Response field Langevin equations and exact duality in nonequilibrium statistical mechanics Ivan Dornic, Hugues Chat´ e, & Miguel A. Mu˜ noz SPEC, CEA Saclay, France & Instituto Carlos I, Univ. Granada, Spain arxiv:0807.XXX (to be posted very soon...)
Outline I. Three examples of “Duality” in stat. mech. and two puzzles 1. Ising model 2. Directed Percolation: Puzzle #1 3. Voter model: Puzzle #2 II. Response field and generating functional: a “reminder” 1. Phenomenological approach 2. Doi-Peliti/Master equation approach III. Duality and Response field Langevin equations 1. Hubbard-Stratonovich transformations 2. Solution of puzzles # 1 & 2 3. Interlude: numerical scheme for Voter Langevin equation 4. The duality big formula IV. An application: the PC-GV class, and some conclusions
Kramers-Wannier self-duality for the 2d Ising model (1941) • Compare a high-temperature expansion of the partition function: T S i S j = (cosh J/T ) ℓ Σ { S i } Π <i,j> (1 + S i S j tanh ( J/T )) J Z = Σ { S i } Π <i,j> e (cosh J/T ) ℓ 2 L 2 Σ #plaquettes P (tanh J/T ) P , ℓ = #links = + + + − − + + − − + + + • vs. the low-temperature one in terms of # bonds joining opposite spins . . . ⇒ The same diagrams iff e − 2 J/T < = tanh ( J/T > ) ⇒ duality locates exactly Onsager critical temperature!
Symmetries of Directed Percolation (DP) • Directed Percolation = Percolation with an extremely anisotropic direction = Time (from Hinrichsen’s 2001 review) isotropic bond percolation directed bond percolation • DP paradigm of (continuous) out-of-equilibrium absorbing phase transition from a fluctuating state where ρ sta > 0 for p > p c to a “dead” absorbing state ρ = 0 below p c
• Two possible experiments: – quench from homogeneous initial state ρ sta ∝ ( p − p c ) β – survival probability from a seed of activity P surv ∝ ( p − p c ) β ′ x A B ξ ξ ξ || ξ || || ξ t ξ || ξ C D p<p p>p p<p p>p c c c c ⇒ quartet β, β ′ , ν ⊥ , ν � of critical exponents ( ν ⊥ , � resp. for spatial and temporal correlation lengths) • DP ubiquitous : Janssen-Grassberger’s conjecture (1981-1982) DP encompasses all continuous phase transitions from a fluctuating active phase towards a unique absorbing state (without additionnal symmetry or conservation law, or quenched disorder)
With (at last!) an unambiguous experimental realization • Idea (Pomeau 1986): Transition to turbulence through spatio-temporal intermittency: laminar regions ↔ absorbing state ⇒ Phase transition between two topologically different turbulent states of electrodynamic convection in nematic liquid crystals (Takeuchi, Kuroda, Chat´ e, Sano, PRL 2007)
Duality at the microscopic level: bond DP • Start with a seed/ a single particle: ⇒ Also equivalent to a reaction-diffusion model A ↔ 2 A , A → ∅ + diffusion • ...If one runs time backwards starting from a fully occupied initial state: ρ sta = P surv ⇒ β = β ′ : Duality reduces the number of independent critical exponents to three
“Rapidity symmetry” for Reggeon field theory • To the Langevin eq. is associated an “action” S = S [˜ φ, φ ] : ֒ → φ “density” field, its average gives the density of particles: � φ � ∝ ρ → ˜ ֒ φ conjugated “response” field, associated to the noise λ σ λ → ∅ A → 2 A , 2 A → A : For the r.d. process A √ � � ∂ t − D ∇ 2 − ( σ − µ ) φφ 2 − ˜ φ 2 φ 2 � S DP [˜ ˜ 2 σλ (˜ φ 2 φ ) + λ ˜ d d x d t � � φ, φ ] = φ φ + • “Rapidity symmetry”: S DP invariant under → − ˜ φ ( x, − t ) & ˜ φ ( x, t ) − φ ( x, t ) − → − φ ( x, − t ) , Puzzle #1 Relationship between the symmetry at the microscopic level fixing β = β ′ and the invariance of the action under φ ↔ ˜ φ ?
The Voter Model • A Voter S x = ± 1 picks up the opinion of one of its 2 d randomly chosen neighbors • One of the simplest but most versatile model of noneq. stat. mech. (mathematical genetics, ecology, sociophysics) • Basic dichotomy: opinions become unanimous in space dimensions d ≤ 2 , where interface density ρ I ( t ) = Proba { S x ( t ) � = S x + e ( t ) } ∝ t 1 − d/ 2 − → 0 Why ? • Note that the successive ancestors of a given voter follow the path of a random walker in reverse time ⇒ Duality with a system of coalescing/annihilating random walkers going backwards in time ρ I ( t ) = Proba { two r.w. have not met up to time t }
• In d = 2 critical coarsening without surface tension: L max ( t ) ∝ t 1 / 2 ρ I ( t ) ∝ 1 / ln t, while behavior representative of a whole universality class of absorbing phase transition with two symmetric absorbing states and solely interfacial noise Dornic, Chat´ e, Chave, Hinrichsen, PRL 2001
Puzzle #2 • On the one hand, Phenomenological Langevin equation for Voter class 1 − ψ 2 η ( x, t ) , ρ I = � 1 − ψ 2 � η � ∂ t ψ ( x, t ) = D ∇ 2 ψ + (Dickman , Janssen) • On the other hand, Exact “imaginary” noise Langevin eq. for annihilating λ − → ∅ : walkers 2 A √ ∂ t φ ( x, t ) = D ∇ 2 φ − λφ 2 + (B . P . Lee ′ 94) − 1 λ φ η ( x, t ) , ρ ( x, t ) = � φ ( x, t ) � η (explanations to follow soon) ⇒ Given the microscopic duality, can the two Langevin equations be related and if yes, how?
Outline I. Three examples of “Duality” in stat. mech. and two puzzles 1. Ising model 2. Directed Percolation: Puzzle #1 3. Voter model: Puzzle #2 II. Response field and generating functional: a “reminder” 1. Phenomenological approach 2. Doi-Peliti/Master equation approach III. Duality and Response field Langevin equations 1. Hubbard-Stratonovich transformations 2. Solution of puzzles # 1 & 2 3. Interlude: numerical scheme for Voter Langevin equation 4. The duality big formula IV. An application: the PC-GV class, and some conclusions
Response field I): Phenomenological viewpoint See e.g. U. C. T¨ auber, arxiv/0707.0794 • Assume some “slow” variables { ϕ } = { ϕ ( x, t ) } obey a (Itˆ o) Langevin equation: ∂ϕ ∂t = F [ ϕ ] + ζ where the “fast” (Gaussian) degrees of freedom ζ have a correlator � ζ ( x, t ) ζ ( x ′ , t ′ ) � = 2 L [ ϕ ] δ ( x − x ′ ) δ ( t − t ′ ) ( F, L may depend on ϕ, ∇ ϕ, . . . ) • We want to compute observables averaged over the noise: � � ϕ − ϕ (sol . E . L . ) � �O [ ϕ ] � ζ = D G [ ζ ] D [ ϕ ] O [ ϕ ] Π ( x,t ) δ
• the response field ˜ ϕ is simply (!) the conjugated parameter in the integral representation of the delta function: � � � � � ϕ − ϕ (sol . E . L . ) � d d x d t ˜ D [ i ˜ − ϕ ( ∂ t ϕ − F − ζ ) Π ( x,t ) δ = ϕ ] exp • This allows the Gaussian integral over the noise to be performed: � �O [ ϕ ] � ζ = D [ ϕ ] D [ i ˜ ϕ ] O [ ϕ ] exp {− S [ ˜ ϕ, ϕ ] } ⇒ Observables can be computed with a statistical weight ∝ e − S given by the “dynamic action”/Janssen-De Dominicis “response functional” � d d x d t ϕ 2 L [ ϕ ] � � S = S [ ˜ ϕ, ϕ ] = ϕ ( ∂ t ϕ − F [ ϕ ]) − ˜ ˜ • ˜ ϕ helpful for causality/FDT but hardly more than a convenient book-keeping device (all its own correlation functions = 0 )
Response field II): Doi-Peliti formalism λ • For a reaction-diffusion process, say 2 A → ∅ , if P ( n, t ) is the proba. to have n particles on a single site annihil . = − λ 2 n ( n − 1) P ( n, t ) + λ � ∂ t P ( n, t ) 2 ( n + 2)( n + 1) P ( n + 2 , t ) � a † such that [ˆ a † ] = ˆ ֒ → introduce annihilation and creation operators ˆ a, ˆ a, ˆ 1 a † ˆ a ) | n > = n | n > for the particle numbers’ state and (ˆ ⇒ the ket | Ψ( t ) > = � n P ( n, t ) | n > satisfies a (imaginary-time) Schr¨ odinger eq. with a non-hermitian “Hamiltonian” H = − λ a 2 − ˆ ˆ a † 2 ˆ a 2 � � ˆ 2
• Then go over a path integral by introducing (overcomplete) basis of coherent states at each time slices a † = < φ | ¯ a | φ > = φ | φ >, < φ | ˆ ˆ φ Note that φ a priori complex • Observables O ( { n } ) can be computed after normal-ordering and a † → ¯ replacing ˆ φ = 1 , ˆ a → φ a † ˆ ρ ( x, t ) = � ˆ a � = � φ ( x, t ) � S but � φ 2 � S = � ρ 2 � − � ρ � can be < 0 . . . • Action for pairwise annihilation (modulo initial + projection state boundary terms) � � � φ − λ ¯ 2 (1 − ¯ d d x d t ∂ t − D ∇ 2 � φ 2 ) φ 2 � S annihil . = φ ⇒ ¯ φ = 1 − ˜ φ is the unshifted response field in the Janssen-de Dominicis’ formalism
Hubbard-Stratonovich transformation(s) Recall this is just a means of completing the square in a Gaussian integral... • The usual transformation is for an action quadratic in the (shifted) response field: � � φφ 2 + ( λ/ 2) ˜ ϕ 2 φ 2 � S annihil . [˜ ˜ φ − λ ˜ d d x d t ∂ t − D ∇ 2 � � φ, φ ] = φ This amounts to perform the inverse route which led from a Langevin eq. to an action: √ ∂ t φ ( x, t ) = D ∇ 2 φ − λφ 2 + − 1 λ φ η ( x, t ) • Physical interpretation of noise term with variance − φ 2 : diffusion-limited reaction with anticorrelations and density decay ρ ( t ) ∝ ( Dt ) − d/ 2 for d < 2 ρ = − λρ 2 slower than classical mean-field rate equation ˙
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