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Entropy and the Second Law of Thermodynamics

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The second law of thermodynamics can be stated quantitatively using the concept of entropy. Entropy is the measure of disorder of the system.
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Entropy Interpretation of Hadamard Type Fractional Operators: Fractional Cumulative Entropy.

Vasily E Tarasov^1,2

¹Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia.

Entropy (Basel, Switzerland)

|December 23, 2022

Summary

New Hadamard-type fractional operators generalize standard entropy, fractional entropies, and cumulative entropies. This study introduces a novel family of fractional cumulative entropies derived from these operators.

Keywords:

Hadamard-type fractional pperator cummulative entropy entropy fractional calculus fractional entropy fractional integrals

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Area of Science:

Fractional Calculus
Information Theory
Mathematical Analysis

Background:

Standard entropy measures provide foundational insights into information content.
Existing fractional entropies offer advanced analytical capabilities.
Cumulative entropies capture information over intervals, but lack fractional generalization.

Purpose of the Study:

To propose novel interpretations for Hadamard-type fractional integral and differential operators.
To generalize standard, fractional, and cumulative entropies using these operators.
To introduce a new family of fractional cumulative entropies.

Main Methods:

Development and interpretation of Hadamard-type fractional integral operators.
Application of these operators to generalize existing entropy concepts.
Formulation of a new class of fractional cumulative entropies.

Main Results:

Hadamard-type fractional integrals are shown to generalize standard, fractional, and cumulative entropies.
A novel family of fractional cumulative entropies is successfully derived.
The proposed operators provide a unified framework for entropy generalization.

Conclusions:

The proposed Hadamard-type fractional operators offer a powerful tool for extending entropy theory.
This work bridges fractional calculus and information theory with new entropy formulations.
The novel fractional cumulative entropies have potential applications in various analytical fields.