Renewal aging and linear response Page: 1
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Renewal aging and linear response
Paolo Allegrini', Giainluca Ascolani' . Ilauro Bologna', and Paolo Grigolni:3'4
1Dipartirnento di Fisica "E.Fermi" - Universita di Pisa, Largo Pontecorvo. 3 56127 Pisa. Italy
Departamento de Fisica, ULniversidad de Tarapacd,
Carmp'us Veldsq'a z, Veldsq'uez 1775. Casilla 7-D, Arica, Chile
Center for Nonlinear Science. University of North Texas,
P.O. Box 311427, Denton, Texas 76203-1427. USA and
4Istituto dei Processi Chirnmico Fisici del CNR, Area della Ricerca di Pisa, Via G. Moruzzi, 56124, Pisa. Italy
(Dated: February 6, 2008)
'e study the linear response to an external p)crturbation of a renewal process, in an aging ondi-
tion that, with no perturbation, would yield super-diffusion. We use the phenomenological approach
to the linear response adopted in earlier work of other groups, and wc find that aging may have
the effect of annihilating any sign of coherent response to harmonic perturbation. We also derive
the linear response using dynamic argumncllts and we find a ('oherent response, although with an
intensity dying out very slowly. In the case of a step-like perturbation the dynamic arguments
yield in the long-time limit a steady signal whose intensity may be significantly smaller than the
phenomenological approac h prcdic(tion.
PACS numbers: 05.40.Fb,02.50.-r,82.20. Uv
The problem of linear response' of a dynamic system to an external pirtlurbation  is one topic of general interest in
the field of statistical thermodynamics. Many authors  have studied the linear response emergence from microscopic
chaos, and thus from a condition that is expected to generate ordinary statistical physics . In the last few years
an increasing ilunber of investigators haxve been addressing the interesting issue of the response to pertlurbatiolns of
dynamical systems departing from the condition of ordinary statistical thermodynamics [4, 5]. A great attention has
been devoted to the case of sub-diffusion that makes it possible to fulfill the Einstein relation  even if the dynamic
systirni gencratcs anomalous diffusion .
It is important to point oout that the same dynarnic generator of interminttence yields either super- or sulb-diffusion
according to whether the long sojourn times refer to the velocity or to the position state, respectively. Let us consider
a regular one-dimnensional lattice, with the fixed distance a between the nearest neighbors. a.nd a particle jumping
from one to another site of this lattice. Let Ius imagine that the particle sojolurns for a. long t.inu' in one site and
that at the c(',nd of this sojoulrrn makes a jlmrnp from the position na. with ri being an integer number ranging from
-oo to +oo, either to the position (n + 1)a or to the position (n- - 1)a, according to the coin-tossing prescription.
This condition can be realized with a dynamic generator of events, occurring at times to = 0, t-, t2. .., ti,.. , with
the waiting times = t'+1 - ti corresponding to a given distribution '-(7). Notice that a single jump involves the
occurrence of two events. One cve'nt is the drawing of a mllunbrber 7i fro the distribution ' ;,(7), and the other event is
the coin tossing. It is worthwhile to stress that in this article we refer to time t as an integer number. The adoption
of the continuous time picture is a good approximation made possible by the choice of conditions that involve t > 1.
Wvith the same dynamic model and the same coin-tossing prescription, we can generate a diffcrcrnt ralndomrr walk
process, producing sluper-diffusion. Here' the random walker inoves with a velocity of fixed mnodhulus I., whose sign
is determined by two events occurring at the times t;,. As in the earlier example, the two events are the drawing of
the time i;, from the distribution i!'(v) and the coin tossing that fixes the velocity sign of the walker between time
t. and time ti+l. This corresponds to defining two states, the state I1 >. with velocity WI and the state 2 >, with
velocity - TV. X\'e refer to the time interval 7; - t;+l - t, as waiting tirite. Of course', the walker can repeatedly j]ullnp
to the right (left), in the first case, or maintain the same positive (negative) velocity sign, in the second case. The
two proccssc's are tliff.erceit, insofar as in + the first case, trhe' large r the sojourn tinmc trhe slower tihe diffusion process,
while' in tihe second tihe opposite conditions applies: the larger tihe sojourn times tihe faster the diffusion pro'e'ss. In
this paper we adopt the velocity rather tha.n the position picture, the reason for this choice being that, as we shall
see, it establishes a connection with the interesting phenomenon of non-Poisson stochastic resonance . We shall be
referring tihrolghout to the wide time regions between t; and ,+l, where no unpredictable cve'nts occur, as qeuiescenat
regions. We shall adlo)lpt a dylamic' model inspired to turbulnce and to thce M\annv'ille ma.p ,where tihe qcriese'sc.rrt
regions are usually denoted as laminar regions. This dyna ic model [9, 10] yields the waiting time distribution !(T)
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Allegrini, Paolo; Ascolani, Gianluca; Bologna, Mauro & Grigolini, Paolo. Renewal aging and linear response, paper, February 6, 2008; (https://digital.library.unt.edu/ark:/67531/metadc174702/m1/1/: accessed March 25, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.