13 research outputs found
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
-generalization of Gauss' law of error
Based on the -deformed functions (-exponential and
-logarithm) and associated multiplication operation (-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
-product. This -generalized maximum likelihood principle leads
to the {\it so-called} -Gaussian distributions.Comment: 9 pages, 1 figure, latex file using elsart.cls style fil
Advantages of -logarithm representation over -exponential representation from the sense of scale and shift on nonlinear systems
Addition and subtraction of observed values can be computed under the obvious
and implicit assumption that the scale unit of measurement should be the same
for all arguments, which is valid even for any nonlinear systems. This paper
starts with the distinction between exponential and non-exponential family in
the sense of the scale unit of measurement. In the simplest nonlinear model
, it is shown how typical effects such as rescaling and shift
emerge in the nonlinear systems and affect observed data. Based on the present
results, the two representations, namely the -exponential and the
-logarithm ones, are proposed. The former is for rescaling, the latter for
unified understanding with a fixed scale unit. As applications of these
representations, the corresponding entropy and the general probability
expression for unified understanding with a fixed scale unit are presented. For
the theoretical study of nonlinear systems, -logarithm representation is
shown to have significant advantages over -exponential representation.Comment: 13 pages, 3 figure
Nonextensive Entropies derived from Form Invariance of Pseudoadditivity
The form invariance of pseudoadditivity is shown to determine the structure
of nonextensive entropies. Nonextensive entropy is defined as the appropriate
expectation value of nonextensive information content, similar to the
definition of Shannon entropy. Information content in a nonextensive system is
obtained uniquely from generalized axioms by replacing the usual additivity
with pseudoadditivity. The satisfaction of the form invariance of the
pseudoadditivity of nonextensive entropy and its information content is found
to require the normalization of nonextensive entropies. The proposed principle
requires the same normalization as that derived in [A.K. Rajagopal and S. Abe,
Phys. Rev. Lett. {\bf 83}, 1711 (1999)], but is simpler and establishes a basis
for the systematic definition of various entropies in nonextensive systems.Comment: 16 pages, accepted for publication in Physical Review