The unique non self-referential q-canonical distribution and the physical temperature derived from the maximum entropy principle in Tsallis statistics

The maximum entropy principle in Tsallis statistics is reformulated in the mathematical framework of the q-product, which results in the unique non self-referential q-canonical distribution. As one of the applications of the present formalism, we theoretically derive the physical temperature which coincides with that already obtained in accordance with the generalized zeroth law of thermodynamics.Comment: The title, representations and references are revise

Large deviation estimates involving deformed exponential functions

We study large deviation properties of probability distributions with either a compact support or a fat tail by comparing them with q-deformed exponential distributions. Our main result is a large deviation property for probability distributions with a fat tail.Comment: typos corrected, some addtional explanations, version to appear in Physica A, http://dx.doi.org/10.1016/j.physa.2015.05.09

κ\kappa-generalization of Gauss' law of error

Based on the $\kappa$-deformed functions ($\kappa$-exponential and $\kappa$-logarithm) and associated multiplication operation ($\kappa$-product) introduced by Kaniadakis (Phys. Rev. E \textbf{66} (2002) 056125), we present another one-parameter generalization of Gauss' law of error. The likelihood function in Gauss' law of error is generalized by means of the $\kappa$-product. This $\kappa$-generalized maximum likelihood principle leads to the {\it so-called} $\kappa$-Gaussian distributions.Comment: 9 pages, 1 figure, latex file using elsart.cls style fil

Multiplicative duality, q-triplet and (mu,nu,q)-relation derived from the one-to-one correspondence between the (mu,nu)-multinomial coefficient and Tsallis entropy Sq

We derive the multiplicative duality "q1/q" and other typical mathematical structures as the special cases of the (mu,nu,q)-relation behind Tsallis statistics by means of the (mu,nu)-multinomial coefficient. Recently the additive duality "q2-q" in Tsallis statistics is derived in the form of the one-to-one correspondence between the q-multinomial coefficient and Tsallis entropy. A slight generalization of this correspondence for the multiplicative duality requires the (mu,nu)-multinomial coefficient as a generalization of the q-multinomial coefficient. This combinatorial formalism provides us with the one-to-one correspondence between the (mu,nu)-multinomial coefficient and Tsallis entropy Sq, which determines a concrete relation among three parameters mu, nu and q, i.e., nu(1-mu)+1=q which is called "(mu,nu,q)-relation" in this paper. As special cases of the (mu,nu,q)-relation, the additive duality and the multiplicative duality are recovered when nu=1 and nu=q, respectively. As other special cases, when nu=2-q, a set of three parameters (mu,nu,q) is identified with the q-triplet (q_{sen},q_{rel},q_{stat}) recently conjectured by Tsallis. Moreover, when nu=1/q, the relation 1/(1-q_{sen})=1/alpha_{min}-1/alpha_{max} in the multifractal singularity spectrum f(alpha) is recovered by means of the (mu,nu,q)-relation.Comment: 20 page