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
arXiv:cond-mat/0702103 · doi:10.1016/j.physa.2007.07.074
Abstract
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 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.
arXiv comment20 pages
References in corpus (15)
- Nonextensivity and multifractality in low-dimensional dissipative systems
- A possible deformed algebra and calculus inspired in nonextensive thermostatistics
- A step beyond Tsallis and Renyi entropies
- Generalized algebra within a nonextensive statistics
- Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive
- What should a statistical mechanics satisfy to reflect nature?
- Triangle for the entropic index q of non-extensive statistical mechanics observed by Voyager 1 in the distant heliosphere
- Dynamical scenario for nonextensive statistical mechanics
- Numerical indications of a q-generalised central limit theorem
- Generalized thermostatistics based on deformed exponential and logarithmic functions
- Connections between Tsallis' formalisms employing the standard linear average energy and ones employing the normalized -average energy
- A non self-referential expression of Tsallis' probability distribution function
- The unique non self-referential q-canonical distribution and the physical temperature derived from the maximum entropy principle in Tsallis statistics
- Combinatorial Information Theory: I. Philosophical Basis of Cross-Entropy and Entropy
- Scaling property and the generalized entropy uniquely determined by a fundamental nonlinear differential equation