Generalized entropies and logarithms and their duality relations

Duality

In contrast to the images of generalized logarithms, which need not span ℝ completely and can differ from one another, the domain of generalized logarithms themselves is always all of ℝ₊. For these reasons, one may classify generalized logarithms according to the minimum and maximum values of their images and consider the group G of order-preserving automorphisms on ℝ₊ that keep an infinitesimal neighborhood of 1 ∈ ℝ₊ invariant, as the means to generate these classes. In the following, we call the elements g of this automorphism group scale transformations. More precisely, g ∈ is a scale transformation if g is differentiable and maps ℝ₊ to ℝ₊ one-to-one, g′ > 0, g(1) = 1, and g′(1) = 1. From these properties it follows that g(0) = 0 and lim_x→∞g(x) = ∞. Finally, we use the notation f ○ g(x) = f (g(x)).

Scale transformations leave the image of a generalized logarithm invariant, which allows us to parameterize classes in the following way. Given a proper generalized logarithm Λ ∈ ℒ, we write for its maximum and minimum valuesand define two functionalswhich associate numbers ν₊ and ν₋ to any Λ. For their sum we write ν* = ν₊ + ν₋. Next, we define sets of proper generalized logarithms,

Members of all have the same maximum and minimum values. In fact, the are exactly the equivalence classes in ℒ generated by G: Two generalized logarithms Λ^(A) and Λ^(B) are considered equivalent if there exists a scale transformation g ∈ G such that Λ^(B) = Λ^(A) ○ g. The space of generalized logarithms can be written as the union of these sets, .

With these definitions we now analyze the relation between the HT and TS approaches. Assuming that Λ_HT is given, Eq. 7 implies

T_ν is a shift operator with the property T_ν ○ T_μ = T_ν+μ. We have of course Λ_TS,0 = Λ_HT. The fact that Λ_HT is a proper generalized logarithm does not imply that Λ_TS is also proper for all choices of ν.

In fact, given that Λ ∈ ℒ, it can be shown (SI Materials and Methods) that T_ν ○ Λ ∈ ℒ if and only if ν₋[Λ] ≤ ν ≤ ν₊[Λ]. Moreover, for and for T_ν ○ Λ being a proper generalized logarithm, it follows that . As a consequence Λ_TS,ν (x) = T_ν ○ Λ_HT (x) is proper only for ν₋[Λ_HT] ≤ ν ≤ ν₊[Λ_HT], and

This equation does not uniquely determine a duality relation * on ℒ, yet by imposing the condition that * commute with scale transformations g ∈ , it can be shown (SI Materials and Methods) that * is given bywith the property

Thus, for each Λ_HT there exists a unique value ν* = ν₊[Λ_HT] + ν₋[Λ_HT] such that Λ_TS,ν* is a proper generalized logarithm. The duality map * gives . Furthermore, because * and g commute ((Λ ○ g)* = Λ* ○ g), any proper generalized logarithm Λ can be decomposed into a specific representative , and a scale transformation g, so thatwhich implies that any Λ_HT or Λ_TS,ν can be decomposed in this way, and that the dual logarithms Λ_HT and transform identically under scale transformations.

Functional Form of the Generalized Logarithms

Eq. 14 implies the existence of transformations that map members of to members of . These maps can be used to represent the duality * on specific families . Λ(x) → −Λ(1/x) is exactly such a map, because . The same holds for , which allows us to construct with the properties

By using Eq. 13 and inserting into Eq. 7, we get

This equation may have many solutions , but we can restrict ourselves to finding a particular one; all of the others can be obtained by scale transformations, which is seen as follows: Suppose and are both solutions of Eq. 17; then according to Eq. 15 for any pair (ν₊, ν₋) there exists a scale transformation such that . Because must leave Eq. 16 invariant (this is not the case for arbitrary scale transformations g ∈ ), these scale transformations have two properties. The first property is , which makes them members of a subgroup of all possible scale transformations g ∈ . The second property is and follows from the fact that * commutes with scale transformations.

A particular solution of Eq. 17 is given bywith h: ℝ → [−1, 1] a continuous, monotonically increasing, odd function, with lim_x→∞ h(x) = 1 and h′(0) = 1. It can easily be verified that this solution has all of the required properties: is a proper logarithm with (correct minimum and maximum), , Λ_ν,−ν(x) = h(ν log(x))/ν is self-dual, and .

The above argument means that we can generate a specific family of logarithms , following Eq. 16, by choosing one particular function h [e.g., h(x) = tanh(x)] and then using scale transformations to reach all other possibilities. In particular, some family with the property can be reached by a family of scale transformations , where are generalized exponential functions (inverse functions of logarithms). Moreover, if also follows Eq. 16, then .

The family of dual logarithms discussed in Hanel et al. (7) is obtained in the framework presented here by setting either ν₊ = 0 or ν₋ = 0. These classes correspond to logarithms that are unbounded either from below or from above, whereas the duality maps . Moreover, in Hanel et al. (7), only pairs of dual logarithms have been considered such that Λ*(x) = −Λ(1/x), and the part that scale transformations play in the unique definition of * had not yet been described.

We are now in a position to understand all observable distribution functions emerging from the two approaches in terms of a single two-parameter family of generalized logarithms and a scale transformation. This result now raises the question of how is related to the two-parameter logarithms associated with the (c, d) entropies in Eq. 1 (6), and will further clarify the role of the scale transformations.

Logarithm and (c, d) Entropy

Generalized entropies can be classified with respect to their asymptotic scaling behavior in terms of two scaling exponents, c and d, where 0 < c ≤ 1 and d is a real number (6); they are obtained from the scaling relationswhere s is the summand in Eq. 2. Using Eq. 19, de l’Hôpital’s rule, and the fact that s′(x) = −Λ(x), we find the exponents (c, d) for a given Λ ∈ ℒwhere we represent Λ as . In this way we get the dependence of (c, d) as a function of (ν₊, ν₋), h, and the scale transformation g. We first compute the asymptotic properties of h and g, defining the exponents c_h,g and d_h,g bywhere φ_h,g = 1 + h ○log ○g(x). Note that log ○g ∈ ℒ_0,0. By defining Λ₀ ≡ log ○g, we compute its scaling exponents c₀ and d₀

With these preparations one can derive the resultswhich demonstrate clearly that, given a fixed h, c is controlled by ν₋, (for ν₊ = 0 and c₀ = 1) and d is determined by the scale transformation.

Examples