11,973 research outputs found

### Conceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems

It is by now well known that the Boltzmann-Gibbs-von Neumann-Shannon
logarithmic entropic functional ($S_{BG}$) is inadequate for wide classes of
strongly correlated systems: see for instance the 2001 Brukner and Zeilinger's
{\it Conceptual inadequacy of the Shannon information in quantum measurements},
among many other systems exhibiting various forms of complexity. On the other
hand, the Shannon and Khinchin axioms uniquely mandate the BG form
$S_{BG}=-k\sum_i p_i \ln p_i$; the Shore and Johnson axioms follow the same
path. Many natural, artificial and social systems have been satisfactorily
approached with nonadditive entropies such as the $S_q=k \frac{1-\sum_i
p_i^q}{q-1}$ one ($q \in {\cal R}; \,S_1=S_{BG}$), basis of nonextensive
statistical mechanics. Consistently, the Shannon 1948 and Khinchine 1953
uniqueness theorems have already been generalized in the literature, by Santos
1997 and Abe 2000 respectively, in order to uniquely mandate $S_q$. We argue
here that the same remains to be done with the Shore and Johnson 1980 axioms.
We arrive to this conclusion by analyzing specific classes of strongly
correlated complex systems that await such generalization.Comment: This new version has been sensibly modified and updated. The title
and abstract have been modifie

### Is the entropy Sq extensive or nonextensive?

The cornerstones of Boltzmann-Gibbs and nonextensive statistical mechanics
respectively are the entropies $S_{BG} \equiv -k \sum_{i=1}^W p_i \ln p_i$ and
$S_{q}\equiv k (1-\sum_{i=1}^Wp_i^{q})/(q-1) (q\in{\mathbb R} ; S_1=S_{BG})$.
Through them we revisit the concept of additivity, and illustrate the (not
always clearly perceived) fact that (thermodynamical) extensivity has a well
defined sense {\it only} if we specify the composition law that is being
assumed for the subsystems (say $A$ and $B$). If the composition law is {\it
not} explicitly indicated, it is {\it tacitly} assumed that $A$ and $B$ are
{\it statistically independent}. In this case, it immediately follows that
$S_{BG}(A+B)= S_{BG}(A)+S_{BG}(B)$, hence extensive, whereas
$S_q(A+B)/k=[S_q(A)/k]+[S_q(B)/k]+(1-q)[S_q(A)/k][S_q(B)/k]$, hence
nonextensive for $q \ne 1$. In the present paper we illustrate the remarkable
changes that occur when $A$ and $B$ are {\it specially correlated}. Indeed, we
show that, in such case, $S_q(A+B)=S_q(A)+S_q(B)$ for the appropriate value of
$q$ (hence extensive), whereas $S_{BG}(A+B) \ne S_{BG}(A)+S_{BG}(B)$ (hence
nonextensive).Comment: To appear in the Proceedings of the 31st Workshop of the
International School of Solid State Physics ``Complexity, Metastability and
Nonextensivity", held at the Ettore Majorana Foundation and Centre for
Scientific Culture, Erice (Sicily) in 20-26 July 2004, eds. C. Beck, A.
Rapisarda and C. Tsallis (World Scientific, Singapore, 2005). 10 pages
including 1 figur

### On the extensivity of the entropy Sq, the q-generalized central limit theorem and the q-triplet

First, we briefly review the conditions under which the entropy $S_q$ can be
extensive (Tsallis, Gell-Mann and Sato, Proc. Natl. Acad. Sc. USA (2005), in
press; cond-mat/0502274), as well as the possible $q$-generalization of the
central limit theorem (Moyano, Tsallis and Gell-Mann, cond-mat/0509229). Then,
we address the $q$-triplet recently determined in the solar wind (Burlaga and
Vinas, Physica A {\bf 356}, 375 (2005)) and its possible relation with the
space-dimension $d$ and with the range of the interactions (characterized by
$\alpha$, the attractive potential energy being assumed to decay as
$r^{-\alpha}$ at long distances $r$).Comment: 9 pages including 3 figures. Invited lecture at the International
Conference on Complexity and Nonextensivity - New Trends in Statistical
Mechanics (Yukawa Institute for Theoretical Physics, Kyoto, 14-18 March
2005). To appear in Prog. Theor. Phys. Supplement, eds. M. Sakagami, N.
Suzuki and S. Ab

- β¦