Generalized Entropy Relations Overview

• Generalized entropy relations are a unified framework that parameterizes conventional entropy measures like Tsallis, Rényi, and Barrow entropies.
• They employ functional expansions and ensemble averages to capture non-extensive and quantum corrections in statistical mechanics.
• These relations extend traditional thermodynamic laws by offering generalized protocols for complex systems, quantum estimation, and information theory.

Generalized entropy relations unify and extend classical, quantum, and statistical entropy concepts by parameterizing the functional dependence of entropy on phase-space distributions, ensembles, and dynamical invariants. These relations encompass one- and multi-parameter deformational frameworks (Tsallis, Rényi, Barrow, Kaniadakis, Sharma-Mittal, loop quantum gravity, and others) and provide the basis for generalized thermodynamic laws, information measures, and operational interpretations across statistical mechanics, quantum theory, dynamical systems, and gravitation.

1. Unified Parameterizations and Structural Foundations

Generalized entropy, denoted $S_\mathrm{gen}$, is formulated to encapsulate a hierarchy of known entropic forms via parameter-driven functional expansions. The most general construction utilizes Taylor-series or functional forms parameterized by $\{\alpha_\pm, \delta, \gamma, \dots\}$: $$S_{\mathrm{gen}} = \sum_{n=0}^\infty \frac{f_n(\text{parameters})}{n!} S^n$$ where the coefficients $f_n$ are functions of the various deformation parameters and $S$ is a reference entropy (e.g., area/volume, surprisal average). This framework reduces to:

• Tsallis entropy: $S_{q} = \frac{S^{1-q} - 1}{1-q}$ for $\alpha_-=0, \alpha_+=1, \delta=1-q, \gamma=1$
• Rényi entropy: $S_R = \frac{1}{1-q} \log[1 + (1-q)S]$ in the limit $\gamma \rightarrow 0$
• Barrow entropy: $S_B = S^{1+\Delta/2}$ for $\Delta \ge 0$
• Kaniadakis entropy: $S_\kappa = (S^{1+\kappa} - S^{1-\kappa})/(2\kappa)$ for symmetric $\alpha_\pm$
• Sharma-Mittal entropy: $$S_{SM}^{(r,q)} = \left[(1 + (1-q)S)^{\frac{1-r}{1-q}} - 1\right]/(1-r)$$

Specializations and limits recover all classically and quantum-motivated entropy expressions (Nojiri et al., 2023, Nojiri et al., 2023).

2. Microscopic and Ensemble-Average Representations

Generalized entropy admits a precise statistical mechanics interpretation via ensemble averages of powers of the surprisal $-k_B \ln \rho$ and fluctuations of energy and number operators: $$S_\mathrm{gen} = \sum_{n=0}^\infty \frac{f_n}{n!} \langle (-k_B \ln \rho)^n \rangle - \sum_{n=0}^\infty \sum_{i=2}^n \frac{f_n}{i!(n-i)!} (k_B \beta)^i \sigma_i (k_B \ln Z)^{n-i}$$ where $\rho$ is the phase-space or Hilbert space density, $Z$ the partition function, and $\sigma_i$ the $i$th central moment of the Hamiltonian (or $H-\mu N$ in grand-canonical ensembles). Fluctuation terms encode all non-additive and non-Gaussian corrections, unifying quantum, thermal, and nonextensive sources of entropy (Nojiri et al., 2023).

3. Geometric, Dynamical, and Dimensional Extensions

Generalized entropy measures can be constructed in fully dimensional, physical-variable frameworks. For a distribution $f(X)$ on phase-space $P$: $$S_w[f] = N \, w^{-1} \left( \frac{1}{N} \int_P f(X) w(f(X)) \, dX \right )$$ with $w$ an invertible weight function and $N=\int f(X)$ normalization. Power-law choices ($w(f) = f^{q-1}$) generate Tsallis and Rényi forms, which link directly to phase-space volumes and scaling relations. Composite entropy measures arise via linear or functional superpositions of $S_w$ with different weightings, adapting entropy probes to multifractal or multiscale irreversibility (Zhdankin, 2023).

Connections to Dynamical Systems

Generalized entropy provides a scale-sensitive diagnostic of complexity beyond topological entropy. For a system $T: X \rightarrow X$, generalized entropy characterizes the polynomial or super-polynomial growth of separated or spanning sets—this enables distinguishing "zero-entropy" systems by their generalized order of growth: $$o(T) = \sup_{\varepsilon > 0} [\mathrm{Span}(T, n, \varepsilon)] \in \overline{\mathbb{O}}$$ Induced maps on hyperspaces or measure spaces amplify complexity, with precise lower and upper bounds (Lacerda, 29 Mar 2025).

4. Generalized Entropy in Quantum and Statistical Estimation

Quantum extensions utilize the framework of hypothesis-testing entropy, expressed via the optimal error-exponents for discriminating states: $$2^{-D_H^\varepsilon(\rho \|\sigma)} = \frac{1}{\varepsilon} \min_{0 \le Q \le I, \mathrm{Tr}[Q\rho] \ge \varepsilon} \mathrm{Tr}[Q\sigma]$$ yielding conditional entropies, operationally tight bounds to min/max/smooth entropies, and exact chain rules. These measures generalize the von Neumann and Gibbs–Shannon entropy in one-shot and asymptotic regimes, preserving composability and data-processing (Dupuis et al., 2012, Sinha et al., 2023).

5. Deformed Thermodynamic and Extensivity Laws

Generalized entropy relations underpin extended thermodynamic laws, including variants of the Clausius relation, generalized first and second laws, and entropy–extremality constraints for black holes and gravitating systems. The generalized mass-to-horizon relation and entropy formula: $$S_G(L) = 2\pi k_B \gamma \left \{ \frac{m}{m+1} x^{m+1} \mp \beta \frac{m(\sigma-1)}{\sigma} x^\sigma \right \}$$ with $x=L/\ell_\mathrm{Pl}$, unifies area, power-law, and quantum-corrected entropy forms; parameter choices recover Tsallis, Barrow, LQG, and other corrections (Gohar, 8 Oct 2025, Anand, 22 Apr 2025).

6. Information-Theoretic and Resource-Theoretic Interrelations

Generalized entropy relations encode resource monotones via majorization extensions (e.g., "c-trumping") fully determined by entropy measures; Shannon entropy emerges as the unique monotone governing possible bistochastic transitions under auxiliary correlations. Extensions to Rényi and Tsallis entropy structure catalytic and non-catalytic resource conversion protocols in quantum information theory (Mueller et al., 2015, Enciso et al., 2017).

7. Statistical Inference, Estimation, and Concentration Phenomena

Maximization principles for generalized entropy provide the foundation for inference under constraints (MaxGEnt framework), optimal estimation bounds via generalized Fisher information, and concentration inequalities in high-dimensional or count-vector settings: $$H_\mathrm{gen}(\nu) = -\sum_{i=1}^m \nu_i \ln \nu_i + N \ln N$$ with sharp non-asymptotic bounds relating solution counts and combinatorial structures (Oikonomou, 2016, Bercher, 2013).

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Together, generalized entropy relations constitute a mathematically rigorous, physically motivated overarching theory, unifying statistical, quantum, geometric, thermodynamic, and information-theoretic perspectives. Parameterized frameworks support cross-disciplinary applications from complex systems, nonequilibrium statistical mechanics, quantum thermodynamics, black-hole physics, and data science.