Entropies Galore!

Abstract

The functional properties of the entropy gives rise to 6 possible types of thermodynamics. Additivity or superadditivity or subadditivity are closely related to extensivity and this is one of the characteristics used to distinguish the 6 types. We give examples some are somewhat academic of all of these, except one. For this purpose we draw also on black hole systems which h a ve been proposed. Some of these systems have subadditive e n tropies, i.e. they tend to fragment rather than clump. After proposing a new entropy function we raise the problem of how to select from these entropy functions. Are some better than others?

On the Uniqueness Theorem for Pseudo-Additive Entropies

“…Types SHC areSHC are logically impossible [19]; see Figure 2. This classification is, however, based entirely on entropies for independent events.…”

Resolving statistical uncertainty in correlation dimension estimation

“…Since ''disorder'' may be interpreted as an inverse measure of predictability, this indicates that it might be possible to establish the mean connectivity with appropriate statistics, although the usual statistics of the information-theoretic entropy are obviously not sufficient since they yield a diverging variance for scale-free networks. Recent work [9] on the ''disorders'' corresponding to the extended entropies of R enyi [10], Tsallis [11] and Landsberg-Vedral [12][13][14][15] (''normalized Tsallis entropies'' [16]) show that the connectivity properties of scale-free networks may be made predictable by a judicious choice of entropy and entropy index. This is confirmed by a finite mean and finite variance for the corresponding escort distributions [17].…”

“…We will overcome this problem by turning to nontraditional statistics associated with the extended entropies of R enyi [10], Tsallis [11] and Landsberg-Vedral [12][13][14][15][16]), which are defined in various ways in terms of Q P k p q k , where p k is the probability of state k, and the parameter q may be referred to as the entropy index. We use the corresponding ''disorders'' [9] as inverse measures of predictability, since they circumvent any problems caused by any extensivity of the entropies.…”

“…We consider R enyi [10], Tsallis [11] and Landsberg-Vedral entropies [12][13][14][15][16]). The R enyi entropies are perhaps best know as the basis of the definition of multifractals [25,26]; the Tsallis entropies are particularly suited for the study of nonextensive systems and arise naturally for power law probability distributions p k / k Àa [27]; the LandsbergVedral entropies were introduced as another example of the set of possible extended entropies.…”

Lambert function and a new non-extensive form of entropy

“…Apart from this common feature shared by all definitions of entropy at two ends of the scale, variations are possible in particularizing the functional form in between [1]. They lead to different forms of the probability distributions for states with different energies or some other conserved attribute.…”