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

Thermodynamic stability conditions for nonadditive composable entropies

The thermodynamic stability conditions (TSC) of nonadditive and composable entropies are discussed. Generally the concavity of a nonadditive entropy with respect to internal energy is not necessarily equivalent to the corresponding TSC. It is shown that both the TSC of Tsallis' entropy and that of the $\kappa$-generalized Boltzmann entropy are equivalent to the positivity of the standard heat capacity.Comment: 6pages; Contribution to a topical issue of Continuum Mechanics and Thermodynamics (CMT), edited by M. Sugiyam

Boltzmann-Gibbs entropy is sufficient but not necessary for the likelihood factorization required by Einstein

In 1910 Einstein published a crucial aspect of his understanding of Boltzmann entropy. He essentially argued that the likelihood function of any system composed by two probabilistically independent subsystems {\it ought} to be factorizable into the likelihood functions of each of the subsystems. Consistently he was satisfied by the fact that Boltzmann (additive) entropy fulfills this epistemologically fundamental requirement. We show here that entropies (e.g., the $q$-entropy on which nonextensive statistical mechanics is based) which generalize the BG one through violation of its well known additivity can {\it also} fulfill the same requirement. This fact sheds light on the very foundations of the connection between the micro- and macro-scopic worlds.Comment: 5 pages including 2 figure

Fractal geometry, information growth and nonextensive thermodynamics

This is a study of the information evolution of complex systems by geometrical consideration. We look at chaotic systems evolving in fractal phase space. The entropy change in time due to the fractal geometry is assimilated to the information growth through the scale refinement. Due to the incompleteness of the state number counting at any scale on fractal support, the incomplete normalization $\sum_ip_i^q=1$ is applied throughout the paper, where $q$ is the fractal dimension divided by the dimension of the smooth Euclidean space in which the fractal structure of the phase space is embedded. It is shown that the information growth is nonadditive and is proportional to the trace-form $\sum_ip_i-\sum_ip_i^q$ which can be connected to several nonadditive entropies. This information growth can be extremized to give power law distributions for these non-equilibrium systems. It can also be used for the study of the thermodynamics derived from Tsallis entropy for nonadditive systems which contain subsystems each having its own $q$. It is argued that, within this thermodynamics, the Stefan-Boltzmann law of blackbody radiation can be preserved.Comment: Final version, 10 pages, no figures, Invited talk at the international conference NEXT2003, 21-28 september 2003, Villasimius (Cagliari), Ital

Beyond the Shannon-Khinchin Formulation: The Composability Axiom and the Universal Group Entropy

The notion of entropy is ubiquitous both in natural and social sciences. In the last two decades, a considerable effort has been devoted to the study of new entropic forms, which generalize the standard Boltzmann-Gibbs (BG) entropy and are widely applicable in thermodynamics, quantum mechanics and information theory. In [23], by extending previous ideas of Shannon [38], [39], Khinchin proposed an axiomatic definition of the BG entropy, based on four requirements, nowadays known as the Shannon-Khinchin (SK) axioms. The purpose of this paper is twofold. First, we show that there exists an intrinsic group-theoretical structure behind the notion of entropy. It comes from the requirement of composability of an entropy with respect to the union of two statistically independent subsystems, that we propose in an axiomatic formulation. Second, we show that there exists a simple universal class of admissible entropies. This class contains many well known examples of entropies and infinitely many new ones, a priori multi-parametric. Due to its specific relation with the universal formal group, the new family of entropies introduced in this work will be called the universal-group entropy. A new example of multi-parametric entropy is explicitly constructed.Comment: Extended version; 25 page

Temperature of nonextensive system: Tsallis entropy as Clausius entropy

The problem of temperature in nonextensive statistical mechanics is studied. Considering the first law of thermodynamics and a "quasi-reversible process", it is shown that the Tsallis entropy becomes the Clausius entropy if the inverse of the Lagrange multiplier, $beta$, associated with the constraint on the internal energy is regarded as the temperature. This temperature is different from the previously proposed "physical temperature" defined through the assumption of divisibility of the total system into independent subsystems. A general discussion is also made about the role of Boltzmann's constant in generalized statistical mechanics based on an entropy, which, under the assumption of independence, is nonadditive.Comment: 14 pages, no figure

Composition law of κ\kappa-entropy for statistically independent systems

The intriguing and still open question concerning the composition law of $\kappa$-entropy $S_{\kappa}(f)=\frac{1}{2\kappa}\sum_i (f_i^{1-\kappa}-f_i^{1+\kappa})$ with $0<\kappa<1$ and $\sum_i f_i =1$ is here reconsidered and solved. It is shown that, for a statistical system described by the probability distribution $f=\{ f_{ij}\}$, made up of two statistically independent subsystems, described through the probability distributions $p=\{ p_i\}$ and $q=\{ q_j\}$, respectively, with $f_{ij}=p_iq_j$, the joint entropy $S_{\kappa}(p\,q)$ can be obtained starting from the $S_{\kappa}(p)$ and $S_{\kappa}(q)$ entropies, and additionally from the entropic functionals $S_{\kappa}(p/e_{\kappa})$ and $S_{\kappa}(q/e_{\kappa})$, $e_{\kappa}$ being the $\kappa$-Napier number. The composition law of the $\kappa$-entropy is given in closed form, and emerges as a one-parameter generalization of the ordinary additivity law of Boltzmann-Shannon entropy recovered in the $\kappa \rightarrow 0$ limit.Comment: 14 page

Black hole thermodynamical entropy

As early as 1902, Gibbs pointed out that systems whose partition function diverges, e.g. gravitation, lie outside the validity of the Boltzmann-Gibbs (BG) theory. Consistently, since the pioneering Bekenstein-Hawking results, physically meaningful evidence (e.g., the holographic principle) has accumulated that the BG entropy $S_{BG}$ of a $(3+1)$ black hole is proportional to its area $L^2$ ($L$ being a characteristic linear length), and not to its volume $L^3$. Similarly it exists the \emph{area law}, so named because, for a wide class of strongly quantum-entangled $d$-dimensional systems, $S_{BG}$ is proportional to $\ln L$ if $d=1$, and to $L^{d-1}$ if $d>1$, instead of being proportional to $L^d$ ($d \ge 1$). These results violate the extensivity of the thermodynamical entropy of a $d$-dimensional system. This thermodynamical inconsistency disappears if we realize that the thermodynamical entropy of such nonstandard systems is \emph{not} to be identified with the BG {\it additive} entropy but with appropriately generalized {\it nonadditive} entropies. Indeed, the celebrated usefulness of the BG entropy is founded on hypothesis such as relatively weak probabilistic correlations (and their connections to ergodicity, which by no means can be assumed as a general rule of nature). Here we introduce a generalized entropy which, for the Schwarzschild black hole and the area law, can solve the thermodynamic puzzle.Comment: 7 pages, 2 figures. Accepted for publication in EPJ