Uwe C. Täuber , Department of Physics, Virginia Tech

1. Stochastic fluctuations and emerging correlations in simple reaction-diffusion systemsIPAM, UCLA – May 5, 2010 Uwe C. Täuber, Department of Physics, Virginia Tech Diffusion-limited pair annihilation: depletion zones Two-species pair annihilation: segregation, reaction fronts Active to absorbing phase transitions: critical phenomena Predator-prey competition: activity fronts, oscillations Research supported through NSF-DMR 0308548, IAESTE

2. Reaction-diffusion systems Particles (e.g., on d-dimensional lattice), subject to: • nearest-neighbor hopping→diffusive transport • upon encounter: reactions Σ m A ↔ Σ n A with specified rates (not necessarily in equilibrium) →effective models for many stochastic processes describing complex cooperative non-equilibrium phenomena chemical kinetics, population dynamics, but also: domain wall kinetics, epidemic and opinion spreading … • spatial fluctuations non-negligible • internal reaction noise prominent • dynamical correlations crucial i i k k i k

3. (1) Diffusion-limited pair annihilation A + A → 0 (inert state), rate λ, irreversible particle number: n = 0,1,2,…; n = 0 special: absorbing state Master equation: ∂ P (t) = λ [(n+2)(n+1) P (t) – n(n-1) P (t)] moments: <n (t)> = Σ n P (t) → ∂ <n(t)> = –2λ <[n(n-1)](t)> mean-field approximation: factorize <[n(n-1)](t)> ≈ (<n(t)>) → rate equation: ∂ <n(t)> ≈ –2λ (<n(t)>) solution: <n(t)> = n(0) / [1 + 2λ n(0) t] → 1/(2λ t) as t → ∞ add diffusion; in low dimensions: depletion zones emerge o o o o o o osurviving particles ↓ t anti-correlated o o ← L(t) → o scaling argument: L(t) ~ (2D t) → n(t) ~ 1/L(t) ~ 1/ (D t) → slower than 1/t in dimensions d<2: random walk recurrent t n n+2 n k k n t n 2 2 t 1/2 d d/2

4. mathematical description: • Smoluchowski theory (1916) solve diffusion equation with appropriate boundary conditions • operator representations - bosonic non-Hermitian “quantum” many-particle system (Doi / Peliti) → coherent-state path integrals, dynamical renormalization group → critical dimension for k A → 0: d > d = 2 / (k-1): mean-field scaling d < d : λ (t) ~ t , n(t) ~ t - mapping to quantum spin chains → bosonization, Bethe ansatz, … • tools specific to one dimension empty-interval method, integrability, … → exact results (density, correlations) pair annihilation: k = 2 d < 2 : n(t) ~ (D t) d = 2 : n(t) ~ (D t) ln(D t) d > 2 : n(t) ~ (λ t) triplet annihilation: k = 3 d = 1 : n(t) ~ [(D t) ln(D t)] d > 1 : n(t) ~ (λt) -d/2 c -1+d/d -d/2 -1 c c eff -1 1/2 -1 -1/2

5. (2) Two-species pair annihilation A B A + B → 0 (inert state), rate λ, irreversible conservation law: n (x,t) – n (x,t) = const → diffusive mode rate equation: ∂ <n (t)> ≈ –λ <n (t)> <n (t)> • n (0) - n (0) = n (∞) > 0 : initial densities different d > 2 : n (t) ~ exp{– λ [n (0) – n (0)] t} ~ n (t) – n (∞) d < 2 : ln n (t) ~ – (D t) ; d = 2 : ln n (t) ~ – t / ln(D t) • n (0) = n (0) : initial densities equal spatial inhomogeneities grow ~ L(t) → species segregation, n (t) ~ (D t) slower than 1/ t for d < d = 4 sharp reaction fronts separate domains • correlated initial conditions: … AB ABAB … → ordering preserved in d = 1, scaling as for A + A → 0 t A/B A B A B A B A B A A d/2 B B A B d/2 -d/4 A/B seg

6. Generalization: q-species pair annihilation i j seg A + A → 0 , 1 ≤ i < j ≤ q no conservation law for q > 2, d = 4 / (q–1) symmetric case: equal rates, n (0) • d ≥ 2 : as A + A → 0 (clear for q → ∞) confirmed by Monte Carlo simulations effective exponent: α(t) = - d ln n(t) / d ln t • d = 1 : species segregation density: n(t) ~ t + c t , α(q) = (q–1) / 2q domain width: w(t) ~ t , β(q) = (2q–1) / 4q i d = 2 (1000 x 1000) -α(q) -1/2 special initial state: e.g., … ABCDABCD … β(q) d = 1 (100000) d = 1 H. Hilhorst, O. Deloubriere, M.J. Washenberger, U.C.T., J. Phys. A 37 (2004) 7063

7. (3) Active to absorbing phase transitions σ μ λ reactions (+ diffusive transport) A → A+A , A → 0 , A+A → A reaction-diffusion (rate) equation: ∂ n(x,t) = D (∆ – r) n(x,t) – λ n(x,t) → phase transition at r = (μ–σ) / D continuous active / absorbing transition: ∂ n(x,t) = D ∆ n(x,t) + R[n(x,t)] + ς(x,t) reactions: R[n] = – r n – λ n + … Langevin noise: < ς(x,t) > = 0 , < ς(x,t) ς(x’,t’) > = L[n] δ(x-x’) δ(t-t’) absorbing state: L[n] = σ n + … → directed percolation (DP) universality class, d = 4 (H.K. Janssen 1981; P. Grassberger 1982) 2 t → t t DP cluster: 2 Critical exponents (ε = 4-d expansion): c

8. (4) Lotka-Volterra predator-prey competition • predators: A→ 0 death, rate μ • prey: B → B+B birth, rate σ • predation: A+B → A+A, rate λ mean-field rate equations for homogeneous densities: da(t) / dt = - µ a(t) + λ a(t) b(t) db(t) / dt = σ b(t) – λ a(t) b(t) a* = σ / λ , b* = μ/ λ K = λ (a + b) - σ ln a – μ ln b conserved→ limit cycles, population oscillations (A.J. Lotka, 1920; V. Volterra, 1926)

9. Model with site restrictions (limited resources) “individual-based” lattice Monte Carlo simulations: σ = 4.0 , μ = 0.1 , 200 x 200 sites mean-field rate equations: da / dt = – µ a(t) + λ a(t) b(t) db / dt = σ[1 – a(t) – b(t)] b(t) – λ a(t) b(t) • λ< μ: a → 0, b → 1; absorbing state • active phase: A / B coexist, fixed point node or focus→ transient erratic oscillations • prey extinction threshold: active / absorbing transition expect DP universality class (A. Lipowski 1999; T. Antal, M. Droz 2001) finite system in principle always reaches absorbing state, but survival times huge ~ exp(cN) for large N

10. Predator / prey coexistence: Stable fixed point is node

11. Predator / prey coexistence: Stable fixed point is focus Oscillations near focus: resonant amplification of stochastic fluctuations (A.J. McKane & T.J. Newman, 2005)

12. Population oscillations in finite systems (A. Provata, G. Nicolis, F. Baras, 1999)

13. oscillations for large system:compare with mean-field expectation: Correlations in the active coexistence phase M. Mobilia, I.T. Georgiev, U.C.T., J. Stat. Phys. 128 (2007) 447

14. abandon site occupation restrictions: M.J. Washenberger, M. Mobilia, U.C.T., J. Phys. Cond. Mat.19 (2007) 065139

15. stochastic Lotka-Volterra model in one dimension • no site restriction: • σ = μ = λ = 0.01: diffusion-dominated • site occupation restriction: species segregation; effectively A+A → A a(t) ~ t → 0 -1/2 σ = μ = λ = 0.1: reaction-dominated

16. Effect of spatially varying reaction rates • 512 x 512 square lattice, up to 1000 particles per site • reaction probabilities drawn from Gaussian distribution, • truncated to interval [0,1], fixed mean, different variances a(t) |a(ω)| • stationary predator and prey densities increase with Δλ • amplitude of initial oscillations becomes larger • Fourier peak associated with transient oscillations broadens • relaxation to stationary state faster U. Dobramysl, U.C.T., Phys. Rev. Lett. 101 (2008) 258102

17. Triplet “NNN” stochastic Lotka-Volterra model “split” predation interaction into two independent steps: A 0 B → A AB , rate δ;AB → 0 A , rate ω - d > 4: first-order transition (as in mean-field theory) - d < 4: continuous DP phase transition, activity rings Introduce particle exchange, “stirring rate” Δ: - Δ < O(δ): continuous DP phase transition - Δ > O(δ): mean-field scenario, first-order transition M. Mobilia, I.T. Georgiev, U.C.T., Phys. Rev. E 73 (2006) 040903(R) Cyclic competition: “rock-paper-scissors game” A+B → A+A , B+C → B+B , A+C → C+C well described by rate equations: species cluster, transient oscillations fluctuations and disorder have little effect Q. He, M. Mobilia, U.C.T., in preparation (2010)

18. Conclusions • stochastic spatial reaction-diffusion systems: models for complex cooperative non-equilibrium phenomena • internal reaction noise, emerging correlations often crucial: - depletion zones (anticorrelations) → renormalized rates - species segregation, formation of reaction fronts - active to absorbing state transitions: DP universality class - fluctuations can stabilize spatio-temporal structures • stochastic spatial systems may display enhanced robustness • deviations from mean-field rate equation predictions: - qualitative: in low dimensions d < d , d = 1 usually special - quantitative: potentially always… how do we know → criterion ? c