Oscillating Entropy

TL;DR

This paper introduces and analyzes an oscillating entropy measure based on a log-periodic function, connecting it with known entropies and exploring its properties in systems with increasing mass.

Contribution

It presents a novel oscillating entropy form derived from the Sharma-Taneja measure and explores its properties, limiting cases, and physical interpretations.

Findings

The entropy exhibits log-periodic oscillations and non-additivity.

Connections are established with Tsallis, Rényi, Boltzmann-Gibbs, and Shannon entropies.

Oscillating entropy can occur in systems with linearly increasing mass.

Abstract

The log-periodic equation for the entropy $S = - (k/a) \sum_{i=1}^{N} p_{i} \sin(a \ln p_{i})$, based on the forgotten Sharma-Taneja entropy measure, is studied for the first time with $N$ the total number of system states and $p_{i}$ the associated probabilities. It is argued that an oscillatory regime for $S$ could in principle be understood in terms of a linear time-dependent behavior for the associated probabilities in analogy with a spring system of frequency $w$ gaining momentum from the surroundings. The physical meaning for the production of entropy given by the parameter $a$ relates the angle $\omega t$. We discuss its properties, and make a connection with the non-extensive Tsallis, R\'{e}nyi, Boltzmann-Gibbs and Shannon entropies as special limiting cases for systems with constant mass. Our non-trivial form of entropy displays peculiar concavity and addition properties. Log-periodicity in $S$, concavity lost, and non-additivity are manifested by increasing the value of the coefficient $a$, which sets the variations with respect to the behavior of the monotonic Gibbs entropy function. We have considered systems with linearly increasing mass and have explained how "oscillating entropy" in such systems may appear.