[Paper Review] Scaling property and the generalized entropy uniquely determined by a fundamental nonlinear differential equation

Hiroki Suyari, Tatsuaki Wada|arXiv (Cornell University)|Jul 31, 2006

Statistical Mechanics and Entropy18 references3 citations

TL;DR

This paper establishes that the generalized entropy uniquely identified as Tsallis entropy arises from a fundamental nonlinear differential equation $ dy/dx = y^q $, whose solutions are $ q $-exponential functions. The scaling invariance of this equation's solutions leads to a consistent algebraic structure—via the $ q $-product—that uniquely supports Tsallis entropy through two independent derivations: from $ q $-multinomial coefficients and $ q $-Stirling's formula, and from generalized Shannon additivity.

ABSTRACT

We derive a scaling property from a fundamental nonlinear differential equation whose solution is the so-called q-exponential function. A scaling property has been believed to be given by a power function only, but actually more general expression for the scaling property is found to be a solution of the above fundamental nonlinear differential equation. In fact, any power function is obtained by restricting the domain of the q-exponential function appropriately. As similarly as the correspondence between the exponential function and Shannon entropy, an appropriate generalization of Shannon entropy is expected for the scaling property. Although the q-exponential function is often appeared in the optimal distributions of some one-parameter generalized entropies such as Renyi entropy, only Tsallis entropy is uniquely derived from the algebra of the q-exponential function, whose uniqueness is shown in the two ways in this paper.

Motivation & Objective

• To establish the mathematical foundation of Tsallis entropy as the unique information measure corresponding to the $ q $-exponential function.
• To demonstrate that the scaling property of the fundamental nonlinear differential equation $ dy/dx = y^q $ leads to a generalized algebraic structure via the $ q $-product.
• To show that Tsallis entropy is uniquely derived through two independent mathematical pathways: $ q $-multinomial coefficients and generalized Shannon additivity.
• To prove the self-consistency of the entire formalism of Tsallis statistics based on the $ q $-product and the $ q $-exponential function.

Proposed method

• Derive the $ q $-exponential function as the solution to the nonlinear differential equation $ dy/dx = y^q $, with $ q \in \mathbb{R} $.
• Introduce the $ q $-logarithm $ \ln_q x = \frac{x^{1-q} - 1}{1 - q} $ as the inverse of the $ q $-exponential, recovering the standard logarithm as $ q \to 1 $.
• Establish a scale-invariant transformation $ y' = y / \exp_q(C) $, $ x' = x / \exp_q(C)^{1-q} $, showing the solution remains $ y' = \exp_q(x') $ for any $ C $ satisfying $ 1 + (1 - q)C > 0 $.
• Define the $ q $-product and $ q $-multinomial coefficient based on the $ q $-exponential, enabling a generalized algebraic framework.
• Use $ q $-Stirling’s formula and the $ q $-multinomial coefficient to derive the Tsallis entropy form $ S_q = \frac{1}{1-q} \left(1 - \sum p_i^q \right) $.
• Prove uniqueness of Tsallis entropy by showing that generalized Shannon additivity holds only when $ \phi(q) $ satisfies $ \phi'(q) = 1/k $, which corresponds to the original Tsallis entropy definition.

Experimental results

Research questions

• RQ1 Can the scaling property of the $ q $-exponential function be derived from a fundamental nonlinear differential equation, and does it extend beyond power-law scaling?
• RQ2 Is Tsallis entropy uniquely determined by the algebraic structure of the $ q $-exponential function and its associated $ q $-product?
• RQ3 Does the generalized Shannon additivity axiom uniquely select Tsallis entropy among one-parameter generalized entropies?
• RQ4 How does the $ q $-exponential function's behavior on restricted domains relate to power-law functions in physical systems?
• RQ5 What is the role of the $ q $-product in ensuring mathematical consistency across key results in Tsallis statistics?

Key findings

• The nonlinear differential equation $ dy/dx = y^q $ generates a scale-invariant solution via the $ q $-exponential function, with scaling transformations defined for any $ C $ satisfying $ 1 + (1 - q)C > 0 $.
• The $ q $-exponential function reduces to a power function $ x^{1/(1-q)} $ on the restricted domain $ \{ x \mid (1 - q)x \gg 1 \} $, explaining the ubiquity of power laws in complex systems.
• Tsallis entropy is uniquely derived from the $ q $-multinomial coefficient and $ q $-Stirling’s formula, confirming its foundational role in nonextensive statistical mechanics.
• Generalized Shannon additivity holds only when $ \phi(q) $ satisfies $ \phi'(q) = 1/k $, which uniquely identifies the Tsallis entropy form.
• The $ q $-product algebra is mathematically consistent across all core results in Tsallis statistics, including the $ q $-canonical distribution and central limit theorem.
• The entire formalism of Tsallis statistics is self-consistent and originates from the fundamental nonlinear differential equation $ dy/dx = y^q $, establishing its foundational role.