Applications of generalized statistics in critical phenomena and financial markets

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The aim of the present project is to apply methods and techniques known from nonequilibrium statistical physics and information theory to systems exhibiting generalized statistics. Broadly speaking, generalized statistics characterizes processes which have broad (or heavy-tail) distributions. The difficulty in working with generalized statistics lies in the fact that the Central Limit Theorem alongside with the usual methods of statistical physics cannot be applied. Theoretical qualification for such “non-canonical” distributions is provided by means of the generalized Central Limit Theorem of P. Lévy. Phenomena obeying generalized statistics are very diverse and structurally rich including fractional diffusion processes, multiparticle hadronic systems, multifractals, or volatility fluctuations. Primary focus of the project will be in presently intensely studied systems represented by distributions emerging either from the Rényi and Tsallis Maximum-Entropy prescription or from Superstatistics. Central applications will be in financial markets. The other goal is to develop Information Geometry for systems with a generalized statistics and apply it to theory of critical phenomena with a particular emphasize on defect-mediated phase transitions.