Noise-induced transition from anomalous to ordinary diffusion: The crossover time as a function of noise intensity Page: 5,910
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VOLUME 52, NUMBER 6
Noise-induced transition from anomalous to ordinary diffusion:
The crossover time as a function of noise intensity
Elena Floriani and Riccardo Mannella
Dipartimento di Fisica dell'Universitl di Pisa, Piazza Torricelli 2, 56100 Pisa, Italy
Dipartimento di Fisica dell'Universitd di Pisa, Piazza Torricelli 2, 56100 Pisa, Italy;
Department of Physics, University of North Texas, P.O. Box 5368, Denton, Texas 76203;
and Istituto di Biofisica del Consiglio Nazionale delle Ricerche, Via San Lorenzo 28, 56127 Pisa, Italy
(Received 10 July 1995)
We study the interplay between a deterministic process of weak chaos, responsible for the anoma-
lous diffusion of a variable x, and a white noise of intensity E. The deterministic process of anomalous
diffusion results from the correlated fluctuations of a statistical variable ( between two distinct val-
ues +1 and -1, each of them characterized by the same waiting time distribution # (t), given by
b(t) ~ t-" with 2 < pi < 3, in the long-time limit. We prove that under the influence of a weak
white noise of intensity E, the process of anomalous diffusion becomes normal at a time t, given
by t, ~ 1/(' (). Here P3(p) is a function of ~t which depends on the dynamical generator of the
waiting-time distribution b(t). We derive an explicit expression for 3(fp) in the case of two dynam-
ical systems, a one-dimensional superdiffusive map and the standard map in the accelerating state.
The theoretical prediction is supported by numerical calculations.
PACS number(s): 05.40.+j, 05.60.+w
The deterministic approach to anomalous diffusion has
been intensively studied in the last few years by several
groups [1-3]. The map originally introduced by Geisel
and co-workers  is now regarded  as the prototype
for deterministic dynamics leading to anomalous diffu-
sion faster than ordinary Brownian motion. It is a one-
dimensional map driving the motion of a variable, (,
which can be thought of as the velocity of a diffusing
particle whose position is given by the variable x itself.
In the continuous time limit we have
d x -
= ((t) (1)
and the map of Ref.  makes the variable ( fluctuate
between the two states, - 1 and = -1. The sojourn
time in each state is characterized by the distribution
Lb(t), which has the inverse power-law structure
lim 0b(t) = const (2)
in the long-time limit, with p > 1. Here we focus our
attention on the range
2 < < 3. (3)
This choice is determined by two distinct but related
reasons: (i) The range of Eq. (3) has been proved [2,3]
to correspond to the dynamical realization of an a-
stable Levy diffusion process. We remind the reader that
Levy processes of diffusion are [5,6] a generalization of
Brownian motion, characterized by distributions p(x)
whose Fourier transforms have the form
,S(k) - e-kla, (4)
with the parameter a in the range 0 < a < 2. The de-
terministic approach of Refs.  and  connects a and
pt via the relation a = t - 1. Thus, the region given
by Eq. (3) corresponds to the parameter a ranging from
a - 2 (Gaussian diffusion) to a = 1 (ballistic motion)
and this range of the parameter has been shown to be
compatible with an equilibrium dynamical realization of
L6vy processes. Moreover, the recent investigations of
many groups  suggest that L6vy processes are as ubiq-
uitous as the Brownian motion itself. (ii) The theoretical
investigations of Refs. [2,3] have shown that the asymp-
totic properties of a process of anomalous diffusion are
independent of the details of the dynamical generator of
the distribution O(t) and that they depend only on the
time asymptotic property of (2). The standard map in
the accelerating state is proved [7,8] to result in a waiting-
time distribution in each of the two accelerating modes
with the same inverse power structure as that of (2) with
the index ,a fulfilling the condition (3). Thus, the anoma-
lous diffusion generated by the standard map "coincides"
with the one generated by the map of Geisel, Nierwet-
berg, and Zacherl  if the index t is the same, and both
are a dynamical realization of a Levy process.
The main purpose of this paper is to study the tran-
sition from anomalous to normal diffusion triggered by
environmental fluctuations. In the literature there are
already investigations of this kind [9,10] and the interest
for this problem in our case is dictated by the following
@ 1995 The American Physical Society
PHYSICAL REVIEW E
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Floriani, Elena; Grigolini, Paolo & Mannella, Riccardo. Noise-induced transition from anomalous to ordinary diffusion: The crossover time as a function of noise intensity, article, December 1995; [College Park, Maryland]. (digital.library.unt.edu/ark:/67531/metadc139501/m1/1/: accessed January 19, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.