The kinetic foundations of Tsallis' nonextensive thermostatistics are investigated through Boltzmann's transport equation approach. Our analysis follows from a nonextensive generalization of the "molecular chaos hypothesis". For q > 0, the q-transport equation satisfies an H-theorem based on Tsallis entropy. It is also proved that the collisional equilibrium is given by Tsallis' q-nonextensive velocity distribution.PACS numbers: 05.45.+b; 05.20.-y; 05.90.+m In 1988 Tsallis proposed a striking generalization of the Boltzmann-Gibbs entropy functional given by [1],

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-The San Andreas fault (SAF) in the USA is one of the most investigated selforganizing systems in nature. In this paper, we studied some geophysical properties of the SAF system in order to analyze the behavior of earthquakes in the context of Tsallis's q-Triplet. To that end, we considered 134,573 earthquake events in magnitude interval 2 ≤ m < 8, taken from the Southern Earthquake Data Center (SCEDC, 1932(SCEDC, -2012. The values obtained ("qTriplet"≡{qstat,qsen,q rel }) reveal that the qstat-Gaussian behavior of the aforementioned data exhibit long-range temporal correlations. Moreover, qsen exhibits quasi-monofractal behavior with a Hurst exponent of 0.87.

The laws of thermodynamics constrain the formulation of statistical mechanics at the microscopic level. The third law of thermodynamics states that the entropy must vanish at absolute zero temperature for systems with nondegenerate ground states in equilibrium. Conversely, the entropy can vanish only at absolute zero temperature. Here we ask whether or not generalized entropies satisfy this fundamental property. We propose a direct analytical procedure to test if a generalized entropy satisfies the third law, assuming only very general assumptions for the entropy S and energy U of an arbitrary N-level classical system. Mathematically, the method relies on exact calculation of β=dS/dU in terms of the microstate probabilities p(i). To illustrate this approach, we present exact results for the two best known generalizations of statistical mechanics. Specifically, we study the Kaniadakis entropy S(κ), which is additive, and the Tsallis entropy S(q), which is nonadditive. We show that the Kaniadakis entropy correctly satisfies the third law only for -1<κ<+1, thereby shedding light on why κ is conventionally restricted to this interval. Surprisingly, however, the Tsallis entropy violates the third law for q<1. Finally, we give a concrete example of the power of our proposed method by applying it to a paradigmatic system: the one-dimensional ferromagnetic Ising model with nearest-neighbor interactions.