New Trends in Statistical Mechanical Foundations of Complexity: Applications in high energy and plasma physics, long-range interactions, edge-of-chaos and elsewhere, to be held at the Ettore Majorana Foundation and Centre for Scientific Culture in Erice, Italy during the period July 27 to August 3, 2015This course belongs to the series International School on Complexity, directed by A. Zichichi, and co-directed by G. Benedek, M. Gell-Mann, L. Pietronero and C. Tsallis. It will primarily focus on the connections between complex systems on one hand, and nonadditive entropies, nonextensive statistical mechanics, as well as any other related approaches, on the other hand. The discussion of systems in physics and elsewhere which constitute possible candidates for analytical, computational and experimental applications of such theories is absolutely welcome. This event will cover some new aspects of complex systems, mainly concerning its statistical-mechanical foundations and various applications. Although no general rigorous first-principle proof (i.e., just using mechanics and theory of probabilities, with no other hypothesis) yet exists for classical (or quantum) Hamiltonian systems, there remains — after 140 years of impressive success — no reasonable doubt that the Boltzmann-Gibbs (BG) entropy is the correct one to be used for a wide and important class of physical systems, basically those whose (nonlinear) dynamics is strongly chaotic (meaning, for classical systems, positive maximal Lyapunov exponent), hence mixing, hence ergodic. Among those very many that violate this hypothesis, there is an important class, namely those that are weakly chaotic, meaning that the maximal Lyapunov exponent vanishes. A vanishing maximal Lyapunov exponent means sub-exponential sensitivity to the initial conditions, which corresponds of course to an infinite number of mathematical behaviors. There is however one of those which can be considered as the most simple and natural one. We refer to a power-law time-dependence of the sensitivity to the initial conditions. It has been proposed in 1988 that the current statistical-mechanical methods can be extended to a wide class of physical systems by just generalizing the BG entropy SBG into nonadditive entropies such as Sq (with S1 = SBG), where the index q is a real number. It turned out that q different from 1 basically corresponds to the power-law class referred above, and is consistently associated with a hierarchical or (multi) fractal geometry. The aim of the present event is to cover a wide class of systems among those very many which violate hypothesis such as ergodicity, under which the BG theory is expected to be valid. It is now known that Sq and similar entropies have large applicability; more specifically speaking, even outside Hamiltonian systems and their thermodynamical approach. Some relevant entropic aspects will be focused on, namely (i) Additivity versus extensivity; (ii) Probability distributions that constitute attractors in the sense of Central Limit Theorems; (iii) Large deviation theory; (iv) Analysis of paradigmatic low-dimensional nonlinear dynamical systems near the edge of chaos; and (v) Analysis of paradigmatic long-range-interacting many-body classical Hamiltonian systems. We intend to address recent as well as typical predictions, verifications and applications of these concepts in natural, artificial, and social systems, as shown through theoretical, experimental, observational and computational results, in high energy physics, plasma physics, nonlinear dynamical systems, free-scale networks, econophysics, granular matter, geophysics, astrophysics, among others. |

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