Two-Parameter Generalized Entropic Functional

• Two-parameter generalized entropic functional is a class of entropy measures using two deformation parameters to extend Boltzmann–Gibbs–Shannon entropy for nonadditivity and generalized averaging.
• It encompasses distinct families—such as Sharma–Taneja–Mittal, Sharma–Mittal, and Hanel–Thurner—each defined by unique trace-form or kernel-form constructions and specialized reduction limits.
• Methodological frameworks employ generalized logarithms, functional equations, and deformed calculus to derive composition laws, stability criteria, and applications in turbulence, holography, and optimization.

A two-parameter generalized entropic functional is an entropy depending on two deformation or scaling parameters that extends the Boltzmann–Gibbs–Shannon entropy in order to accommodate nonadditivity, generalized averaging, or nonstandard asymptotic growth of accessible states. In the modern literature it denotes not a single formula but a class of inequivalent constructions. The main examples are the Sharma–Taneja–Mittal entropies of degree $(\alpha,\beta)$, the Sharma–Mittal entropies of order $(\alpha,\beta)$, and the Hanel–Thurner entropies of class $(c,d)$, to which later work adds scale-invariant, logarithmic, deformed-calculus, and application-driven two-parameter forms such as $S_{q,\delta}$, $S_{\delta,\epsilon}$, and $\mathcal{E}^{LN}_{\alpha,\beta}$ (Ilić et al., 2021).

1. Principal families and their mathematical archetypes

The literature classifies the main two-parameter entropies into distinct structural families rather than simple reparameterizations of one master formula. The most important division is between trace-form functionals, kernel-form functionals, and asymptotic-scaling classes (Ilić et al., 2021).

Family	Representative form	Distinguishing principle
Sharma–Taneja–Mittal, degree $(\alpha,\beta)$	$$S_{\alpha,\beta}[p]=\sum_{i=1}^W \frac{p_i^\beta-p_i^\alpha}{\alpha-\beta}$$	Trace-form; generalized logarithm and generalized composability
Sharma–Mittal, order $(\alpha,\beta)$	Kernel-form entropy of order $(\alpha,\beta)$	Generalized averaging; interpolates between Rényi and Tsallis
Hanel–Thurner, class $(\alpha,\beta)$0	$(\alpha,\beta)$1	Trace-form; asymptotic scaling class (Ghikas et al., 2017)
Logarithmic $(\alpha,\beta)$2-norm entropy	$(\alpha,\beta)$3	Scale-invariant on sub-probabilities (Ghosh et al., 2019)

For the Sharma–Taneja–Mittal family, the two parameters deform the elementary information content through the generalized logarithm

$(\alpha,\beta)$4

so that

$(\alpha,\beta)$5

In this family, $(\alpha,\beta)$6 and $(\alpha,\beta)$7 directly control the trace kernel and the nonlinear composition law (Ilić et al., 2021).

In the Sharma–Mittal family, the two parameters play different roles: $(\alpha,\beta)$8 controls the internal order of the power sum, while $(\alpha,\beta)$9 controls the external nonadditive deformation. The review literature treats it as a kernel-form entropy rather than a trace-form one (Ilić et al., 2021).

In the Hanel–Thurner family, the pair $(c,d)$0 is not introduced primarily as a deformation of additivity. Instead, it characterizes the asymptotic scaling behavior

$(c,d)$1

so that $(c,d)$2 labels universality classes of complex systems (Ilić et al., 2021).

Later constructions broaden the landscape. The two-parameter turbulence entropy

$(c,d)$3

assigns distinct roles to $(c,d)$4 and $(c,d)$5: the first controls the deformation of the logarithm and the second the stretching exponent in the associated canonical distributions (Beck et al., 2 Jun 2026). The entropy

$(c,d)$6

was introduced as a microscopic functional whose equiprobable form yields two independent area contributions in holographic applications (Luciano et al., 11 Mar 2026).

2. Kernel characterizations by functional equations

A major line of research treats two-parameter entropies through functional equations. The central result of Furuichi is a characterization theorem for the one-point kernel $(c,d)$7 of the entropy

$(c,d)$8

If a differentiable nonnegative function $(c,d)$9 satisfies

$S_{q,\delta}$0

then necessarily

$S_{q,\delta}$1

and in the diagonal case,

$S_{q,\delta}$2

This gives a uniqueness theorem for the differentiable nonnegative solutions of a simple multiplicative functional equation and identifies the two-parameter entropy kernel exactly (Furuichi, 2010).

The same paper shows that Tsallis entropy is recovered by the choices $S_{q,\delta}$3 or $S_{q,\delta}$4. The corresponding specialized equation,

$S_{q,\delta}$5

has unique solution

$S_{q,\delta}$6

and the limit $S_{q,\delta}$7 gives the Shannon kernel $S_{q,\delta}$8 (Furuichi, 2010).

A related functional-equation analysis gives the general solution of

$S_{q,\delta}$9

namely

$S_{\delta,\epsilon}$0

and, in the degenerate case,

$S_{\delta,\epsilon}$1

with $S_{\delta,\epsilon}$2; under regularity, $S_{\delta,\epsilon}$3 (Gselmann, 2013). This is another route to the same difference-of-powers kernel.

The axiomatic characterization of two-parameter trace-form entropies was later corrected in the context of the two-parameter generalized Shannon–Khinchin axioms. The corrected class is

$S_{\delta,\epsilon}$4

with a denominator $S_{\delta,\epsilon}$5 constrained by continuity, sign, vanishing on the diagonal, and the limit condition

$S_{\delta,\epsilon}$6

This correction showed that the stronger differentiability assumptions used in the earlier Wada–Suyari theorem were neither necessary nor sufficient (Ilic et al., 2013).

3. Construction principles: generalized logarithms, calculus, and deformed kernels

Two-parameter entropies are often generated from generalized logarithms. In the Sharma–Taneja–Mittal setting, the basic object is $S_{\delta,\epsilon}$7, and the entropy remains trace-form. In other constructions, the logarithm itself is the primary deformation.

One influential example is the Euler two-parameter logarithm

$S_{\delta,\epsilon}$8

which generates the generalized Borges–Roditi entropy

$S_{\delta,\epsilon}$9

This formalism unifies several known one-parameter logarithms and entropies inside a two-parameter deformation, while also serving as the mirror map in generalized exponentiated-gradient algorithms (Cichocki, 21 Feb 2025).

A different construction begins from $\mathcal{E}^{LN}_{\alpha,\beta}$0-calculus. With

$\mathcal{E}^{LN}_{\alpha,\beta}$1

the corresponding entropy is defined by

$\mathcal{E}^{LN}_{\alpha,\beta}$2

It can be rewritten as

$\mathcal{E}^{LN}_{\alpha,\beta}$3

This additive-shift calculus differs from the multiplicative $\mathcal{E}^{LN}_{\alpha,\beta}$4-calculus used in Jackson, Abe, and Borges–Roditi constructions (Kang et al., 2019).

Another trace-form proposal uses the deformed logarithm

$\mathcal{E}^{LN}_{\alpha,\beta}$5

and defines

$\mathcal{E}^{LN}_{\alpha,\beta}$6

Here the factor $\mathcal{E}^{LN}_{\alpha,\beta}$7 is not incidental: it is tailored to the exact product rule of $\mathcal{E}^{LN}_{\alpha,\beta}$8, which in turn yields exact chain rules and pseudo-additivity (Dutta et al., 2019).

The logarithmic $\mathcal{E}^{LN}_{\alpha,\beta}$9-norm entropy follows yet another principle. Instead of deforming a pointwise logarithm, it compares logarithms of $(\alpha,\beta)$0- and $(\alpha,\beta)$1-norms: $(\alpha,\beta)$2 This construction is scale-invariant on sub-probabilities and is therefore structurally distinct from the Sharma–Taneja–Mittal and Sharma–Mittal families (Ghosh et al., 2019).

4. Limiting cases and reductions to standard entropies

The reduction patterns of two-parameter entropies are one of their main organizing principles. For the Sharma–Taneja–Mittal family, the point $(\alpha,\beta)$3 yields the Boltzmann–Gibbs–Shannon entropy, the line $(\alpha,\beta)$4 yields the Havrda–Charvát–Daróczy–Tsallis entropy, the choice $(\alpha,\beta)$5 yields the Kaniadakis entropy, the choice $(\alpha,\beta)$6 yields the Abe entropy, and the diagonal limit $(\alpha,\beta)$7 gives

$(\alpha,\beta)$8

(Ilić et al., 2021).

The Sharma–Mittal family reduces to Rényi entropy on the line $(\alpha,\beta)$9, to Tsallis entropy on the line $$S_{\alpha,\beta}[p]=\sum_{i=1}^W \frac{p_i^\beta-p_i^\alpha}{\alpha-\beta}$$0, and to Boltzmann–Gibbs–Shannon entropy at $$S_{\alpha,\beta}[p]=\sum_{i=1}^W \frac{p_i^\beta-p_i^\alpha}{\alpha-\beta}$$1. Its diagonal limit also produces a weighted logarithmic kernel of the form $$S_{\alpha,\beta}[p]=\sum_{i=1}^W \frac{p_i^\beta-p_i^\alpha}{\alpha-\beta}$$2 (Ilić et al., 2021).

For the Hanel–Thurner family, $$S_{\alpha,\beta}[p]=\sum_{i=1}^W \frac{p_i^\beta-p_i^\alpha}{\alpha-\beta}$$3 gives Boltzmann–Gibbs–Shannon entropy, while the class $$S_{\alpha,\beta}[p]=\sum_{i=1}^W \frac{p_i^\beta-p_i^\alpha}{\alpha-\beta}$$4 includes the Tsallis family in the scaling classification, and $$S_{\alpha,\beta}[p]=\sum_{i=1}^W \frac{p_i^\beta-p_i^\alpha}{\alpha-\beta}$$5 includes the Kaniadakis family (Ilić et al., 2021).

The $$S_{\alpha,\beta}[p]=\sum_{i=1}^W \frac{p_i^\beta-p_i^\alpha}{\alpha-\beta}$$6-calculus entropy is explicitly designed to recover known one-parameter and two-parameter entropies. The limits and parameter choices

$$S_{\alpha,\beta}[p]=\sum_{i=1}^W \frac{p_i^\beta-p_i^\alpha}{\alpha-\beta}$$7

recover respectively the Boltzmann–Gibbs, Tsallis, Abe, Kaniadakis, and Shafee entropies (Kang et al., 2019).

For $$S_{\alpha,\beta}[p]=\sum_{i=1}^W \frac{p_i^\beta-p_i^\alpha}{\alpha-\beta}$$8, the cases

$$S_{\alpha,\beta}[p]=\sum_{i=1}^W \frac{p_i^\beta-p_i^\alpha}{\alpha-\beta}$$9

yield Boltzmann–Gibbs entropy, Tsallis entropy, and the $(\alpha,\beta)$0-entropy

$(\alpha,\beta)$1

The point $(\alpha,\beta)$2 is singled out in earlier work as a thermodynamic black-hole entropy and reappears in the turbulence application as the endpoint value $(\alpha,\beta)$3 (Beck et al., 2 Jun 2026).

For the logarithmic $(\alpha,\beta)$4-norm entropy, if either parameter equals $(\alpha,\beta)$5, then on $(\alpha,\beta)$6 it reduces to Rényi entropy; on the diagonal it becomes

$(\alpha,\beta)$7

the Shannon entropy of the escort distribution $(\alpha,\beta)$8 (Ghosh et al., 2019).

5. Axioms, composability, stability, and geometry

The original Shannon–Khinchin axioms single out the Boltzmann–Gibbs–Shannon entropy. Generalized two-parameter families emerge by modifying the conditional composition axiom. Replacing the conditional term by a $(\alpha,\beta)$9-weighted one yields Tsallis entropy; a further two-parameter generalization due to Wada and Suyari leads to the Sharma–Taneja–Mittal family with kernel $(\alpha,\beta)$0 (Ilić et al., 2021).

For statistically independent systems, these modifications induce nonlinear composition laws. Tsallis entropy satisfies the standard pseudo-additivity

$(\alpha,\beta)$1

while the normalized Tsallis entropy satisfies a similar law with coefficient $(\alpha,\beta)$2 instead of $(\alpha,\beta)$3. Furuichi showed that the functional equation for the Tsallis kernel encodes both pseudoadditivity structures at once (Furuichi, 2010).

The Sharma–Taneja–Mittal family has a more general composition law involving the power sums $(\alpha,\beta)$4 and $(\alpha,\beta)$5, whereas the Sharma–Mittal family retains a Tsallis-type pseudo-additivity at the level of independent-system composition (Ilić et al., 2021). By contrast, the Hanel–Thurner family is organized by thermodynamic-limit scaling rather than a single universal composition rule.

Stability properties are not uniform across the landscape. The review literature states that trace-form families such as Sharma–Taneja–Mittal and Hanel–Thurner satisfy the Lesche criterion, whereas the situation for Sharma–Mittal is more problematic, and some members such as Rényi entropy appear not to be Lesche stable (Ilić et al., 2021).

Some later two-parameter proposals were designed to restore stronger information-theoretic properties. The entropy $(\alpha,\beta)$6 satisfies exact chain rules, pseudo-additivity for independent variables, sub-additivity, and strong sub-additivity, and its associated divergence is nonnegative, jointly convex, and information monotone under stochastic maps (Dutta et al., 2019). The logarithmic $(\alpha,\beta)$7-norm entropy is extensive on products,

$(\alpha,\beta)$8

and is scale-invariant on sub-probabilities,

$(\alpha,\beta)$9

(Ghosh et al., 2019).

The Hanel–Thurner family also supports an information-geometric program. The associated maximizing distributions form a $(\alpha,\beta)$00-exponential family

$(\alpha,\beta)$01

from which one obtains a generalized divergence, a Fisher-like metric, a generalized Cramér–Rao inequality, and a scalar curvature used to discriminate complexity classes (Ghikas et al., 2017).

6. Applications and extensions

Two-parameter entropies have become application-specific modeling tools. In turbulence, maximizing

$(\alpha,\beta)$02

under ordinary normalization and linear energy constraints produces generalized canonical distributions of stretched $(\alpha,\beta)$03-exponential form. For turbulent velocity differences $(\alpha,\beta)$04, the resulting continuum density is

$(\alpha,\beta)$05

and the theory yields the scale-dependent relation

$(\alpha,\beta)$06

which the authors compare with Taylor–Couette data (Beck et al., 2 Jun 2026).

In holographic cosmology, the entropy

$(\alpha,\beta)$07

induces a generalized area law

$(\alpha,\beta)$08

and hence a generalized holographic dark-energy density

$(\alpha,\beta)$09

This framework embeds standard holographic dark energy and $(\alpha,\beta)$10CDM as limiting cases (Luciano et al., 11 Mar 2026).

In optimization, the Euler logarithm

$(\alpha,\beta)$11

defines a mirror map

$(\alpha,\beta)$12

whose Bregman divergence regularizes generalized exponentiated-gradient and mirror-descent updates. The resulting simplex-constrained update is

$(\alpha,\beta)$13

and was developed for online portfolio selection (Cichocki, 21 Feb 2025).

The two-parameter idea also extends beyond scalar probability functionals. In operator theory, the Tsallis relative operator $(\alpha,\beta)$14-entropy

$(\alpha,\beta)$15

and the relative operator $(\alpha,\beta)$16-entropy

$(\alpha,\beta)$17

were introduced so as to preserve joint convexity under explicit parameter restrictions (Nikoufar, 2017). In quantum information, the $(\alpha,\beta)$18-$(\alpha,\beta)$19-Rényi relative entropy is a continuous, real-valued, two-parameter family of contractive distinguishability measures used to derive generalized entropic quantum speed limits (Sousa et al., 19 Jan 2025).

7. Corrections, controversies, and conceptual cautions

The theory of two-parameter entropies contains several technical cautions. First, uniqueness theorems are sensitive to regularity assumptions. The correction of the Wada–Suyari theorem showed that the earlier differentiability conditions on $(\alpha,\beta)$20 were not calibrated correctly: admissible entropies existed without them, and the old conditions did not guarantee the Shannon limit (Ilic et al., 2013). Related work on entropy functional equations further showed that measurability or local boundedness often suffices to eliminate pathological additive or logarithmic solutions (Gselmann, 2013).

Second, non-trace-form entropies preserve Legendre reciprocity more generally than was sometimes assumed. For the class

$(\alpha,\beta)$21

including the Rényi case

$(\alpha,\beta)$22

the canonical MaxEnt treatment still yields the reciprocity relations

$(\alpha,\beta)$23

The anomalies that appear in some multiplier formulas arise when one varies a normalization-reduced entropy representation rather than the full unconstrained functional (Plastino et al., 2018).

Third, superstatistical reconstruction does not single out a unique entropy functional from a distribution. Under the first three Shannon–Khinchin axioms, only two reconstruction schemes survive: the HT scheme without escort probabilities and the TS scheme with escort probabilities. These are related by a duality at the level of generalized logarithms,

$(\alpha,\beta)$24

or equivalently

$(\alpha,\beta)$25

The choice between escort and non-escort reconstruction is therefore empirical rather than purely formal (Hanel et al., 2011).

A plausible implication is that two-parameter entropies should be distinguished not only by their values on normalized distributions, but also by their variational representatives, their composability rules, and the scaling or stability principle from which they are derived. That is why the literature does not treat the Sharma–Taneja–Mittal, Sharma–Mittal, and Hanel–Thurner families as equivalent parameterizations of the same object, even when they share limiting cases such as Boltzmann–Gibbs, Tsallis, Rényi, or Kaniadakis (Ilić et al., 2021).