Open Problems within Nonextensive Statistical Mechanics

Abstract

Nonextensive Statistical Mechanics has developed into an important framework for modeling the thermodynamics of complex systems and the information of complex signals. Upon the 80th birthday of the field's founder, Constantino Tsallis, a review of open problems that can stimulate future research is provided. Over the thirty-year development of NSM a variety of criticisms have been published ranging from questions about the justification for generalizing the entropy function to interpretation of the generalizing parameter q. While these criticisms have been addressed in the past and the breadth of applications has demonstrated the utility of the NSM methodologies, this review provides insights on how the field can continue to improve the understanding and application of complex system models. The review starts by grounding q-statistics within scale-shape distributions and then frames a series of open problems for investigation. The open problems include using the degree of freedom to quantify the difference between entropy and its generalization; clarifying the physical interpretation of the parameter q; improving the definition of the generalized product using multidimensional analysis; defining a generalized Fourier transform applicable to signal processing applications; and re-examination of the normalization of nonextensive entropy. The review concludes with a proposal that the shape parameter is a candidate for defining the statistical complexity of a system.

19 pages; 1 figure; 2 tables; recommended for publication in Entropy, special issue Nonadditive Entropies and Nonextensive Statistical Mechanics

References in corpus (11)

• A possible deformed algebra and calculus inspired in nonextensive thermostatistics
• Analysis of return distributions in the coherent noise model
• A note on q-Gaussians and non-Gaussians in statistical mechanics
• A generalization of the central limit theorem consistent with nonextensive statistical mechanics
• Analysis of self-organized criticality in Ehrenfest's dog-flea model
• Fluctuations, correlations and the nonextensivity
• On a representation of the inverse Fq transform
• Entropy of financial time series due to the shock of war
• Non-Gaussian distributions under scrutiny
• The limit distribution in the $q$-CLT for $q\ge 1$ is unique and can not have a compact support
• A definition of the coupled-product for multivariate coupled-exponentials