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 S-q

摘要

We derive the multiplicative duality "q <-> 1/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 "q <-> 2 - 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 S-q, 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. (c) 2007 Published by Elsevier B.V.