Aging in the one-dimensional coagulation-diffusion process Xavier - - PowerPoint PPT Presentation

Aging in the one-dimensional coagulation-diffusion process

Xavier Durang, Jean Yves Fortin, Malte Henkel

IJL, Universit´ e Henri Poincar´ e Nancy I

XD, Fortin, Del Biondo, Henkel, Richert, J. Stat. Mech 2010 XD, Fortin, Henkel J. Stat. Mech 2011

Dresden, MPI, LAFNES11

15 juillet 2011

Xavier Durang, Jean Yves Fortin, Malte Henkel

Contents

1 Introduction

Ageing phenomena : from simple magnets to directed percolation Two-time observables, Fluctuation-Dissipation ratio Model

2 One-time quantities

Training example (Method used) Influence of the initial conditions

3 Two-time functions

Generalisation of the empty-interval method Ageing exponents

4 Fluctuation-dissipation ratio 5 Conclusion Xavier Durang, Jean Yves Fortin, Malte Henkel

I.1 Ageing

Struik 1978

Three defining properties of ageing :

• 1. observe slow relaxation after quenching PVC from melt to low T
• 2. creep curves depend on waiting time te (or s) and creep time t
• 3. find master curve for all (t, te) −

→ dynamical scaling

Quench

Constraint Measure s t time Xavier Durang, Jean Yves Fortin, Malte Henkel

t = t1 t = t2 > t1

magnet T < Tc − → ordered cluster magnet T = Tc − → correlated cluster critical contact process

diffusion, A → 2A, A → φ

= ⇒ cluster dilution

voter model, contact process,. . .

Characteristic length scale : L(t) ∼ t1/z

Xavier Durang, Jean Yves Fortin, Malte Henkel

I.2 Two-time observables

time-dependent order-parameter φ(t, r) (Directed percolation : φ =part. density) two-time correlator C(t, s) := φ(t, r)φ(s, r) − φ(t, r) φ(s, r) two-time response R(t, s) := δ φ(t, r) δh(s, r)

• h=0

(Directed percolation : h(t) = creation of part.) t : observation time, s : waiting time Scaling regime : t ≫ s ≫ τmicro (For simple magnets) C(t, s) = s−bfC t s

• , R(t, s) = s−1−afR

t s

• Asymptotics : fC,R(y) ∼ y−λC,R/z for y ≫ 1

λC : autocorrelation exponent, λR : autoresponse exponent, z : dynamical exponent, a, b : ageing exponents

Xavier Durang, Jean Yves Fortin, Malte Henkel

I.3 Fluctuation dissipation ratio

Usually λR = λC

The fluctuation-dissipation ratio (fdr)

Cugliandolo, Kurchan, Parisi ’94

X(t, s) := TR(t, s) ∂C(t, s)/∂s X∞ = lim

s→∞

• lim

t→∞ X(t, s)

• Godr`

eche & Luck 00

measures the distance to the equilibrium : Xeq = X(t − s) = 1. a = b valid when systems satisfy detailed balance Contact process 1 + a = b : ⇐ = rapidity-reversal symmetry of stationary state

• f cp ⇒ specific property !

= ⇒ try new form of FDR !

Enss et. al. 04

Ξ(t, s) := R(t, s) C(t, s) = fR(t/s) fC(t/s) , Ξ∞ := lim

s→∞

• lim

t→∞ Ξ(t, s)

• Universality of Ξ∞ proven to one-loop order.

Baumann & Gambassi 07 Xavier Durang, Jean Yves Fortin, Malte Henkel

I.4 Coagulation-diffusion process

AIM : to test these scaling predictions on an exactly solvable model without detailed balance Model : One dimensional lattice of spacing a

D D Coagulation Diffusion

(diffusion and coagulation can occur in both directions) Space translation invariance Absence of detailed balance Absorbing phase Stationary state

Xavier Durang, Jean Yves Fortin, Malte Henkel

II.1 Empty interval method : training example

Particle concentration : c(t) = Pr({•; t})

• D. ben Avraham et al. 90

n empty sites

En(t) : time-dependent probability of having an interval of n consecutive empty sites at time t c(t) = E1(t) − E0(t) continuum limit (x=na) − → c(t) = − ∂xE(x, t)|x=0 Equation of motion For n > 1 ∂tEn(t) = (2D/a2) (En−1 − 2En + En+1) For n = 1 ∂tE1(t) = (2D/a2)

• 1 − 2E1(t) + E2(t)
• This gives the constraint : E0(t) = 1

Equation of motion in the continuum limit (x = na) ∂tE(x, t) = 2D∂xxE(x, t), and E(0, t) = 1.

Xavier Durang, Jean Yves Fortin, Malte Henkel

II.2 Solution by analytical continuation

Assume that the differential equation is valid for n ≤ 0 E(x, t) = ∞

−∞

dx′ √πℓ0 exp

• − 1

ℓ02 (x − x′)2 E(x′, 0). where ℓ0 is the scaling length ℓ0 := √ 8Dt . Take into account the constraint : E0(t) = 1. For n = 0 ∂tE0(t) = (2D/a2) (E−1 − 2E0 + E1) = 0 E−1(t) = 2E0(t) − E1(t) = 2 − E1(t) Redefine the meaning of E(n, 0) for negative n such that E−n(t) = 2 − En(t) and E(−x, t) = 2 − E(x, t)

Xavier Durang, Jean Yves Fortin, Malte Henkel

II.3 General expression for the particle concentration

One-empty-interval probability E(x, t) = erfc(x/ℓ0) + +∞ dx′ √πℓ0 E(x′, 0)

• e

−

1 ℓ02 (x−x′)2

− e

−

1 ℓ02 (x+x′)2

. Hierarchy Particle concentration c(t) = 2 √πℓ0

• 1 −

∞ dxE(xℓ0, 0)2xe−x2 c(t) = 2 √πℓ0 + o(1/ℓ0) ∼ t−1/2 Independent of initial condition → very well known result

Xavier Durang, Jean Yves Fortin, Malte Henkel

• II. 4 One-time Correlation funtion

Connected correlator : C(d, t) = Pr(• d •, t) − Pr(•, t)Pr(•, t) Two-interval probability

n d m

En1,n2,d(t) : time-dependent probability of having two intervals of n1 and n2 consecutive empty sites distant from d at time t Continuum limit (x = n1a, y = n2a, z = da) C(z, t) = ∂2

xyE(x, y, z, t)

• x=0,y=0 − ∂xE(x, t)|x=0 ∂yE(y, t)|y=0

Xavier Durang, Jean Yves Fortin, Malte Henkel

II.5 One-time correlation function

Equation of motion : (only for x, y and z positive) ∂tE(x, y, z, t) = 2D

• ∂2

x + ∂2 y + ∂2 z −

• ∂x∂z + ∂y∂z
• E(x, y, z, t)

with compatibility conditions (ex : E(x, 0, d, t) = E(x, t)). Decomposition of the solution as a sum of three terms : E(x, y, z, t) = E (0)(x, y, z, t) + E (1)(x, y, z, t) + E (2)(x, y, z, t)

1

E (0)(x, y, z, t) is independent of the initial conditions

2

E (1)(x, y, z, t) depends on the initial one-interval probability

3

E (2)(x, y, z, t) depends on the initial two-intervals probability

→ System initially filled with particles

E(x, y, z; t) = E (0)(x, y, z, t) = erfc x ℓ0

• erfc

y ℓ0

• +erfc

z ℓ0

• erfc

x + y + z ℓ0

• − erfc

x + z ℓ0

• erfc

y + z ℓ0

• Xavier Durang, Jean Yves Fortin, Malte Henkel

II.6 One-time correlation function

Correlation function C(z, t) = ∂2

xyE(x, y, z, t)

• x=0,y=0 − ∂xE(x, t)|x=0 ∂yE(y, t)|y=0

In the case of an initially completely filled system, E(x, 0) = 0 and E(x, y, z, 0) = 0, we obtain dynamical scaling with ℓ0 = √ 8Dt. Connected correlator C(z, t) =

• 2

√πℓ0 2 f (z/ℓ0) with f (y) = −e−2y2 + √πye−y2erfc(y)

• D. ben Avraham, 1998

exact in asymptotic regime for all initial conditions.

Xavier Durang, Jean Yves Fortin, Malte Henkel

II.7 Correction to the leading behaviour

Time evolution of C(z; t) for an initial one-interval probability E(x; 0) = exp(−x) and z = 1/2

10 100

t

0,01 0,1

• C(z=1/2,t)

C

(0)(z,t)

C

(0)(z,t)+C (1)(z,t)

C

(0)(z,t)+C (1)(z,t)+C (2)(z,t)

Full black solid line : leading contribution Algebraic behaviour when t large |C(z, t)| ∼ t−1. Red dashed line includes the effect of one-interval contribution Dashed-dotted line includes all contributions

Hierarchy :

• C (0)(1/2, t)
• ≫
• C (1)(1/2, t)
• ≫
• C (2)(1/2, t)
• Xavier Durang, Jean Yves Fortin, Malte Henkel

III.1 Two-times functions

We want to evaluate connected correlation C(z; t, s) and response R(z; t, s) functions (t ≥ s) using the interval probability method In the discrete space, their definition are C(d; t, s) = Pr({•; t} d {•; s}) − Pr(•; t)Pr(•; s) If we add a particle at a given site at time s : R(d; t, s) = Pr({•; t} d {•; s}) − Pr(•; t) = δ φ(t, r) δh(s, r)

• h=0

φ = part. density and h(t) = creation of part.

n d

Mixed-interval probability F(n, d; t, s) = Pr({ n ; t} d {•; s})

Xavier Durang, Jean Yves Fortin, Malte Henkel

III.2 Mixed-interval probability

Two-time correlation C(z; t, s) = lim

a→0

C(d; t, s) a = −∂xF(x, z; t, s)|x=0 ∼ 1/L2 Two-time response, G has the same definition than F R(z; t, s) = lim

a→0

R(d; t, s) a = −∂xG(x, z; t, s)|x=0 ∼ 1/L Initial conditions at t = s F(x, z; s, s) = lim

a→0

1 aF(n, d; s, s) = −∂yE(x, y, z; s)|y=0 G(x, z; s, s) = lim

a→0 G(n, d; s, s) = E(x; s)

F and G are different because of their initial conditions at t = s

Xavier Durang, Jean Yves Fortin, Malte Henkel

III.3 Equation of motion

In the continuum limit and for x and z positive variables ∂tF(x, z; t, s) = 2D

• ∂2

x + 1

2∂2

z − ∂x∂z

• F(x, z; t, s)

Symmetries between positive and negative variables F(−x < 0, z; s, s) = 2c0(s) − F(x, z − x; s, s) F(x, −z < 0; s, s) = θ(z − x)F(x, z − x; s, s)) F(−x < 0, −z < 0; s, s) = 2c0(s) − F(x, z; s, s) with c0(s) = c(s) for F and c0(s) = 1 for G

Xavier Durang, Jean Yves Fortin, Malte Henkel

III.4 Response function

Expression for an initially filled system

R(z; t, s) = 2 πℓ1ℓ0

• R+ dx′
• erfc

x′ − 2z ℓ1

• + erfc

x′ + 2z ℓ1

• exp
• − x′2(ℓ2

0 + ℓ2 1)

ℓ2

0ℓ2 1

• + 2

πℓ2

1

e−2z2/ℓ2

1

• R+ dx′
• exp
• − 2(z − x′)2

ℓ2

1

• + exp
• − 2(z + x′)2

ℓ2

1

• erfc

x′ ℓ0

• −

2 √πℓ0

Analogous expression for the correlation function but much more lengthy.

Xavier Durang, Jean Yves Fortin, Malte Henkel

III.5 Response function

Discrete numerical simulations on a chain of 512 sites with s = 10 and D = 1/2 versus analytical expressions

10 100 1000 t 0,0001 0,001 0,01 0,1 1 R(t,s;r) s=10 100 t 0,001 0,01 0,1 1 t s=100

Full line : analytical solution ; symbols : simulation r = [0, 1, 2, 3] from top to bottom

Xavier Durang, Jean Yves Fortin, Malte Henkel

III.6 Auto-correlator and auto-response

Autocorrelator

y = t/s C(ys, s) = 4 πℓ2

• 1
• y 2 − 1

+

• 2(y − 1)

π(1 + y) tan−1

• √

2 √y − 1

• −

2 π√y tan−1 1 √y

• −

1 1 + y

• := 1

s fC(y) = 1 sb fC(y)

in agreement with P.Mayer and P. Sollich 2007

Autoresponse

R(ys, s) = 2 √πℓ0

• √

2 π√y − 1 tan−1

• √

2 √y − 1

• −

2 π√y tan−1 1 √y

• := 1

√s fR(y) = 1 s1+a fR(y)

Exponents λC = λR = 4, a = −1/2 and b = 1

Xavier Durang, Jean Yves Fortin, Malte Henkel

IV.1 Fluctuation-dissipation ratio

Question What’s the relation between R and C ? a = b detailed balance → Usual definition of the fluctuation-dissipation ratio X(t, s) = TR(t, s) ∂sC(t, s)

L.F. Cugliandolo and al. 1994

1 + a = b directed percolation → Ξ(t, s) := R(t, s) C(t, s) = fR(t/s) fC(t/s) , Ξ∞ := lim

s→∞

• lim

t→∞ Ξ(t, s)

• here b = 1 and a = −1/2 → 3/2 + a = b, therefore we define

another ratio Ξ(t, s) = R(t, s) √ 8D∂s−1/2C(t, s)

Xavier Durang, Jean Yves Fortin, Malte Henkel

IV.2 New forms of fluctuation-dissipation relations ?

Return to the non-equilibrium scaling forms

C(t, s) = L(s)−bzFC t s

• = s−bfC

t s

• R(t, s)

= L(s)−az−zFR t s

• = s−1−afR

t s

• time-dependent length scale :

L(s) = (ϑs)1/z

Xavier Durang, Jean Yves Fortin, Malte Henkel

IV.3 Generalisation ?

Ξ(t, s) := ϑ1+a−b R(t, s) ∂1+a−b

s

C(t, s) use Riemann-Liouville fractional derivative ∂α

s

ϑ−1 is a characteristic time for the passage into the scaling behaviour. contains two situations :

1

critical systems with detailed balance (a = b holds true and ϑ → T becomes the equilibrium temperature)

2

critical particle-reaction models with a + 1 = b

in the limit y = t/s ≫ 1, the function Ξ tends towards an universal limit Ξ∞. for the coagulation diffusion process : Ξ∞ =

3π 6π−16

Xavier Durang, Jean Yves Fortin, Malte Henkel

Conclusion

Ageing in the 1D coagulation-diffusion process quite analogous to what have been seen in simple magnets Exactly solvable model allow us to verify the expected scaling forms Tool : generalized empty interval method We find the one-time and two-time correlation functions for any given initial conditions Proposal of a new FDR Physical meaning of Ξ ?

Xavier Durang, Jean Yves Fortin, Malte Henkel

Xavier Durang, Jean Yves Fortin, Malte Henkel

Xavier Durang, Jean Yves Fortin, Malte Henkel

III.3 Quantum representation

Empty site |0 >i= 1

• Filled site |1 >i=

1

• Operators of creation and destruction

d†

i =

1 1

• , di =

1

• di|0 >i= 0, di|1 >i= |0 >i, d†

i |0 >i= |1 >i, d† i |1 >i= |1 >i

Particle number operator ni = d†

i di and hole operator

hi = 1 − ni Initial state : |P(0) >=

• {αi=0,1}

P({αi}, t = 0) ⊗i |αi > Ground state : < G| = ⊗i{ 1

1

• }

Xavier Durang, Jean Yves Fortin, Malte Henkel

III.4 Quantum representation

Hamiltonian H = −Γ

• i
• d†

i+1di + d† i−1di

• + µ
• i

ni µ = 2Γ is the chemical potential such that < G|H = 0 and P(t) =< G|P(t) >= 1 Empty-interval operator X(1, n) :=

• i=1,n

(1 − ni) =

• i=1,n

hi. Mixed function is represented by the expectation value G(n, d; t, s) =< G|X(1, n)e−H(t−s)d†

n+d+1e−Hs|P(0) >

Xavier Durang, Jean Yves Fortin, Malte Henkel

III.3 Equations of motion

Same equation for F and G ∂tF(n, d; t, s) = Γ [F(n + 1, d; t, s) + F(n + 1, d − 1; t, s) + F(n − 1, d; t, s) + F(n − 1, d + 1; t, s) − 4F(n, d; t, s)] In the continuum limit ∂tF(x, z; t, s) = 2D

• ∂2

x + 1

2∂2

z − ∂x∂z

• F(x, z; t, s)

General solution F(x, z; t, s) =

• R2

2dx′dz′ πℓ2

1

Wℓ1(x − x′, z − z′)F(x′, z′; s, s), Wℓ1(x − x′, z − z′) = e

− 2

ℓ2 1 [(x−x′+z−z′)2−(z−z′)2]

with ℓ1 =

• 8D(t − s) and ℓ0 =

√ 8Ds

Xavier Durang, Jean Yves Fortin, Malte Henkel

III.4 Symmetries

We have to rewrite the former integrals in terms of physical domain (x > 0, z > 0) We obtain, using the constraints imposed by the differential equation of motion F(−x < 0, z; s, s) = 2c0(s) − F(x, z − x; s, s) F(x, −z < 0; s, s) = θ(z − x)F(x, z − x; s, s)) F(−x < 0, −z < 0; s, s) = 2c0(s) − F(x, z; s, s) with c0(s) = c(s) for F and c0(s) = 1 for G

Xavier Durang, Jean Yves Fortin, Malte Henkel

III.8 Correlation function

Discrete numerical simulations on a chain of 512 sites with s = 10 and D = 1/2 versus analytical expressions

100 1000 t 0,0001 0,001 0,01 0,1 s=100 10 100 1000 t 0,0001 0,001 0,01 0,1 C(t,s;r) s=10

The full line are the analytical solution while the symbols are the simulations Black curves are z = 0 and red curves are for z = 3

Xavier Durang, Jean Yves Fortin, Malte Henkel

III.9 Response function

Expression for an initially filled system

R(z; t, s) = 2 πℓ1ℓ0

• R+ dx′
• erfc

x′ − 2z ℓ1

• + erfc

x′ + 2z ℓ1

• exp
• − x′2(ℓ2

0 + ℓ2 1)

ℓ2

0ℓ2 1

• + 2

πℓ2

1

e−2z2/ℓ2

1

• R+ dx′
• exp
• − 2(z − x′)2

ℓ2

1

• + exp
• − 2(z + x′)2

ℓ2

1

• erfc

x′ ℓ0

• −

2 √πℓ0

Xavier Durang, Jean Yves Fortin, Malte Henkel

III.12 Auto-correlation and auto-response functions

C(0; t, s) = C(t, s) = 1 sb fC(y = t/s) := 1 s fC(y = t/s) R(0; t, s) = R(t, s) = 1 s1+a fR(y = t/s) := 1 √s fR(y = t/s) Asymptotically fC(y) ≃ (2D)−1 (1 − 8/(3π))π−1y−2 + ( 16

5 π − 1/2)π−1y−3

which gives the exponent λC = 4. Asymptotically fR(y) ≃ (2D)−1/2 4

3π−3/2y−2 + 8 15π−3/2y−3

which gives the exponent λR = 4.

Xavier Durang, Jean Yves Fortin, Malte Henkel

III.13 Asymptotic behavior

fC behaves asymptotically like fC(y) ≃ (2D)−1

• (1 − 8/(3π))π−1y−2 + (16

5 π − 1/2)π−1y−3

• with the dynamical exponent z = 2, which gives the exponent

λC = 4. fR behaves asymptotically like fR(y) ≃ (2D)−1/2 4 3π−3/2y−2 + 8 15π−3/2y−3

• which gives the exponent λR = 4

Xavier Durang, Jean Yves Fortin, Malte Henkel

IV.4 Behaviour of Ξ(t, s) for the coagulation-diffusion process

1 10 100 1000

t/s

~ 3.307 3 2 3.5 2.5

Ξ(t,s)

The behaviour is not monotonous while in models such that a + 1 = b or a = b it seems to be generically the case.

Xavier Durang, Jean Yves Fortin, Malte Henkel

IV.3 Model with a + 3/2 = b (Coagulation-diffusion)

We use the Riemann representation for fractional derivatives with negative arguments ∂−ν ∂s−ν ϕ(s) = 1 Γ(ν) s (s − u)ν−1ϕ(u) du Asymptotic limits when t ≫ s ≫ 1 C(t, s) ≃ aC t−2s and R(t, s) ≃ aR t−2s3/2 with aC = (2D)−1(3π − 8)/3π−2 and aR = (2D)−1/24/3π−3/2 then the FDR in the asymptotic limit t ≫ s ≫ 1 reads X∞ = lim

y≫1 X(t, s) =

3π 6π − 16 ≃ 3.307

Xavier Durang, Jean Yves Fortin, Malte Henkel

IV.2 Models

model reactions Diffusion nmax BCPD A → 2A A → ∅ diffusion ∞ BCPL A → 2A A → ∅ L´ evy flight ∞ FA A → 2A 2A → A none 1 contact process A → 2A A → ∅ diffusion 1 NEKIM 2A → ∅ A ↔ 3A diffusion 1

Xavier Durang, Jean Yves Fortin, Malte Henkel

model d b, a Ξ∞ Ref. BCPD > 0 b = a 1/2

• F. Baumann

BCPL > 0 b = a 1/2

• X. Durang

FA 1 b = a −3π/(6π − 16)

• P. Mayer

contact process 1 b = 1 + a 1.15(5) (directed perco.) 4 − ε b = 1 + a 2 − ε( 119

240 − π2 60 )

• F. Baumann

NEKIM 1 b = 1 + a ≈ 0.1

• G. Odor

coag.-diff. 1 b = 3/2 + a 3π/(6π − 16) this work

Xavier Durang, Jean Yves Fortin, Malte Henkel

Models with a = b contains critical systems with detailed balance (FA model) in systems without detailed balance, we use X(t, s) = R(s, s) ∂sC(s, s) −1 R(t, s) ∂sC(t, s) = X t s

• −

→ X∞ since there is no longer reference to an equilibrium temperature Models with 1 + a = b Limit value Ξ∞ is directly universal since the dependence one the scale ϑ drops out In the stationary state, one has Ξ−1

stat = 0

In the directed percolation class, the universality of Ξ∞ has been proven to one-loop order.

• F. Baumann and al. 2007

Xavier Durang, Jean Yves Fortin, Malte Henkel