Conformable Derivative Approach to Granular Gases

TL;DR

This paper introduces a conformable derivative approach to model granular gases, deriving new equations and functions that align with classical kinetic theory and provide insights into their thermodynamic properties.

Contribution

It develops a novel mathematical framework using deformed derivatives to analyze granular gases, deriving generalized equations and thermodynamic quantities.

Findings

Stretched exponential emerges naturally from maximum entropy constraints.

Generalized Langevin equations are obtained for deformed derivatives.

Partition function and thermodynamic properties are consistent with classical theory.

Abstract

Proper modeling of complex systems requires innovative mathematical tools. In this sense, we sought to use deformed or fractal derivatives for studying the dynamics of systems, particularly those, such as granular gases, in which the description of the dynamics can be done by using the stretched exponential probability densities. In this contribution we draw up three results of this application of mathematical tools. The first result shows that when we use constraints with finite momentum and the principle of maximum entropy, the Kohlrausch--Williams--Watts function, known as stretched exponential, emerges naturally and in a simpler way, when compared to results in the literature. Next, we obtain generalized expressions for the Langevin equation, as well as its solutions for three different deformed derivatives, including those connected with nonaddictive statistical mechanics. The Haff's-like law for granular gases are obtained. Next, we calculate the partition function $Z$ for a granular gas system by building up the probability density in terms of the stretched exponential function. From this partition function, we determine the internal energy of the system as well as the specific heat, both dependent on temperature. The consistency with classical approach of kinetic theory for ideal gases was verified.