Some Open Points in Nonextensive Statistical Mechanics

Abstract

We present and discuss a list of some interesting points that are currently open in nonextensive statistical mechanics. Their analytical, numerical, experimental or observational advancement would naturally be very welcome.

Weak chaos from Tsallis entropy

“…Naturally, these modified Lyapunov exponents will explicitly depend on the value of the non-entropic parameter q. Since, as is conjectured, the systems described by the Tsallis entropy may possess more than just one value of the non-entropic parameter q [2], [11], depending on which property of the system is described, such modified Lyapunov exponents would explicitly depend on the value of q characterising the sensitivity of the system to infinitesimal perturbations q sen [2].…”

“…This fact is in agreement with all currently known results pertaining to the microscopic dynamics of systems described by the Tsallis entropy. As a by-product we also comment on modified definitions of asymptotic numerical invariants of dynamical systems better reflecting the properties of the Tsallis entropy [2], [11]- [16]. Our approach relies on the metric aspects of Riemannian manifolds.. For this reason some arguments can also be applied, or non-trivially generalized, in the much more general classes of (geodesic) metric spaces, such as spaces with a (negative) upper bound k < 0 on their curvature, known as CAT (k) spaces.…”

Distributivity and deformation of the reals from Tsallis entropy

“…The question of whether one can still define a generalized addition and multiplication that obey, in some sense, the distributive property had remained open for a while [8], but a recent proposal [3] has provided a satisfactory answer. If one maintains the form of the generalized multiplication given above, and attempts to modify the generalized addition, an argument exists that satisfying the distributive property may not even be possible [2].…”

“…On the other hand, an algebraic structure requiring the validity of the distributive property, akin to a ring, field or vector space is highly desirable [8]. This a basic motivation guiding the algebraic construction of [3] and of the present work.…”

Statistical crossover and nonextensive behavior of neuronal short-term depression

“…Nonextensive Statistical Mechanics (NSM) describes systems in which the entropy is not proportional to the system size, a property frequently observed in complex systems that display long-range interactions or that are out of equilibrium [11]. In this framework, Tsallis Statistics is successfully employed to investigate a variety of phenomena due to its ability to model power law phenomena [12]. Although its widespread application, NSM remains scarcely applied in studies of the physiology of neurotransmission, despite the confirmation of nonextensivity associated to spontaneous release at the mammalian neuromuscular junction [13,14].…”