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VOLUME 62, NUMBER 3
Towards the thermodynamics of localization processes
Paolo Grigolini,1'2'3 Marco G. Pala,3 Luigi Palatella,3'4 and Roberto Roncaglia4
'Istituto di Biofisica CNR, Area della Ricerca di Pisa, Via Alfieri 1, San Cataldo 56010 Ghezzano-Pisa, Italy
2 Center for Nonlinear Science, University of North Texas, P.O. Box 5368, Denton, Texas 76203-5368
3Dipartimento di Fisica dell'Universita di Pisa and INFM, Piazza Torricelli 2, 56127 Pisa, Italy
4lstituto Nazionale per la Fisica della Materia, UdR di Pisa, Via Buonarroti 2, 56126 Pisa, Italy
(Received 29 July 1999; revised manuscript received 25 January 2000)
We study the entropy time evolution of a quantum mechanical model, which is frequently used as a
prototype for Anderson's localization. Recently Latora and Baranger [Phys. Rev. Lett. 82, 520 (1999)] found
that there exist three entropy regimes, a transient regime of passage from dynamics to thermodynamics, a
linear-in-time regime of entropy increase, that is, a thermodynamic regime of Kolmogorov kind, and a satu-
ration regime. We use the nonextensive entropic indicator advocated by Tsallis [J. Stat. Phys. 52, 479 (1988)]
with a mobile entropic index q, and we find that the adoption of the "magic" value q= Q= 1/2, compared to
the traditional entropic index q= 1, reduces the length of the transient regime and makes earlier the emergence
of the Kolmogorov regime. We adopt a two-site model to explain these properties by means of an analytical
treatment and we argue that Q= 1/2 might be a typical signature of the occurrence of Anderson localization.
PACS number(s): 05.45.Mt, 05.20.-y, 03.65.BzI. INTRODUCTION
In this paper we focus our attention on the process of
localization discovered by Anderson [1,2], and we discuss
the corresponding time evolution using the nonextensive
thermodynamics view of Tsallis [3,4]. The subject of Tsallis
nonextensive thermodynamics is attracting the interest of an
ever increasing number of investigators in different branchs
of complexity theory (see, for instance, Ref. [5]). We want to
apply this new perspective to the delicate problem of the
connection between thermodynamics and quantum dynam-
ics.
According to the new paradigm of deterministic chaos,
study of this connection leads to study of the quantum be-
havior of those systems that would be chaotic in the classical
limit. The subject of the entropy increase of quantum sys-
tems that would be classically chaotic has been addressed in
a number of papers [6-9]. The pioneering work of Ref. [6]
has established that the coarse graining necessary for the
entropy to increase can be produced by a weak fluctuation.
As a consequence of this, after a transient of time duration
inversely proportional to the classical Lyapunov coefficient
the system is expected to reach a thermodynamic condition.
This prediction is based on several conjectures. The first is
that there must be a kind of equivalence between the Gibbs
entropy and the Kolmogorov-Sinai (KS) entropy [10,11]
hKs. The Gibbs entropy is a functional of probability den-
sity, whereas hKS is the entropy of a trajectory and, in prin-
ciple, might not coincide with the entropy expressed in terms
of probability density. Furthermore, the KS entropy is an
entropy per unit time and the existence of a finite value of
hKs implies a steady rate of entropy increase. Consequently,
one must make a second important conjecture: The thermo-
dynamic regime corresponds to the probability entropy in-
creasing as a linear function of time.
Both conjectures are supported by the recent findings of
Latora and Baranger [12]. These authors studied several cha-
otic maps and found that the time evolution of the Gibbsentropy goes through three time regimes: (i) an early regime
of exponential increase, (ii) an intermediate time regime of
linear increase, the Kolmogorov regime, and, finally, (iii) a
saturation regime. On the basis of arguments similar to those
used here earlier, the second time regime is identified with
the thermodynamic regime. In this classical case, the coarse
graining is done by the division of the space into cells.
The quantum case [6-9] is very complex. The results de-
pend on the relations among three fundamental parameters,
expressed in the same units. These are h, I, and D. The first
is the Planck constant, the second the classical action, and
the third the intensity of the coarse grain generating stochas-
tic force. For the quantum system to exhibit ordinary ther-
modynamic behavior it is necessary not only that h I but
also that h < D [6]. In this case the rate of increase of the von
Neumann entropy is found [8] to be proportional to the KS
entropy.
The authors of Ref. [7] focused their attention on the tran-
sition from dynamics to thermodynamics, namely, the first of
the three time regimes discussed in Ref. [12], in the case of
the quantum Arnold's cat [8]. They focused their attention
on the condition I> D> h, where, according to [6], an
exponential-like transition from dynamics to thermodynam-
ics is expected to occur. The numerical results of these au-
thors provide a satisfactory support to the heuristic argu-
ments of Zurek and Paz [6]. The authors of Ref. [9] studied
the quantum kicked rotor in the regime where h D, and
found that it yields an entropy increase proportional to ta,
with a> 1. This finding agrees with the result found by Pat-
tanayak and Brumer [7] in the case of the quantum Arnolds's
cat, when the same condition h D applies. The authors of
Ref. [7], however, overlook this interesting aspect, probably
because an algebraic entropy increase in time is interpreted
as a dramatic postponement of the transition to thermody-
namics. The authors of Ref. [9], on the contrary, found that
in this case also a kind of thermodynamic behavior, namely,
a regime of linear increase in time, shows up quickly when
they adopt a nonextensive form of entropy indicator.2000 The American Physical Society
PHYSICAL REVIEW E
SEPTEMBER 2000
1063-651 X/2000/62(3)/3429(8)/$15.00
PRE 62 3429
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Grigolini, Paolo; Pala, Marco G.; Palatella, Luigi & Roncaglia, Roberto. Towards the thermodynamics of localization processes, article, September 2000; [College Park, Maryland]. (https://digital.library.unt.edu/ark:/67531/metadc77163/m1/1/: accessed May 14, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.