Entropy Maximization with Linear Constraints: The Uniqueness of the Shannon Entropy

TL;DR

This paper proves that maximizing entropy with linear constraints uniquely results in Shannon entropy, cautioning against extending this method to generalized entropies like Tsallis or Rényi, and highlighting the limitations of current approaches.

Contribution

It establishes the uniqueness of Shannon entropy in the entropy maximization framework with linear constraints and discusses implications for generalized entropies.

Findings

Maximization with linear constraints leads uniquely to Shannon entropy.

Using this method with generalized entropies causes contradictions or reduces to Shannon entropy.

The scope of different averaging schemes in entropy maximization is beyond this work.

Abstract

Within a framework of utmost generality, we show that the entropy maximization procedure with linear constraints uniquely leads to the Shannon-Boltzmann-Gibbs entropy. Therefore, the use of this procedure with linear constraints should not be extended to the generalized entropies introduced recently. In passing, it is remarked how the forceful use of the entropy maximization for the Tsallis and R\'enyi entropies implies either the Shannon limit of these entropies or self-referential contradictions. Finally, we note that the utilization of the entropy maximization procedure with different averaging schemes is beyond the scope of this work.