Experimental Quenching of Harmonic Stimuli: Universality of Linear Response Theory Page: 1
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PRL 103, 030602 (2009)
PHYSICAL REVIEW LETTERS
Experimental Quenching of Harmonic Stimuli: Universality of Linear Response Theory
Paolo Allegrini,1 Mauro Bologna,2'3 Leone Fronzoni,1 Paolo Grigolini,1'2'4 and Ludovico Silvestri1'5
iDipartimento di Fisica "E. Fermi," Universita di Pisa and INFM CRS-SOFT, Largo Pontecorvo 3, 56127 Pisa, Italy
2Center for Nonlinear Science, University of North Texas, PO. Box 311427, Denton, Texas 76203, USA
3Instituto de Alta Investigacidn, Universidad de Tarapacd-Casilla 6-D Arica, Chile
4lstituto dei Processi Chimico Fisici del CNR Area della Ricerca di Pisa, Via G. Moruzzi 1, 56124 Pisa, Italy
5LE.N.S., University of Florence, via Nello Carrara 1, 50019 Sesto Fiorentino (FI), Italy
(Received 26 May 2009; revised manuscript received 16 June 2009; published 15 July 2009)
We show that liquid crystals in the weak turbulence electroconvective regime respond to harmonic
perturbations with oscillations whose intensity decay with an inverse power law of time. We use the results
of this experiment to prove that this effect is the manifestation of a form of linear response theory (LRT)
valid in the out-of-equilibrium case, as well as at thermodynamic equilibrium where it reduces to the
ordinary LRT. We argue that this theory is a universal property, which is not confined to physical processes
such as turbulent or excitable media, and that it holds true in all possible conditions, and for all possible
systems, including complex networks, thereby establishing a bridge between statistical physics and all the
fields of research in complexity.
DOI: 10 1103/PhysRevLett 103 030602
The linear response theory (LRT)  is a theoretical tool
of general interest in physics. In spite of some criticism ,
the experimental work done over a time span of about
52 years has not revealed any breakdown of the theory.
There is general agreement that the LRT is one of the
fundamental accomplishments of statistical physics. In
addition to affording an invaluable guideline for experi
mental investigation in condensed matter, the LRT led Lee
 (see also ) to the definition of an inner operator
product and hence to the foundation of an operator basis
set that made it possible for him to design a rigorous,
efficient, and versatile approach to the relaxation of
Hamiltonian systems . Unfortunately, no general theory
exists yet to extend in the same elegant manner the LRT
from equilibrium to nonequilibrium conditions .
The conventional LRT rests on two basic assumptions:
(i) The time evolution of the system variable es is driven
by Hamiltonian operators (Liouville equation); (ii) The
external perturbation sr has the effect of making the
system depart from canonical equilibrium so weakly as
to render the linear response function X(t, tj) identical to
the derivative of the correlation function C(t, t') ,
X(t, t') = C(t, t), (1)
where C(t, I') = C(t - t') is stationary. The response is
(s(t)) = E dt'x(t, t)p(t), (2)
with E < 1 being the interaction strength.
The recent literature on non Poisson renewal processes
is raising increasing interest on the action of nonergodic
renewal events [8 10], with a wide set of applications,
ranging from quantum mechanics  to the brain dynam
PACS numbers: 05 60 Cd, 02 50 Ey, 47 52 +j, 61 30 v
ics , thereby casting deep doubt on the possibility of
using the conventional LRT to study the effects of pertur-
bation in these cases. This is so for two main reasons: (i) It
is very difficult, if not impossible , to describe the time
evolution of the event driven systems by means of
Hamiltonian operators (classical or quantum Liouville
equation); (ii) It is not yet well understood how to use
the linear response structure of Eq. (2) when a stationary
correlation function is not available, in spite of the fact that
some prescriptions already exist [14,15]. These are proba
bly the reasons why Sokolov and Klafter [16,17] coined the
term "death of linear response" to denote the fading away
response of a complex system to a harmonic stimulus. This
interesting phenomenon raised the interest of other re
searchers and also a debate on the best way to generate it
with surrogate sequences [18-20].
The purpose of this Letter is twofold: we give the first
experimental evidence of this interesting effect, task No. 1,
but we argue that surprisingly the LRT is in action also in
this case, task No. 2. We use real experiments on liquid
crystals, not only to support our theoretical arguments, but,
more importantly, to make our conclusions accessible to an
audience of readers as wide as possible. Our arguments do
not require a Hamiltonian formalism: this is not a limita
tion, but rather a significant extension of the LRT that is
expected to apply not only to physical systems but to
neurophysiological and sociological processes as well
, thereby involving an interdisciplinary audience.
The guidelines for this wide audience of readers to
understand the new LRT are indicated by the following
three steps: preparation, perturbation, and experiment.
Preparation.-There are systems, and liquid crystals
electroconvective belongs to this group, whose nonequi
librium nature is determined by a cascade of renewal
2009 The American Physical Society
17 JULY 2009
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Allegrini, Paolo; Bologna, Mauro; Fronzoni, Leone; Grigolini, Paolo & Silvestri, Ludovico. Experimental Quenching of Harmonic Stimuli: Universality of Linear Response Theory, article, July 15, 2009; [College Park, Maryland]. (digital.library.unt.edu/ark:/67531/metadc40394/m1/1/: accessed June 27, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.