Tsallis entropy and nonextensive generalization of Boltzmann-Gibbs statistical mechanics

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In 2000 and 2001, I have been addressing to the above research project from the viewpoints of thermodynamics and information theory. I have derived the most general composition law of entropy, which is compatible with the concept of "composability" and the zeroth law of thermodynamics. Starting with thus obtained nonadditive entropy, I have established the thermodynamic Legendre transform structure. This enables one to calculate thermodynamic properties of nonextensive systems. I have also studied the general properties of macroscopic thermodynamical equilibrium, and have found that such a state can be characterized not only by the standard Boltzmann factor but also by the Tsallis-type power-law factor. Then, nonadditive quantum information theory has been developed. I have found that this new theory has in some points superior to the ordinary additive von Neumann theory. Finally, I have discussed the information entropic uncertainty relation associated with the measurements of the position and momentum observables in the power-law wave packets, which may have the divergent variances. I could show that this information theoretic approach can reveal the properties of the wave packets, which cannot be discussed by using the standard Heisenberg-type formulation of the uncertainty relation.