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 t e (or s ) and creep time t 3. find master curve for all ( t , t e ) − → dynamical scaling Quench Constraint Measure time 0 s t Xavier Durang , Jean Yves Fortin, Malte Henkel
t = t 1 t = t 2 > t 1 magnet T < T c − → ordered cluster magnet T = T c − → correlated cluster critical contact process diffusion, A → 2 A , A → φ = ⇒ cluster dilution voter model, contact process,. . . Characteristic length scale : L ( t ) ∼ t 1 / 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 ) � � R ( t , s ) := δ � φ ( t , r ) � � two-time response � δ h ( s , r ) � h =0 (Directed percolation : h ( t ) = creation of part.) t : observation time, s : waiting time Scaling regime : t ≫ s ≫ τ micro (For simple magnets) � t � t � � C ( t , s ) = s − b f C , R ( t , s ) = s − 1 − a f R s s Asymptotics : f C , R ( y ) ∼ y − λ C , R / z for y ≫ 1 λ C : autocorrelation exponent, λ R : autoresponse exponent, z : dynamical exponent, a , b : ageing exponents Xavier Durang , Jean Yves Fortin, Malte Henkel
I.3 Fluctuation dissipation ratio Usually λ R = λ C The fluctuation-dissipation ratio ( fdr ) Cugliandolo, Kurchan, Parisi ’94 TR ( t , s ) � � X ( t , s ) := ∂ C ( t , s ) /∂ s X ∞ = lim t →∞ X ( t , s ) lim s →∞ Godr` eche & Luck 00 measures the distance to the equilibrium : X eq = X ( t − s ) = 1. a = b valid when systems satisfy detailed balance Contact process 1 + a = b : ⇐ = rapidity-reversal symmetry of stationary state of cp ⇒ specific property ! = ⇒ try new form of FDR ! Enss et. al. 04 Ξ( t , s ) := R ( t , s ) C ( t , s ) = f R ( t / s ) � � f C ( t / s ) , Ξ ∞ := lim t →∞ Ξ( t , s ) lim 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 Diffusion D Coagulation (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 E n ( t ) : time-dependent probability of having an interval of n consecutive empty sites at time t c ( t ) = E 1 ( t ) − E 0 ( t ) continuum limit ( x = na ) − → c ( t ) = − ∂ x E ( x , t ) | x =0 Equation of motion (2 D / a 2 ) ( E n − 1 − 2 E n + E n +1 ) For n > 1 ∂ t E n ( t ) = � � (2 D / a 2 ) For n = 1 ∂ t E 1 ( t ) = 1 − 2 E 1 ( t ) + E 2 ( t ) This gives the constraint : E 0 ( t ) = 1 Equation of motion in the continuum limit ( x = na ) ∂ t E ( x , t ) = 2 D ∂ xx E ( 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 � ∞ dx ′ − 1 � ℓ 02 ( x − x ′ ) 2 � E ( x ′ , 0) . √ πℓ 0 E ( x , t ) = exp −∞ √ where ℓ 0 is the scaling length ℓ 0 := 8 Dt . Take into account the constraint : E 0 ( t ) = 1. For n = 0 (2 D / a 2 ) ( E − 1 − 2 E 0 + E 1 ) = 0 ∂ t E 0 ( t ) = E − 1 ( t ) = 2 E 0 ( t ) − E 1 ( t ) = 2 − E 1 ( t ) Redefine the meaning of E ( n , 0) for negative n such that E − n ( t ) = 2 − E n ( t ) E ( − x , t ) = 2 − E ( x , t ) and Xavier Durang , Jean Yves Fortin, Malte Henkel
II.3 General expression for the particle concentration One-empty-interval probability E ( x , t ) = erfc ( x /ℓ 0 ) � + ∞ dx ′ 1 1 ℓ 02 ( x − x ′ ) 2 ℓ 02 ( x + x ′ ) 2 � − − � E ( x ′ , 0) + √ πℓ 0 − e . e 0 Hierarchy Particle concentration � ∞ 2 � dxE ( x ℓ 0 , 0)2 xe − x 2 � √ πℓ 0 1 − c ( t ) = 0 2 + o(1 /ℓ 0 ) ∼ t − 1 / 2 c ( t ) = √ πℓ 0 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 E n 1 , n 2 , d ( t ) : time-dependent probability of having two intervals of n 1 and n 2 consecutive empty sites distant from d at time t Continuum limit ( x = n 1 a , y = n 2 a , z = da ) C ( z , t ) = ∂ 2 � xy E ( x , y , z , t ) x =0 , y =0 − ∂ x E ( x , t ) | x =0 ∂ y E ( 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) � � �� ∂ 2 x + ∂ 2 y + ∂ 2 ∂ t E ( x , y , z , t ) = 2 D 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 ) E (0) ( x , y , z , t ) is independent of the initial conditions 1 E (1) ( x , y , z , t ) depends on the initial one-interval probability 2 E (2) ( x , y , z , t ) depends on the initial two-intervals probability 3 → System initially filled with particles � x � y � � E ( x , y , z ; t ) = E (0) ( x , y , z , t ) = erfc erfc ℓ 0 ℓ 0 � z � � x + y + z � � x + z � � y + z � + erfc − erfc erfc erfc ℓ 0 ℓ 0 ℓ 0 ℓ 0 Xavier Durang , Jean Yves Fortin, Malte Henkel
II.6 One-time correlation function Correlation function C ( z , t ) = ∂ 2 � xy E ( x , y , z , t ) x =0 , y =0 − ∂ x E ( x , t ) | x =0 ∂ y E ( 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 ℓ 0 = 8 Dt . Connected correlator � 2 � 2 √ πℓ 0 C ( z , t ) = f ( z /ℓ 0 ) − e − 2 y 2 + √ π ye − y 2 erfc ( y ) with f ( 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 (0) (z,t) C 0,1 (0) (z,t)+C (1) (z,t) C (0) (z,t)+C (1) (z,t)+C (2) (z,t) C -C(z=1/2,t) 0,01 10 100 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 � ≫ � ≫ � C (0) (1 / 2 , t ) � C (1) (1 / 2 , t ) � C (2) (1 / 2 , t ) � � � � � � Hierarchy : � 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 ) � � � = � δ h ( s , r ) � h =0 φ = part. density and h ( t ) = creation of part. d n 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 ( d ; t , s ) = − ∂ x F ( x , z ; t , s ) | x =0 ∼ 1 / L 2 C ( z ; t , s ) = lim a a → 0 Two-time response, G has the same definition than F R ( d ; t , s ) = − ∂ x G ( x , z ; t , s ) | x =0 ∼ 1 / L R ( z ; t , s ) = lim a a → 0 Initial conditions at t = s 1 F ( x , z ; s , s ) = lim aF ( n , d ; s , s ) = − ∂ y E ( x , y , z ; s ) | y =0 a → 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
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