Modified Entropy Functions

• Modified entropy functions are generalized forms of classical entropy that incorporate nonadditivity, quantum corrections, and microphysical structures.
• They employ deformed trace forms and geometric modifications to yield distinctive thermodynamic and cosmological predictions.
• Applications span black hole thermodynamics, entropic gravity, and information theory, impacting stability, concavity, and dynamical complexity.

A modified entropy function is any deformation or generalization of the canonical Boltzmann–Gibbs–Shannon or Bekenstein–Hawking entropy expressions, motivated by the need to encode nonadditive statistical effects, quantum gravity corrections, information-theoretic bounds, or underlying microphysical structures. In both statistical physics and cosmology, as well as in dynamical systems and information theory, a broad array of such modified entropy functions has been constructed, each yielding distinctive consequences for fundamental laws—ranging from thermodynamics and dynamical equations to measures of uncertainty and complexity. This functional diversity is realized through alteration of composability properties (nonextensive, pseudoadditive forms), area-dependence (deformed area laws), structural invariants in probability space, or by imposing stability, regularity, or modified uncertainty constraints.

1. Formal Constructions and Classifications

The construction of modified entropy functions typically takes one of the following forms:

• Deformed trace-form entropies: These involve nonlinear functions of probabilities or phase-space volumes, characterized by a deformation or pathway parameter. Notable examples include:
  • Tsallis entropy: $S_q = \frac{1 - \sum_{i} p_i^q}{1 - q}$ with parameter $q$, yielding nonadditive composition and emerging uniquely from the requirement that the conditional extensivity coefficient is a power-law of the marginal probability (Takatsu, 2020, Gselmann, 2013).
  • Kaniadakis entropy: $$S_\kappa = \frac{\sum (p_i^{1+\kappa} - p_i^{1-\kappa})}{2\kappa}$$, or in the microcanonical case $S_\kappa = \frac{W^\kappa - W^{-\kappa}}{2\kappa}$, with deformation parameter $\kappa$, respecting κ-exponential and relativistic additivity (Ambrósio et al., 2024, Lymperis et al., 2021, Zangeneh et al., 2023).
  • Mathai entropy: $$M_\alpha(p) = \frac{\sum_i p_i^\alpha - 1}{\alpha - q}$$, a two-parameter family with pathway and anchor exponents, which yields classical gamma, beta, and Tsallis distributions depending on limiting processes (Haubold, 2024).
• Geometric and horizon-entropic deformations: In gravitational and cosmological contexts, the entropy-area relation is modified:
  • Barrow entropy: $S_\delta \sim A^{1+\delta/2}$, where $\delta \in [0,1]$ quantifies quantum-gravitational fractality of the horizon (Sheykhi, 2022, Asghari et al., 2021).
  • Fractional entropy: $S_h \sim (\pi r_A^2)^{(2+\alpha)/2\alpha}$, interpolating between area-law ($\alpha=2$) and super-area scaling (Çoker et al., 2023).
  • Modified dispersion/GUP-induced entropy: Gravitational entropy corrections expressed as $$S(A) = \frac{A}{4L_p^2} - \frac{\beta\pi}{4}\ln\left(\frac{A}{4L_p^2}\right) + ...$$, where higher curvature or uncertainty principles induce terms logarithmic or of negative powers in area (Hammad, 2015, Sefiedgar et al., 2010, Liu et al., 2010).
• Combinatorial, polynomial, or dynamical entropic constructions:
  • Symmetric polynomial-based entropy functionals exhibit complete monotonicity and Pick/Bernstein properties and can be expressed analytically in terms of the elementary symmetric polynomials of probabilities (Jozsa et al., 2014).
  • Modified power entropy: A dynamical entropy concept using Hamming averages rather than maximal separation, encoding sensitivity (or insensitivity) to transient dynamical events (Gröger et al., 2015).

2. Mathematical Properties and Uniqueness

Modified entropy functions are often tightly constrained by either axiomatic or dynamical considerations:

• Axiomatic uniqueness: The Tsallis entropy is unique among deformed entropies in satisfying generalized Shannon–Khinchin axioms with power-law extensivity; any attempt to generalize the conditional extensivity coefficient to a non-power function fails (Takatsu, 2020). Similarly, Mathai's entropy recovers trace-form, non-escort structure in a unified scalar, vector, and matrix setting (Haubold, 2024).
• Functional equations: Many entropy types are characterized via cocycle or multiplicative-type functional equations. For instance, Gselmann shows that the branching equation

$f(xy) + f((1-x)y) - f(y) = (f(x) + f(1-x)) y^q$

admits only affine combinations of Tsallis and Shannon forms under weak regularity assumptions (Gselmann, 2013).

• Stability: The modified entropy equation, for exponents $\alpha$, admits Hyers–Ulam stability for $\alpha \le 0$ (i.e., approximate solutions are close to exact ones), and bounded stability in positive exponents on compacta. The stable forms ultimately take the sum-of-powers plus aggregation form at the core of trace entropies (Gselmann, 2013, Gselmann, 2013).

3. Thermodynamic and Cosmological Consequences

Substituting a modified entropy function for the area law in horizon thermodynamics or gravitational field equations leads to structurally novel modifications of cosmological evolution:

• Modified Friedmann equations: For example, replacing semiclassical horizon entropy with the Tsallis, Barrow, or Kaniadakis forms modifies the Friedmann equation's scaling in the Hubble parameter ($H$), introduces new effective dark-energy-like terms, and changes the predicted cosmic acceleration history (Lymperis et al., 2021, Sheykhi, 2018, Sheykhi, 2022, Çoker et al., 2023, Zangeneh et al., 2023).
  • Barrow entropy yields $(H^2)^{1-\delta/2} \propto \rho$, shifting the power-law scaling and reducing the predicted age of the universe for $\delta>0$ (Sheykhi, 2022, Asghari et al., 2021).
  • Tsallis entropy encodes late-time acceleration without explicit dark energy for $\beta < 1/2$, with the age-of-universe problem alleviated (Sheykhi, 2018).
  • Kaniadakis entropy induces extra cubic and higher-order area (or inverse Hubble) terms, resulting in geometric self-acceleration scenarios, phantom effective equations of state, and longer universe ages (Lymperis et al., 2021, Zangeneh et al., 2023).
• Thermodynamic and non-equilibrium constraints: The generalized second law is typically preserved in frameworks based on the first law at the apparent horizon with suitably regular modified entropy, as explicitly demonstrated in multiple settings (Sheykhi, 2018, Çoker et al., 2023).

4. Applications to Black Hole Physics, Entropic Gravity, and Information Theory

• Black hole entropy and information bounds: Modified entropy functions such as Tsallis, Kaniadakis, and GUP-induced forms result in logarithmic or inverse-area corrections to the Bekenstein–Hawking entropy, with direct implications for thermodynamic consistency, the saturation of the Bekenstein entropy bound, and information-theoretic limits such as those imposed by the Landauer principle (Ambrósio et al., 2024, Hammad, 2015, Sefiedgar et al., 2010).
• Entropic gravity and MOND analogues: Utilizing modified entropy in the Verlinde entropic-force framework leads naturally to MOND-like interpolation functions; for instance, the modified Kaniadakis entropy yields an effective force law

$$F_{\rm eff} = \frac{G M m}{R^2} \frac{1}{\sqrt{1+(\kappa \pi R^2/l_p^2)^2}}$$

reproducing MOND's characteristic transition between Newtonian and deep-MOND regimes (Ambrósio et al., 2024).

• Quantum and classical information theory: Modified symmetric polynomial entropies and alternative functionals, such as those tailored to fractional Gaussian noise (fGn) and parameterized by the Hurst index $H$, enable robust analysis of entropy monotonicity and phase transitions in the entropy-rate of long-memory processes. These surrogates trade explicitness for computational tractability in high dimensions while preserving qualitative behaviors (Malyarenko et al., 2023, Jozsa et al., 2014).

5. Regularity, Composability, and Domain Generalization

• Domain-unifying forms: Mathai's entropy $M_\alpha$ applies seamlessly across real and complex scalars, vectors, and positive-definite matrices, generating through optimization all standard beta, gamma, and heavy-tailed families as pathway parameter $\alpha$ traverses the anchor $q$ (Haubold, 2024).
• Pseudoadditivity and composability: Modified entropies often violate or generalize strict additivity. Tsallis entropy is pseudoadditive, yielding $S_q(A+B) = S_q(A) + S_q(B) + (1-q)S_q(A)S_q(B)$. Kaniadakis and Barrow forms display non-additive but smooth composability corrections; concavity and monotonicity constraints are often retained.
• Escort measures and moment constraints: For distributions maximizing Tsallis entropy, escort distributions (reweighting of probabilities via $p_i^q$) are typically needed in optimization, whereas Mathai's entropy admits direct moment-like constraints without escorts, simplifying calculus-of-variations treatments and leading to elementary maximum-entropy characterizations (Haubold, 2024).

6. Dynamical Entropy and Complexity Measures

• Modified dynamical entropy and slow entropy invariants: In dynamical systems, modified entropy functions serve to detect nonhyperbolic complexity not captured by Kolmogorov-Sinai entropy. However, not all modified entropies are equally sensitive; for example, modified power entropy (MPE) is strongly affected by transient dynamics, does not satisfy a measure-theoretic variational principle, and may fail to detect phase transitions in equicontinuity, in contrast to amorphic complexity or power entropy (Gröger et al., 2015).

Entropy Type	Functional Form	Principal Deformation Parameter(s)
Tsallis	$S_q = \frac{1 - \sum p_i^q}{1-q}$	$q$
Kaniadakis	$S_\kappa = \frac{W^\kappa - W^{-\kappa}}{2\kappa}$	$\kappa$
Barrow	$S_\delta \sim A^{1 + \delta/2}$	$\delta$
Mathai	$M_\alpha = \frac{ \sum p_i^\alpha - 1 }{\alpha-q}$	$\alpha, q$

7. Limitations, Open Problems, and Research Directions

• Axiomatic constraints: The impossibility of constructing a non-power-law modified extensivity coefficient in multi-variable entropy functionals tightly restricts admissible deformations once the composition rule is fixed (Takatsu, 2020).
• Concavity and stability: For certain deformed entropies (e.g., Kaniadakis, Barrow), the full domain of concavity and rigorous stability bounds in all deformation parameters remain open (Ambrósio et al., 2024).
• Physical constraints: Astrophysical and cosmological datasets place stringent bounds on parameters such as $\delta$, $\beta$, or $\kappa$, typically finding consistency with the undeformed (Boltzmann–Gibbs–Shannon) or ΛCDM cases within current observational accuracy (Asghari et al., 2021).
• Non-equilibrium and nonadditive thermodynamics: The composition and extensivity properties of modified entropies raise foundational questions for the consistent formulation of non-equilibrium statistical mechanics and for deriving generalized kinetic or hydrodynamic equations.

Modified entropy functions thus offer a flexible and mathematically tractable framework to systematically generalize canonical entropy, analyze information or thermodynamic content under deformed statistics or geometry, and probe the interface of gravity, cosmology, and information theory. The structural, axiomatic, and observational constraints now sharply delimit the viable forms, with future progress dependent on high-precision phenomenology and the elucidation of microscopic origins for the deformation parameters.