f.corberi m. zannetti e.l
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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The generalization of fluctuation-dissipation theorem
and a new algorithm for the computation of the linear
response functionF.Corberi
M. Zannetti
E.L.
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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Ising Model in d=2
LSI VIOLATION
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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
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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
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Numerical results for the quench to T<Tc
LSI predicts aststR 1)(),(
Violation of LSI
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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
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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