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

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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.

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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
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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
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
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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?
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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
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SYSTEM WITH DISCRET SYMMETRY
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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
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Also for order parameter with discrete symmetry
one has
Results generality
No hypothesis on the form of unperturbed
transition rates W
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Algorithm Validation
Comparison with exact results
ISING NCOP d1
New applications
Computation of the punctual response R

• ISING d1 COP E.L., Corberi,Zannetti PRE 2004

• ISING d2 NCOP a Tlt TC Corberi, E.L., Zannetti
  PRE 2005

• ISING d2 e d4 NCOP a TTC E.L., Corberi,
  Zannetti sottomesso a PRE

• Clock Model in d1 Andrenacci, Corberi, E.L.
  PRE 2006

• Clock Model in d2 Corberi, E.L., Zannetti
  PRE 2006

• Local temperature Ising model 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
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Numerical results for the quench to TTc
Ising Model in d4
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
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Ising Model in d2
LSI VIOLATION
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Quench to TltTc
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
interfacesresponse
LSI predicts the same structure as at TTC. The
only difference is in the exponentsvalues
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Numerical results for the quench toTltTc
A comparison with LSI can be acchieved if one
focuses on the short time separation regime
(t-s)ltlts
One expects a time translation invariant and a
power law behavior with a slope 1a larger than 1
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Numerical results for the quench to TltTc
Violation of LSI
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Numerical results for the quench to TltTc
Agreement with the fenomenological picture with
a0.25
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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 d4 and TTC results agree with LSI
  prediction. In d2 for both the quench to TTC
  and to TltTc one observes deviations from LSI