The generalization of fluctuation-dissipation theorem and a new algorithm for the computation of the linear response function F.Corberi M. Zannetti E.L.

Motivations The analysis of the response function R is an efficient tool to characterize non-equilibrium properties of slowly evolving systems R can be related to the overlap probability distribution P(q) of the equilibrium state Franz, Mezard, Parisi e Peliti PRL 1998 R can be used to define an effective temperature Cugliandolo, Kurchan, e Peliti PRE 1998

Numerical computation of R(t,s) In the standard algorithms a magnetic field h is switched-on for an infinitesimal time interval dt. Response function is given by the correlation between the order parameter s and h The signal-noise ratio is of order h2 i.e. to small to be detected In order to improve the signal-noise ratio one looks for an expression of R in terms of unperturbed correlation functions Generalizations of the fluctuation-dissipation theorem

Onsager regression hypothesis (1930) EQUILIBRIUM Onsager regression hypothesis (1930) The relaxation of macroscopic perturbations is controlled by the same laws governing the regression of spontaneous fluctuations of the equilibrium system s(t) t’ s(t) t’ OUT OF EQUILIBRIUM Can be R expressed in term of some correlation controlling non stationary spontaneous fluctuations?

Cugliandolo, Kurchan, Parisi, J.Physics I France 1994 Order Parameter with continuous symmetry Langevin Equation Deterministic Force White noise White noise property Cugliandolo, Kurchan, Parisi, J.Physics I France 1994 From the definition of B t’<t Asimmetry EQUILIBRIUM SYSTEMS Time reversion invariance A(t,t’)=0 Time translation invariance

SYSTEM WITH DISCRET SYMMETRY Dynamical evolution is controlled by the Master-Equation. Conditional probability can be written as Transition rates W satisfy detailed balance condition Constraint on the form of Wh in the presence of the external field

The h dependence is all included in the transition rates W For the computation of R, one supposes that an external field is switched on during the interval [t’,t’+t] The h dependence is all included in the transition rates W E.L., Corberi,Zannetti PRE 2004 With the quantity acting as the deterministic force of Langevin Equation

A New algorithm for the computation of R Also for order parameter with discrete symmetry one has GENERALIZATION OF FLUCTUATION DISSIPATION THEOREM Analogously to the case of Langevin spins Result’s generality No hypothesis on the form of unperturbed transition rates W It holds for any Hamiltonian Quenched disorder Independence of dynamical constraints COP, NCOP Independence of the number of order parameter components Ising Spins di infinite number of components A New algorithm for the computation of R

Algorithm Validation New applications ISING NCOP d=1 Comparison with exact results ISING NCOP d=1 New applications Computation of the punctual response R ISING d=1 COP E.L., Corberi,Zannetti PRE 2004 ISING d=2 NCOP a T< TC Corberi, E.L., Zannetti PRE 2005 ISING d=2 e d=4 NCOP a T=TC E.L., Corberi, Zannetti sottomesso a PRE Clock Model in d=1 Andrenacci, Corberi, E.L. PRE 2006 Clock Model in d=2 Corberi, E.L., Zannetti PRE 2006 Local temperature Ising model Andrenacci, Corberi, E.L. PRE 2006

Analytical results for R in the quench toT c The Ising model quenched to T≤ TC Analytical results for R in the quench toT c Renormalization group and mean field theory provide the scaling form H.K.Janssen, B.Schaub, B. Schmittmann, Z.Phys. B Cond. Mat. (1989) P. Calabrese e A. Gambassi PRE (2002) is the static critical exponent, z is the growth exponent,  is the initial slip exponent and the function fR(x) can be obtained by means of the  expansion Local scale invariance (LSI) predicts fR(x)=1 M.Henkel, M.Pleimling, C.Godreche e J.M. Luck PRL (2001) The two loop  expansion give deviations from (LSI) and suggests that LSI is a gaussian theory P.Calabrese e A.Gambassi PRE (2002) M.Pleimling e A.Gambassi PRB (2206)

Numerical results for the quench to T=Tc Ising Model in d=4 The dynamics is controlled by a gaussian fixed point and one expects R(t,s)=A (t-s)-2 con fR(x)=1 as predicted by LSI. Numerical data are in agreement with the theorical prediction

Ising Model in d=2 LSI VIOLATION

F.Corberi, E.L. e M.Zannetti PRE (2003) Quench to T<Tc Dynamical evolution is characterized by the growth of compact regions (domains) with a typical size L(t)=t1/z The fixed point of the dynamics is no gaussian. One cannot use the powerfull tool of  expansion used at TC. Fenomenological hypothesis There exixts a fenomionenological picture according to which the response is the sum of a stationary contribution related to inside domain response and an aging contribution related to the interfaces’response For the aging contribution one expects the structure F.Corberi, E.L. e M.Zannetti PRE (2003) In agreement with the Otha, Jasnow, Kawasaky approximation LSI predicts the same structure as at T=TC. The only difference is in the exponents’values

Numerical results for the quench toT<Tc A comparison with LSI can be acchieved if one focuses on the short time separation regime (t-s)<<s LSI predicts One expects a time translation invariant and a power law behavior with a slope 1+a larger than 1

Numerical results for the quench to T<Tc LSI predicts Violation of LSI

Numerical results for the quench to T<Tc The fenomen. picture predicts Agreement with the fenomenological picture with a=0.25

CONCLUSIONS We have found an expression of R in term of correlation functions of the unperturbed dynamics. This expression can be considered a generalization of the Equilibrium Fluctuation-Dissipation Theorem We have found a new numerical algorithm for the computation of R The numerical evaluation of R for the Ising model confirms the idea that LSI is a gaussian theory. In d=4 and T=TC results agree with LSI prediction. In d=2 for both the quench to T=TC and to T<Tc one observes deviations from LSI