact solution [Bramson and Griffeaths 80] à = ( 3 4π(3 l)) 1/2 for 3A la in dc = 1 (0.21, 0.26, 0.37) for l = (0, 1, 2) Simulations give (0.26, 0.76, 0.93) 3A data matches Smoluchowski Theory [Oshanin, et al. 95, ben-avraham 93]

31 Reaction-Diffusion Models Doi Second Quantized Representation Field Theory 3A (, A, 2A) Reactions in One Dimension Renormalization Group Calculation Smoluchowski Theory Simulations Summary

32 Loop Expansion for Density Tree level: One-loop: Renormalization: n 0, λ 0 λ R 2π 3 ln(t/τ) Result n(t)(dt) 1/2 Ã ln(t/τ) + B + C ln(t/τ)

33 Universal Leading Corrections! Since ln(t/τ) = ln t + ln τ 2 1 ln t +... n(t)(dt) 1/2 Ã ln t + B+ nonuniversal B = 9 2π(2+l) 128 Corrections large ( 50%) for accessible simulation range Does not happen for 2A (, A), which has ln(t/τ) instead of ln(t/τ)

34 Reaction-Diffusion Models Doi Second Quantized Representation Field Theory 3A (, A, 2A) Reactions in One Dimension Renormalization Group Calculation Smoluchowski Theory Simulations Summary

35 Smoluchowski Theory Correlation function mean-field theory: gets correct exponents for 2A (, A)! Two different generalizations to 3A la, both find log corrections, different amplitudes [Krapivsky 94, Oshanin et al. 95] Both missed various factors. When corrected...

36 Smoluchowski Theory Correlation function mean-field theory: gets correct exponents for 2A (, A)! Two different generalizations to 3A la, both find log corrections, different amplitudes [Krapivsky 94, Oshanin et al. 95] Both missed various factors. When corrected... both agree with RG for à but Smoluchowksi theory has B = 0

37 Reaction-Diffusion Models Doi Second Quantized Representation Field Theory 3A (, A, 2A) Reactions in One Dimension Renormalization Group Calculation Smoluchowski Theory Simulations Summary

38 Simulation Technique 1d lattice, sites, 10 8 time steps synchronous dynamics (D = 1/2): before after linked list of particle sites efficient update 10 to 20 independent runs for each reaction

39 RG: n à ln t/dt Rate Eq: n 1/t 1/2 5 n(t) 2 Dt A 2A 3A A 3A ln t

40 0.5 3A n(t) (Dt/ln t) 1/ simulation data RG asymptote RG with subleading term time Data consistent with Oshanin et al. 95

41 A A n(t) (Dt/ln t) 1/ simulation data RG asymptote RG with subleading term time Data not consistent with ben-avraham 93

42 A 2A n(t) (Dt/ln t) 1/ simulation data RG asymptote RG with subleading term time Data not consistent with ben-avraham 93

43 Rescaling Symmetry rescaling φ bφ, φ b 1 φ changes only c i density for different reactions related by rescaling to all orders in 1/ ln(t/τ) mixed reactions: (p 0, p 1, p 2 ) related to 3A 2A

44 Mixed 3A (, A, 2A) with rates (p 0, p 1, p 2 ) 0.7 n(t) (Dt/ln t) 1/ (1/4, 3/4, 0) (1/9, 5/9, 1/3) 3A 2A time

45 Reaction-Diffusion Models Doi Second Quantized Representation Field Theory 3A (, A, 2A) Reactions in One Dimension Renormalization Group Calculation Smoluchowski Theory Simulations Summary