1. Stochastic fluctuations and emerging correlations in simple reaction-diffusion systems IPAM, UCLA – May 5, 2010 Uwe C. Täuber, Department of Physics, Virginia Tech (1)Diffusion-limited pair annihilation: depletion zones (2)Two-species pair annihilation: segregation, reaction fronts (3)Active to absorbing phase transitions: critical phenomena (4)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 ii k kk

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: = Σ n P (t) → ∂ = –2λ mean-field approximation: factorize ≈ ( ) → rate equation: ∂ ≈ –2λ ( ) solution: = 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 o surviving 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 t t kk nn n n n+2 2 2 dd/21/2

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) c 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) ~ (λ t) -d/2 1/2 -1/2 c eff -1+d/d c -d/2

5. (2) Two-species pair annihilation A + B → 0 (inert state), rate λ, irreversible conservation law: n (x,t) – n (x,t) = const → diffusive mode rate equation: ∂ ≈ –λ 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: … A B A B A B … → ordering preserved in d = 1, scaling as for A + A → 0 A/BAB AB t AB ABAB A A d/2 B B A/B d/2 -d/4 seg AB

6. Generalization: q-species pair annihilation 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 H. Hilhorst, O. Deloubriere, M.J. Washenberger, U.C.T., J. Phys. A 37 (2004) 7063 d = 2 (1000 x 1000) d = 1 (100000) ij special initial state: e.g., … ABCDABCD … d = 1 seg i β(q) -α(q) -1/2

7. (3) Active to absorbing phase transitions 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: = 0, = 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) σλμ 2 t → t 2 tDP cluster: 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) 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) “individual-based” lattice Monte Carlo simulations: σ = 4.0, μ = 0.1, 200 x 200 sites 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. Correlations in the active coexistence phase oscillations for large system: compare with mean-field expectation: 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 site occupation restriction: species segregation; effectively A+A → A a(t) ~ t → 0 no site restriction: σ = μ = λ = 0.01: diffusion-dominated σ = μ = λ = 0.1: reaction-dominated -1/2

16. Effect of spatially varying reaction rates 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 a(t) |a(ω)| U. Dobramysl, U.C.T., Phys. Rev. Lett. 101 (2008) 258102 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

17. Triplet “NNN” stochastic Lotka-Volterra model “split” predation interaction into two independent steps: A 0 B → A A B, rate δ; A B → 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” Q. He, M. Mobilia, U.C.T., in preparation (2010) 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

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