A. M. Mathai1,2 and H. J. Haubold3**
Dedicated to the Memory of Prof. Dr.habil. Dr.h.c.mult. Hans-Juergen
Treder (1928-2006),
1
Department of Mathematics and Statistics, McGil l University,
Montreal, Canada H3A 2K6
2
Centre for Mathematical Sciences, Pala Campus, Arunapuram P.O., Pala
686 574, India
3
Office for Outer Space Affairs, United Nations, Vienna International
Centre,
P.O. Box 500, A-1400 Vienna, Austria
former Director of the Einstein Laboratory for Theoretical Physics,
Caputh
Abstract. Product probability property, known in the literature as statistical independence, is examined first. Then generalized entropies are introduced, all of which give generalizations to Shannon entropy. It is shown that the nature of the recursivity postulate automatically determines the logarithmic functional form for Shannon entropy. Due to the logarithmic nature, Shannon entropy naturally gives rise to additivity, when applied to situations having product probability property. It is argued that the natural process is non- additivity even in product probability property situations and additivity can hold due to the involvement of a recursivity postulate leading to a logarithmic function. Generalized entropies are introduced and some of their properties are examined. Particularly, situations are examined where a generalized entropy of order leads to entropic pathway models, exhibiting exponential and power law behavior. Subsequently it is shown that these models link to distributional and differential pathways. Connection of the generalized entropy of order to Kerridge's measure to create "inaccuracy" is also explored. Further for each of the three pathways their relevance to Tsallis statistics and Beck-Cohen super- statistics is emphasized.
Keywords: pathway model - generalized enthropy measures - Boltzmann- Gibbs entropy - Tsallis entropy