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The generalization of fluctuation-dissipation theorem and a new algorithm for the computation of the linear response function. F.Corberi M. Zannetti E.L. R can be related to the overlap probability distribution P(q) of the equilibrium state. Franz, Mezard, Parisi e Peliti PRL 1998.

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slide1

The generalization of fluctuation-dissipation theorem and a new algorithm for the computation of the linear response function

F.Corberi

M. Zannetti

E.L.

slide2

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

slide3

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

slide4

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?

slide5

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

slide6

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

slide7

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

slide8

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

slide9

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
slide10

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

slide11

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

slide12

Ising Model in d=2

LSI VIOLATION

slide13

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

slide14

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

slide15

LSI predicts

Numerical results for the quench to T<Tc

Violation of LSI

slide16

The fenomen. picture predicts

Numerical results for the quench to T<Tc

Agreement with the fenomenological picture with a=0.25

slide17

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