The generalization of fluctuation-dissipation theorem

and a new algorithm for the computation of the linear

response functionF.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 temperatureCugliandolo, 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

RT

hsh h 2

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

The signal-noise ratio is of order h2 i.e. to small to be

detected

Onsager regression hypothesis (1930)

s(t)

t’

The relaxation of macroscopic perturbations is controlled by the same laws governing the regression of spontaneous fluctuations of the equilibrium system

TR t t C t tt( ' ) ( ' )

OUT OF EQUILIBRIUM

s(t)t’

Can be R expressed in term of some correlation controlling non stationary spontaneous fluctuations?

EQUILIBRIUM

Order Parameter with continuous symmetry

Langevin Equation ts t B t t( ) ( ) ( )

Deterministic ForceWhite noise

White noise property 2T R t t s t t( , ' ) ( ) ( ' )

2T R t t C t t s t B tt( , ' ) ( , ' ) ( ) ( ' )'

From the definition of B

2T R t t C t t C t t A t tt t( , ' ) ( , ' ) ( , ' ) ( , ' )' A t t s t B t B t s t( , ' ) ( ) ( ' ) ( ) ( ' ) Asimmetry

EQUILIBRIUM SYSTEMSTime reversion invariance A(t,t’)=0

Time translation invariance t tC t t C t t' ( , ' ) ( , ' )TR t t C t tt( ' ) ( ' )

B t O t s t O tt( ) ( ' ) ( ) ( ' ) t’<t

Cugliandolo, Kurchan, Parisi, J.Physics I

France 1994

SYSTEM WITH DISCRET SYMMETRY

Transition rates W satisfy detailed balance condition

W C C e W C C eH C H C( ' ) ( ' )( ) ( ')

Constraint on the form of Wh in the presence of the external field

)]'()([

21)'()'( 0 CsCs

T

hCCWCCWh

Dynamical evolution is controlled by the Master-Equation. Conditional probability can be written as

)()'()',(),',( 2totCCWCCtCttCP

For the computation of R, one supposes that an external field is switched on during the interval [t’,t’+t]

)'t,''C(Ph

)'t,''Ct't,'C(P)t't,'Ct,C(P)C(s

)'t(h

)t(s)'t,t(R h

''C,'C,C

2T R t t C t t s t B tt( , ' ) ( , ' ) ( ) ( ' )'

B C s C s c W C CC

( ) [ ( ' ) ( )] ( ' )'

0With the quantity acting as the

deterministic force of Langevin Equation

B t O t s t O tt( ) ( ' ) ( ) ( ' )

E.L., Corberi,Zannetti PRE 2004

t)'CC(W)'C,C()t,'Ctt,C(P hh

)]'C(s)C(s[

T2

h1)C'C(W)'CC(W 0h

The h dependence is all included in the transition rates W

Result’s generalityNo hypothesis on the form of unperturbed transition rates W

A New algorithm for the computation of R )'t(B)t(s)'t,t(C)'t,t(RT2 't

It holds for any Hamiltonian

Quenched disorder

Independence of the number of order parameter componentsIsing Spins di infinite number of components

Independence of dynamical constraints COP, NCOP

2T R t t C t t C t t A t tt t( , ' ) ( , ' ) ( , ' ) ( , ' )' GENERALIZATION OF FLUCTUATION DISSIPATION THEOREM

Analogously to the case of Langevin spins

Also for order parameter with discrete symmetry one has

Algorithm Validation

Comparison with exact results ISING NCOP d=1

New applicationsComputation of the punctual response R

• Local temperature Ising model Andrenacci, Corberi, E.L. PRE 2006

• ISING d=1 COP E.L., Corberi,Zannetti PRE 2004

•ISING d=2 NCOP a T< TC Corberi, E.L., Zannetti PRE 2005

•Clock Model in d=2 Corberi, E.L., Zannetti PRE 2006

•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

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

)/()(),( /)2( stfxstAstR Rzd

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=4The 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

Quench to T<Tc

The fixed point of the dynamics is no gaussian. One cannot use the powerfull tool of expansion used at TC.

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

),()(),( stRstRstR agst

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

Fenomenological hypothesis

For the aging contribution one expects the structure)/()(),( /11/1 ststsstR zaz

ag F.Corberi, E.L. e M.Zannetti PRE (2003)

In agreement with the Otha, Jasnow, Kawasaky approximation

Dynamical evolution is characterized by the growth of compact regions (domains) with a typical size L(t)=t1/z

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 aststR 1)(),(

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 aststR 1)(),(

Violation of LSI

The fenomen. picture predicts zazst stsstRstR /11/1 )()(),(

Agreement with the fenomenological picture with a=0.25

Numerical results for the quench to T<Tc

CONCLUSIONS

• 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

• 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