Applications of generalized statistics in critical phenomena and financial markets

本项目的目的是将非平衡统计物理学和信息论中已知的方法和技术应用于展示广义统计的系统。广义统计描述的是具有宽分布(或重尾分布)的过程。处理广义统计学的困难在于，中心极限定理和统计物理的通常方法都不能应用。借助于P.Lévy的广义中心极限定理，给出了这种“非正则”分布的理论条件。服从广义统计的现象是非常多样和结构丰富的，包括分数扩散过程、多粒子强子系统、多重分形或波动。该项目的主要重点将放在目前正在深入研究的系统中，这些系统由Rényi和Tsallis最大熵规定或超级统计中出现的分布所代表。核心应用程序将出现在金融市场。另一个目标是发展具有广义统计的系统的信息几何，并将其应用于临界现象理论，特别强调缺陷介导的相变。

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.