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. 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

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. 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?

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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)

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. 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

14. 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

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

16. Numerical results for the quench to T<Tc
The fenomen. picture predicts
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