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

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 i k k k

(1) Diffusion-limited pair annihilation

A + A → 0 (inert state), rate λ, irreversibleparticle 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 o surviving particles ↓ t anti-correlated o o ← L(t) → oscaling 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

tk k

n n

nn

n+2

2

2

d d/21/2

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 dimensionempty-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

-1

-1 1/2

-1/2

c eff-1+d/d c -d/2

(2) Two-species pair annihilation

A + B → 0 (inert state), rate λ, irreversibleconservation law: n (x,t) – n (x,t) = const → diffusive moderate 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: … A B A B A B … → ordering preserved in d = 1, scaling as for A + A → 0

A/B A B

A B

t

A B

A B A B

A

Ad/2

B B

A/B

d/2

-d/4

seg

A B

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)

i j

special initial state: e.g., … ABCDABCD …

d = 1

seg

i

β(q)

-α(q) -1/2

(3) Active to absorbing phase transitions reactions (+ diffusive transport) A → A+A , A → 0 , A+A → Areaction-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)

σ λμ

2t

→ t

2t DP cluster:

Critical exponents (ε = 4-d expansion):

c

(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)

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

Predator / prey coexistence: Stable fixed point is node

Predator / prey coexistence: Stable fixed point is focus

Oscillations near focus: resonant amplification of stochastic fluctuations (A.J. McKane & T.J. Newman, 2005)

Population oscillations in finite systems

(A. Provata, G. Nicolis, F. Baras, 1999)

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

abandon site occupation restrictions :

M.J. Washenberger, M. Mobilia, U.C.T., J. Phys. Cond. Mat.19 (2007) 065139

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

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

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 ringsIntroduce 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 oscillationsfluctuations and disorder have little effect

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 ?

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