/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

Real and Complex Analysis

/pdf/introduction-to-the-modern-theory-of-dynamical-systems-1eced70mum.pdf

Introduction to the Modern Theory of Dynamical Systems

What do you do to start reading dimension theory? Searching the book that you love to read first or find an interesting book that will make you want to read? Everybody has difference with their reason of reading a book. Actuary, reading habit must be from earlier. Many people may be love to read, but not a book. It's not fault. Someone will be bored to open the thick book with small words to read. In more, this is the real condition. So do happen probably with this dimension theory.

Aspects of topology: On dimension theory

We present a geometric argument that explains why some systems having vanishing largest Lyapunov exponent have underlying dynamic aspects which can be effectively described by the Tsallis entropy. We rely on a comparison of the generalised additivity of the Tsallis entropy versus the ordinary additivity of the BGS entropy. We translate this comparison, in metric terms, by using an effective hyperbolic metric on the configuration/phase space for the Tsallis entropy versus the Euclidean one in the case of the BGS entropy. Solving the Jacobi equation for such hyperbolic metrics effectively sets the largest Lyapunov exponent computed, with respect to the corresponding Euclidean metric, to zero. This conclusion is in agreement with all currently known results on systems that have a simple asymptotic behaviour, and are described by the Tsallis entropy.

Vanishing largest Lyapunov exponent and Tsallis entropy

We present a path toward determining the statistical origin of the thermodynamic limit for systems with long-range interactions. We assume throughout that the systems under consideration have thermodynamic properties given by the Tsallis entropy. We rely on the composition property of the Tsallis entropy for determining effective metrics and measures on their configuration/phase spaces. We point out the significance of Muckenhoupt weights, of doubling measures and of doubling measure-induced metric deformations of the metric. We comment on the volume deformations induced by the Tsallis entropy composition and on the significance of functional spaces for these constructions.

Long-range interactions, doubling measures and Tsallis entropy

We construct generalized additions and multiplications, forming fields and division algebras inspired by the Tsallis thermo-statistics. We also construct derivations and integrations in this spirit. These operations do not reduce to the naively expected ones, when the deformation parameter approaches zero.

Algebra and calculus for Tsallis thermo-statistics

Searching for the dynamical foundations of the Havrda-Charvat/Daroczy/Cressie-Read/Tsallis non-additive entropy, we come across a covariant quantity called, alternatively, a generalized Ricci curvature, an $N$-Ricci curvature or a Bakry-Emery-Ricci curvature in the configuration/phase space of a system. We explore some of the implications of this tensor and its associated curvature and present a connection with the non-additive entropy under investigation. We present an isoperimetric interpretation of the non-extensive parameter and comment on further features of the system that can be probed through this tensor.

Ricci curvature, isoperimetry and a non-additive entropy

We investigate the possibility of discrete groups furnishing a kinematic framework for systems whose thermodynamic behaviour may be given by non-additive entropies. Relying on the well-known result of the growth rate of balls of nilpotent groups, we see that maintaining extensivity of the entropy of a nilpotent group requires using a non-Boltzmann/Gibbs/Shannon (BGS) entropic form. We use the Tsallis entropy as an indicative alternative. Using basic results from hyperbolic and random groups, we investigate the genericity and possible range of applicability of the BGS entropy in this context. We propose a sufficient condition for phase transitions, in the context of (multi-) parameter families of non-additive entropies.

Nikos Kalogeropoulos

Papers

Vanishing largest Lyapunov exponent and Tsallis entropy

Long-range interactions, doubling measures and Tsallis entropy

Algebra and calculus for Tsallis thermo-statistics

Ricci curvature, isoperimetry and a non-additive entropy

Groups, non-additive entropy and phase transitions