Quasi equilibrium construction for long time glassy dynamics Pierfrancesco Urbani Dipartimento di Fisica Università di Roma, La Sapienza Laboratoire de Physique Théorique et Modèles Statistiques Université Paris-Sud pierfrancesco.u@gmail.com In collaboration with: Silvio Franz, Giorgio Parisi 19 July, 2013 Yukawa Institute, Kyoto P. Urbani (La Sapienza, LPTMS) Boltzmann Pseudodynamics July 2013 1 / 16
Overview Overview of the talk • Dynamics of glassy systems • Mean Field disordered systems • Langevin Dynamics • Supersymmetric approach • The Potential Method • The Boltzmann Pseudodynamics • Application to mean field spin glass systems • Results in the replicated liquid theory • Dynamical Ornstein-Zernike equations • HNC closure • Conclusions and perspectives P. Urbani (La Sapienza, LPTMS) Boltzmann Pseudodynamics July 2013 2 / 16
Dynamics of glassy systems 1 0.8 0.6 T = T MCT C ( t ) 0.4 0.2 0 10 − 2 10 0 10 2 10 4 10 6 t Two exponents: C ( t ) ≃ q EA + c 1 t − a + ... C ( t ) ≃ q EA + c 2 t b + ... The λ exponent is defined by λ = Γ 2 ( 1 − a ) Γ ( 1 − 2 a ) = Γ 2 ( 1 + b ) γ = 1 2 a + 1 τ α ∼ | T − T d | − γ 2 b = f ( λ ) Γ ( 1 + 2 b ) P. Urbani (La Sapienza, LPTMS) Boltzmann Pseudodynamics July 2013 3 / 16
Dynamics from statics: first steps Mean Field models (Caltagirone, Ferrari, Leuzzi, Parisi, Ricci-Tersenghi, Rizzo, 2011) → schematic mode coupling equations: Ingredients (Kurchan 1992): • Langevin Dynamics • Martin-Siggia-Rose functional • Supersymmetric formalism • Dynamical action • Saddle point equations → schematic mode coupling equation • ultrafast motion limit ("replicas = supertimes") λ = w 2 w 1 where w 1 and w 2 are two of the cubic coefficient of a replica field theory action that can be written from the statics. P. Urbani (La Sapienza, LPTMS) Boltzmann Pseudodynamics July 2013 4 / 16
The Potential Method Consider a system with hamiltonia H J ( σ ) where J is an internal quenched disorder and σ is the configuration of the internal degrees of freedom. Define the potential (Franz Parisi 1995) e − β H J [ τ ] 1 � V ( q ) = − lim log Z J [ q , τ ] N β E J Z J N →∞ τ where � � e − β H J [ σ ] δ ( q − q ( σ, τ )) Z J [ q , τ ] = Z J = d qZ J [ q , τ ] σ 0.02 T=T d T=T g 0.015 T=T K V(q) 0.01 0.005 0 0 0.2 0.4 0.6 0.8 1 q P. Urbani (La Sapienza, LPTMS) Boltzmann Pseudodynamics July 2013 5 / 16
The Boltzmann Pseudodynamics Let us consider a generalization of the Franz-Parisi potential. We define the potential of a chain of coupled systems of length L the following way (Franz Parisi, 2012) L − 2 1 1 � � { β k } ; { ˜ � � Z e − β 1 H J [ σ 1 ] � V C ( k − 1 , k ) } = − lim . . . M ( σ k + 1 | σ k ) × N E J N →∞ σ 1 σ L − 1 k = 1 � � � ˜ e − β L H J [ σ L ] δ × ln C ( L − 1 , L ) − q ( σ L − 1 , σ L ) σ L and 1 � � Z ( σ k − 1 ) e − β k H J [ σ k ] δ ˜ M ( σ k | σ k − 1 ) = C ( k − 1 , k ) − q ( σ k − 1 , σ k ) � � � ˜ e − β k H J [ σ k ] δ Z ( σ k − 1 ) = C ( k − 1 , k ) − q ( σ k − 1 , σ k ) . σ k We considered the most general case where the temperature of each system is different one from another. The kernel M ( σ k | σ k − 1 ) defines the Boltzmann Pseudodynamics Markov Chain. P. Urbani (La Sapienza, LPTMS) Boltzmann Pseudodynamics July 2013 6 / 16
Schematic models - p -spin (1) Let us consider the p -spin spherical model. The replica method can be employed to treat the logarithm and the factors Z ( σ k − 1 ) − 1 . In this way we have a replicated chain. For each system we have a certain number of replicas that eventually will be sent to zero. By averaging over the disorder, the potential becomes a function of the overlap N Q ab ( t , s ) = 1 � σ ( a ) ( t ) σ ( b ) ( s ) i i N i = 1 and the action for the potential becomes n t n s L S ( Q ) = β 2 Q ab ( t , s ) p + 1 � � � 2 ln det Q 4 t , s = 1 a = 1 b = 1 We must choose a parametrization for the overlap matrix that is compatible with the constraints. P. Urbani (La Sapienza, LPTMS) Boltzmann Pseudodynamics July 2013 7 / 16
Schematic models - p -spin (2) Replica symmetric parametrization for the overlap matrix Q ab ( t , s ) = C ( t , s ) + δ ab δ su ∆ C ( s , s ) + Θ > ( s − t ) δ a 1 ∆ C ( t , s )+ + Θ > ( t − s ) δ a 1 ∆ C ( s , t ) ∆ C ( t , s ) = ˜ C ( t , s ) − C ( t , s ) We want to optimize over the free parameter of the overlap matrix. However, as in the standard potential method, we will search for a stationary point of the potential with respect to all the constraints. In this way we will optimize over all the parameters of Q . The saddle point equations are L n z β 2 p [ Q ac ( k , z )] p − 1 Q cb ( z , j ) + δ kj δ ac − ν k Q ab ( k , j ) = 0 � � 2 z = 1 c = 1 In the limit in which the chain becomes infinitely long we put the crucial ansatz 1 β R ( u , s ) d s = Θ > ( u − s ) ∆ C ( s , u ) that can be justified by an explicit computation of the response function within the Boltzmann pseudodynamics process. P. Urbani (La Sapienza, LPTMS) Boltzmann Pseudodynamics July 2013 8 / 16
Schematic models - p -spin (3) The crucial result is that L n z � [ Q ac ( k , z )] p − 1 Q cb ( z , j ) = � � d c Q ( a , c ) p − 1 Q ( c , b ) lim { n ( t ) }→ 0 z = 1 c = 1 The resulting equations are ν ( t ) C ( t , u ) = β 2 p C p − 1 ( t , u ) ∆ C ( u , u ) + ∆ C p − 1 ( t , t ) C ( t , u ) + C p − 1 ( t , 0 ) C ( u , 0 )+ � 2 � u � t + 1 d z C p − 1 ( t , z ) R ( u , z ) + p − 1 � d z C p − 2 ( t , z ) R ( t , z ) C ( z , u ) β β 0 0 � 1 β ν ( t ) R ( t , u ) = β 2 p 1 β ∆ C p − 1 ( t , t ) R ( t , u ) + p − 1 C p − 2 ( t , u ) R ( t , u ) ∆ C ( u , u )+ 2 β � t � p − 1 d z C p − 2 ( t , z ) R ( t , z ) R ( z , u ) β 2 u ν ( t ) ∆ C ( t , t ) = β 2 p 2 ∆ C p − 1 ( t , t ) ∆ C ( u , u ) + 1 . By imposing ∆ C ( t , t ) = 1 − q d we recover the dynamical equation in the α regime. P. Urbani (La Sapienza, LPTMS) Boltzmann Pseudodynamics July 2013 9 / 16
The Boltzmann Pseudodynamics for structural glasses -(1) Consider a replicated system of particles so that we can define the fields N � δ ( x − x ( a ) � δ ( x − x ( a ) ) δ ( y − x ( b ) ρ a ( x ) = � ) � ρ ab ( x ; y ) = � ) � i i j i = 1 [ ij ] ρ ab ( x , y ) h ab ( x , y ) = ρ a ( x ) ρ b ( y ) − 1 And the replicated Ornstein-Zernike equations that defines the direct correlation function n � � c ab ( x , y ) = h ab ( x , y ) − d z h ac ( x , z ) ρ c ( z ) c cb ( z , y ) c = 1 We will consider solutions such that ρ a ( x ) = ρ . Note that in the OZ equation there is the product between the matrix h and the matrix c . P. Urbani (La Sapienza, LPTMS) Boltzmann Pseudodynamics July 2013 10 / 16
The Boltzmann Pseudodynamics for structural glasses -(2) Put now the pseudodynamics ansatz h ab ( x ) = h ( s , u ; x ) + δ ab δ su ∆ h ( s , s ; x ) + Θ > ( u − s ) δ a 1 ∆ h ( s , u ; x )+ + Θ > ( s − u ) δ b 1 ∆ h ( s , u ; x ) 1 β R h ( q ; u ; s ) d s = Θ > ( u − s ) ∆ h ( q ; s , u ) and the analogous expression for the direct correlation function. The OZ equations become h ( q ; s , u ) = c ( q ; s , u ) + ρ [ h ( q ; s , 0 ) c ( q ; 0 , u ) + h ( q ; s , u ) ∆ c ( q ; u , u )+ � u � s � + ∆ h ( q ; s , s ) c ( q ; s , u ) + 1 d zh ( q ; s , z ) R c ( q ; u , z ) + 1 d zR h ( q ; s , z ) c ( q ; z , u ) β β 0 0 ∆ h ( q ; s , s ) = ∆ c ( q ; s , s ) + ρ∆ h ( q ; s , s ) ∆ c ( q ; s , s ) R h ( q ; u , s ) = R c ( q ; u , s ) + ρ [ R h ( q ; u , s ) ∆ c ( q ; u , u ) + ∆ h ( q ; s , s ) R c ( q ; u , s )+ � u + 1 � d zR h ( q ; z , s ) R c ( q ; u , z ) . β s P. Urbani (La Sapienza, LPTMS) Boltzmann Pseudodynamics July 2013 11 / 16
The Boltzmann Pseudodynamics for structural glasses -(3) We have to choose a closure for the OZ equations. We choose HNC. ln [ h ab ( x , y ) + 1 ] + βφ ab ( x , y ) = h ab ( x , y ) − c ab ( x , y ) Putting the BPD ansatz in the equation we get ln [ h ( x ; s , u ) + 1 ] = h ( x ; , s , u ) − c ( x ; s , u ) h ( x ; s , u ) R c ( x ; s , u ) = R h ( x ; s , u ) 1 + h ( x ; s , u ) . and moreover that ∆ h ( q , s , s ) and ∆ c ( q , s , s ) are actually s independent. We can now analyze the full set of equations. First we go to the equilibrium regime and we see that the equations admits a solution that satisfies TTI + FDT − β d h ( x ; s − u ) = R h ( x ; s − u ) d s P. Urbani (La Sapienza, LPTMS) Boltzmann Pseudodynamics July 2013 12 / 16
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