Scaling property and the generalized entropy uniquely determined by a fundamental nonlinear differential equation

TL;DR

This paper derives a fundamental nonlinear differential equation that uniquely determines the q-exponential function, revealing a generalized scaling property and entropy that extend beyond traditional power functions and Shannon entropy.

Contribution

It demonstrates that the q-exponential function and its associated entropy are uniquely derived from a fundamental nonlinear differential equation, expanding the understanding of generalized scaling and entropy.

Findings

The scaling property is more general than a power function, derived from the q-exponential solution.

Tsallis entropy is uniquely determined by the algebra of the q-exponential function.

The paper provides two proofs of the uniqueness of Tsallis entropy.

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.