F.Corberi M. Zannetti E.L.

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

2. 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 Motivations The analysis of the response function R is an efficient tool to characterize non-equilibrium properties of slowly evolving systems

3. 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

4. s(t) t’ s(t) t’ 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 OUT OF EQUILIBRIUM Can be R expressed in term of some correlation controlling non stationary spontaneous fluctuations?

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

6. 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 SYSTEM WITH DISCRET SYMMETRY

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

8. 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 GENERALIZATION OF FLUCTUATION DISSIPATION THEOREM Analogously to the case of Langevin spins Also for order parameter with discrete symmetry one has Result’s generality No hypothesis on the form of unperturbed transition rates W

9. Algorithm Validation 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< TCCorberi, E.L., Zannetti PRE 2005 • ISING d=2 e d=4 NCOP a T=TCE.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

10. 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) The Ising model quenched to T≤ TC Analytical results for R in the quench toT c

11. 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

12. Ising Model in d=2 LSI VIOLATION

13. 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 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 LSI predicts the same structure as at T=TC. The only difference is in the exponents’values

14. LSI predicts 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 One expects a time translation invariant and a power law behavior with a slope 1+a larger than 1

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

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

17. 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