[Paper Review] Nongeneralizability of Tsallis Entropy by means of Kolmogorov-Nagumo averages under pseudo-additivity

Ambedkar Dukkipati, M.N. Murty+1 more|ArXiv.org|May 30, 2005

Statistical Mechanics and Entropy8 references3 citations

TL;DR

This paper demonstrates that Tsallis entropy cannot be generalized via Kolmogorov-Nagumo (KN) averages while preserving its defining pseudo-additivity property. By analyzing the functional constraints required for KN-averaged Tsallis entropy to satisfy $ S_q(pr) = S_q(p) +_q S_q(r) $, the authors prove that only linear KN functions satisfy the necessary conditions, reducing the generalized form back to standard Tsallis entropy, thus proving its nongeneralizability under this framework.

ABSTRACT

As additivity is a characteristic property of the classical information measure, Shannon entropy, pseudo-additivity is a characteristic property of Tsallis entropy. Renyi generalized Shannon entropy by means of Kolmogorov-Nagumo averages, by imposing additivity as a constraint.In this paper we show that there exists no generalization for Tsallis entropy, by means of Kolmogorov-Nagumo averages, which preserves the pseudo-additivity.

Motivation & Objective

• To investigate whether Tsallis entropy can be generalized using Kolmogorov-Nagumo (KN) averages while preserving its pseudo-additivity property.
• To determine the functional form of the KN function $ \psi $ that would allow a generalized entropy measure to maintain the $ q $-additive structure of Tsallis entropy.
• To establish the mathematical constraints under which KN-averaged entropies can satisfy pseudo-additivity for independent probability distributions.
• To clarify the fundamental distinction between Rényi entropy (generalized via KN averages with additivity) and Tsallis entropy (which resists such generalization under the same formalism).

Proposed method

• Formalizing the generalized entropy as $ \widetilde{S}_{\psi}(p) = \psi^{-1}\left[\sum_k p_k \psi(\ln_q(1/p_k))\right] $, where $ \psi $ is a KN function.
• Imposing the pseudo-additivity condition $ \widetilde{S}_{\psi}(pr) = \widetilde{S}_{\psi}(p) +_q \widetilde{S}_{\psi}(r) $ for independent distributions $ p $ and $ r $.
• Deriving the functional equation $ \psi^{-1}\left[\sum_{i,j} p_i r_j \psi(\widetilde{H}_i^p +_q \widetilde{H}_j^r)\right] = \psi^{-1}\left[\sum_i p_i \psi(\widetilde{H}_i^p)\right] +_q \psi^{-1}\left[\sum_j r_j \psi(\widetilde{H}_j^r)\right] $.
• Analyzing the resulting equation under the substitution $ \widetilde{H}_j^r = J $, a constant, to derive the condition $ \psi^{-1}\left[\sum_k p_k \psi(\widetilde{H}_k^p +_q J)\right] = \psi^{-1}\left[\sum_k p_k \psi(\widetilde{H}_k^p)\right] +_q J $.
• Reducing the condition to $ \langle x +_q C \rangle_\psi = \langle x \rangle_\psi +_q C $, and further to the linear functional equations $ \langle x + C \rangle_\psi = \langle x \rangle_\psi + C $ and $ \langle Bx \rangle_\psi = B \langle x \rangle_\psi $.
• Concluding that only linear $ \psi $ satisfies both conditions, implying the generalized form reduces to standard Tsallis entropy.

Experimental results

Research questions

• RQ1 Can Tsallis entropy be generalized via Kolmogorov-Nagumo averages while preserving its pseudo-additivity property?
• RQ2 What functional form must the KN function $ \psi $ take for the generalized entropy to satisfy $ \widetilde{S}_{\psi}(pr) = \widetilde{S}_{\psi}(p) +_q \widetilde{S}_{\psi}(r) $?
• RQ3 Why does the Rényi entropy generalization via KN averages not extend to Tsallis entropy, despite both being one-parameter generalizations of Shannon entropy?
• RQ4 Are there any non-linear KN functions that can yield a pseudo-additive entropy generalizing Tsallis entropy?
• RQ5 What are the necessary and sufficient conditions on $ \psi $ for the generalized entropy to maintain the $ q $-additive structure?

Key findings

• The only KN function $ \psi $ that satisfies both the shift-invariance $ \langle x + C \rangle_\psi = \langle x \rangle_\psi + C $ and scaling condition $ \langle Bx \rangle_\psi = B \langle x \rangle_\psi $ is a linear function.
• When $ \psi $ is linear, the generalized entropy $ \widetilde{S}_{\psi}(p) $ reduces to the standard Tsallis entropy $ S_q(p) = \frac{1 - \sum_k p_k^q}{q - 1} $.
• Nonlinear KN functions, such as exponential functions, satisfy the shift-invariance condition but fail the scaling condition, thus violating the required pseudo-additivity.
• The requirement for pseudo-additivity under KN averaging forces the functional form of $ \psi $ to be linear, which eliminates the possibility of nontrivial generalizations of Tsallis entropy.
• This proves that Tsallis entropy is fundamentally incompatible with generalization via KN averages under the pseudo-additivity constraint, unlike Rényi entropy.
• The result establishes a clear mathematical boundary: Tsallis entropy cannot be extended beyond its standard form using KN averaging while preserving its core structural property of pseudo-additivity.