Response field Langevin equations and exact duality in - - PowerPoint PPT Presentation

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:

Z = Σ{Si}Π<i,j>e

J T SiSj = (cosh J/T)ℓΣ{Si}Π<i,j>(1 + SiSj tanh (J/T))

= (cosh J/T)ℓ2L2Σ#plaquettesP (tanh J/T)P , ℓ = #links

+ + + + + + + + − − − −

• vs. the low-temperature one in terms of # bonds joining opposite spins

. . . ⇒ The same diagrams iff e−2J/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)

directed bond percolation isotropic bond percolation

• DP paradigm of (continuous) out-of-equilibrium absorbing phase

transition from a fluctuating state where ρsta > 0 for p > pc to a “dead” absorbing state ρ = 0 below pc

• Two possible experiments:

– quench from homogeneous initial state ρsta ∝ (p − pc)β – survival probability from a seed of activity Psurv ∝ (p − pc)β′

ξ

||

ξ

||

ξ ξ ξ|| ξ

D

||

C x

p>p

c

p<p

c

p>p

c

p<p

c

ξ

A B t

⇒ 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 ↔ 2A, A → ∅ + diffusion

• ...If one runs time backwards starting from a fully occupied initial state:

ρsta = Psurv ⇒ 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 For the r.d. process A

λ

→ ∅ A

σ

→ 2A, 2A

λ

→ A: SDP[˜ φ, φ] =

• ddxdt
• ˜

φ

• ∂t − D∇2 − (σ − µ)
• φ +

√ 2σλ(˜ φφ2 − ˜ φ2φ) + λ˜ φ2φ2

• “Rapidity symmetry”: SDP 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 Sx = ±1 picks up the opinion of one of its 2d 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 ρI(t) = Proba{Sx(t) = Sx+e(t)} ∝ t1−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 ρI(t) = Proba{ two r.w. have not met up to time t}

• In d = 2 critical coarsening without surface tension:

ρI(t) ∝ 1/ ln t, while Lmax(t) ∝ t1/2 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

∂tψ(x, t) = D∇2ψ +

• 1 − ψ2 η(x, t), ρI = 1 − ψ2η

(Dickman, Janssen)

• On the other hand, Exact “imaginary” noise Langevin eq. for annihilating

walkers 2A

λ

− → ∅: ∂tφ(x, t) = D∇2φ − λφ2 + √ −1 λ φ η(x, t), ρ(x, t) = φ(x, t)η (B.P.Lee′94)

(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
• ) Langevin

equation: ∂ϕ ∂t = F[ϕ] + ζ where the “fast” (Gaussian) degrees of freedom ζ have a correlator ζ(x, t) ζ(x′, t′) = 2L[ϕ] δ(x − x′) δ(t − t′) (F, L may depend on ϕ, ∇ϕ, . . . )

• We want to compute observables averaged over the noise:

O[ϕ]ζ =

• DG[ζ]D[ϕ] O[ϕ] Π(x,t)δ
• ϕ − ϕ(sol.E.L.)

• the response field ˜

ϕ is simply (!) the conjugated parameter in the integral representation of the delta function: Π(x,t)δ

• ϕ − ϕ(sol.E.L.)

=

• D[i ˜

ϕ] exp

• −
• ddxdt ˜

ϕ (∂tϕ − F − ζ)

• 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” S = S[ ˜ ϕ, ϕ] =

• ddxdt
• ˜

ϕ (∂tϕ − F[ϕ]) − ˜ ϕ2L[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

• For a reaction-diffusion process, say 2A

λ

→ ∅, if P(n, t) is the proba. to have n particles on a single site ∂tP(n, t)

• annihil. = −λ

2 n(n − 1)P(n, t) + λ 2 (n + 2)(n + 1)P(n + 2, t) ֒ → introduce annihilation and creation operators ˆ a, ˆ a† such that [ˆ a, ˆ a†] = ˆ 1 and (ˆ a†ˆ a)|n >= n|n > for the particle numbers’ state ⇒ the ket |Ψ(t) >=

n P(n, t)|n > satisfies a (imaginary-time)

Schr¨

• dinger eq. with a non-hermitian “Hamiltonian”

ˆ H = −λ 2

• ˆ

a†2ˆ a2 − ˆ a2

• Then go over a path integral by introducing (overcomplete) basis of

coherent states at each time slices ˆ a|φ >= φ|φ >, < φ|ˆ a† =< φ|¯ φ Note that φ a priori complex

• Observables O({n}) can be computed after normal-ordering and

replacing ˆ a† → ¯ φ = 1, ˆ a → φ ρ(x, t) = ˆ a†ˆ a = φ(x, t)S but φ2S = ρ2 − ρ can be < 0 . . .

• Action for pairwise annihilation (modulo initial + projection state boundary

terms)

• Sannihil. =
• ddxdt
• ¯

φ

• ∂t − D∇2

φ − λ 2 (1 − ¯ φ2)φ2

• ⇒ ¯

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: Sannihil.[˜ φ, φ] =

• ddxdt
• ˜

φ

• ∂t − D∇2

φ − λ˜ φφ2 + (λ/2) ˜ ϕ2φ2 This amounts to perform the inverse route which led from a Langevin eq. to an action: ∂tφ(x, t) = D∇2φ − λφ2 + √ −1 λ φ η(x, t)

• Physical interpretation of noise term with variance −φ2: diffusion-limited

reaction with anticorrelations and density decay ρ(t) ∝ (Dt)−d/2 for d < 2 slower than classical mean-field rate equation ˙ ρ = −λρ2

⇒ Solution of Puzzles #1 & #2

• But the action is also quadratic in the “density” field . . . :

⇒ Sannihil.

i.b.p.

=

• ddxdt
• φ
• −∂t − D∇2)¯

φ − λ 2 φ2(1 − ¯ φ2)

• Integrating out the density field and reversing time:

∂tψ(x, t) = D∇2ψ +

• λ(1 − ψ2) η(x, t),

ψ(x, t)

def

= ¯ φ(x, −t) ⇒ matches the phenomenological postulate for the voter universality class!

• For DP realized as A

σ,µ

→ 2A, ∅, and 2A

λ

→ A (∂t − D∇2)φ = (σ − µ)φ − λφ2 +

• σφ − λφ2 η

(∂t − D∇2)˜ φ = (σ − µ)˜ φ − σ ˜ φ2 +

• λ˜

φ(1 − ˜ φ2) η, ˜ φ = 1 − ψ hence ρ(t) = φ(x, t) = (σ/λ)˜ φ(x, −t) ∝ Psurv(t)

What to do with a voter noise (Itˆ

• ) Langevin equation ?
• First 0d problem (no diffusion):

˙ X = g

• 1 − X2 η

X(t) = x0 ∈ (−1, 1) Main properties to be obeyed (analytically AND numerically): ֒ → Interval (−1, 1) stable for ∀t if initially so ֒ → Conservation of any initial “magnetization”: ∂tXη = 0 ⇒ X(t)η = X(0) = x0

• Any Euler scheme (“d → ∆”) is bound to failure. . . :

X(t + ∆t) = X(t) + g

• 1 − X2(t)

√ ∆tGauss.(0; 1) will output |X(t + ∆t)| > 1 with a finite proba. whatever small ∆t

Idea #1

• Rather solve the associated Fokker-Planck equation for the conditional

proba P(x, t) = Prob{X(t) = x|X(0) = x0} ∂P ∂t = g2 2 ∂2[(1 − x2)P] ∂x2 P(x, t) − → δ(x − x0), t → 0+ (continuation of a method introduced in Dornic, Chat´ e, Mu˜ noz PRL 2005 for Langevin eq. with DP-like square-root noise)

• If Gn(x) are the Gegebauer polynomials of index 3/2 (orthogonal on

(−1, 1) with respect to 1 − x2), then the continuous part of the conditional p.d.f. reads: Pc(x, t) =

• n≥0

(n + 3/2) (n + 1)(n + 2)e−g2(n+1)(n+2)t/2 (1 − x2

0)Gn(x0)Gn(x)

But this Pc(x, t) → 0 for long times: where probability has disappeared?

⇒ It has accumulated at the boundaries x = ±1 ! The correct solution is P(x, t) = Pc(x, t) + δ(x − 1)p(x0, t) + δ(x + 1)q(x0, t) where the weights of the delta peaks at the boundaries are determined by the integrated current of proba. J(x, t) = −∂x[(1 − x2)P] which has leaked there: p(x0, t) = q(−x0, t) = 1 + x0 2 −

• n≥0

(n + 3/2) (n + 1)(n + 2)e−g2(n+1)(n+2)t/2(1 − x2

0)Gn(x0)

Now correctly normalized and note in particular that X(t) →t≫1 ±1 with

• proba. 1±x0

2

(magnetization preserved on average)

• Pb.: How to (numerically) sample this awesome-looking p.d.f. ?

Idea #2

• The time-dependence is only contained in the rapidly-decaying

prefactors e−g2(n+1)(n+2)t/2 ⇒ In the strong-noise limit g ≫ 1 even for small t (= ∆t) we discard all the terms in the series with n > 1: P trunc

c

becomes an affine polynomial in x ֒ → sampling this truncated p.d.f. amounts to solving a simple quadratic equation ֒ → every initial magnetization is preserved on average ֒ → this last property remains true even after doing the diffusion step Xi(t + ∆t) = (1 − 2d.D.∆t)X∗

i + D.∆t

• j∈V(i)

X∗

j

as well as the fact that Xi(t + ∆t) ∈ (−1, 1) ∀i, ∀t

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

What are we really doing ?

• For r.d. processes where the reactions involve at most pairs of reactants,

i.e. A → kA, 2A → ℓA, ∀k, ℓ, the Doi-Peliti action will be at most quadratic in φ, and the method (formally) applies: ˆ H = −F(ˆ a†)ˆ a − 1 2G(ˆ a†)ˆ a2 ⇒ (∂t − D∇2)ψ = F(ψ) +

• G(ψ) η
• Since F(1) = G(1) = 0 (by conservation of proba.) this Langevin

equation has (at least) an absorbing state ⇒ two kinds of possible numerical simulations: ֒ → “bulk” properties: quench from homogeneous initial conditions away from ψ = 1 ֒ → seed/survival properties: everybody on the absorbing state but for a few sites

The duality big formula

• Claim/Last puzzle:

bulk response field simulations measure survival properties of standard density field (and vice-versa)

• If you talk fast, the proof is easy: just reverse the spreading cone as for

bond DP (even with β = β′)

• Result actually known by probabilists in the particular case of 2A ↔ A:

Shiga (1986) + translation by Doering et al. (2003): motivation = stochastic Fisher-Kolmogorov front equations

• but their proof go through in our setting

Πx[1 − ψ(x, t)]n(x,0)η = Πx[1 − ψ(x, 0)]n(x,t)r.d.

An application: Langevin equation for the PC-GV class

• Reaction-diffusion processes such as A

σ

− → 3A, 2A

λ

− → 0 (+ diffusion)

• dc = 2 with in space dimension d = 1 a non-trivial transition, non DP:

additional symmetry “Parity Conserved” (PC)

space time

(a)

+ + + + +

A 3A 2A

⇒ Better viewed as a Generalized Voter model (GV): Two Z2-symmetric absorbing states and transition solely induced by interfacial noise

• Phenomenological (Z2-symmetric) Langevin equation proposed:

(Al-Hammal, Chat´ e, Dornic, Mu˜ noz, PRL 2005)

∂tψ(x, t) = D ∇2ψ + (aψ − bψ3)(1 − ψ2) + c

• (1 − ψ2) η(x, t)
• Reproduces all types of transitions observed in microscopic models:

֒ → either a direct one in d = 1: PC-GV class ֒ → or in d = 2 first critical Ising then DP or GV

• 2

2 4

b

• 1

1 2 3 a

GV Ising DP

morphology of PC-clusters/GV-domains well accounted for . . .

This is just the response field Langevin equation for A

σ3

→ 3A, A

σ5

→ 5A, 2A

λ

→ ∅ with a = −(σ3 + σ5), b = σ5, c = √ λ !