Change of entropy for the one-dimensional ballistic heat equation: Sinusoidal initial perturbation

TL;DR

This paper analyzes the entropy evolution in the ballistic heat equation using both classical and extended thermodynamics, revealing differences in entropy behavior and irreversibility compared to classical heat conduction models.

Contribution

It derives a new entropy formula within Extended Irreversible Thermodynamics for the ballistic heat equation and compares its behavior with classical models.

Findings

Entropy is non-monotonic in CIT, with possible negative production.

Entropy is monotonic and nonnegative in EIT.

Ballistic heat conduction shows distinct asymptotic entropy behavior from classical models.

Abstract

This work presents the thermodynamical analysis of the ballistic heat equation from the viewpoint of two approaches: Classical Irreversible Thermodynamics (CIT) and Extended Irreversible Thermodynamics (EIT). A formula for calculation of the entropy within the framework of EIT for the ballistic heat equation is derived in this work. Entropy is calculated for a sinusoidal initial temperature perturbation by using both approaches. The results obtained from CIT show that the entropy is a non-monotonic function and the entropy production can be negative. The results obtained with EIT show that the entropy is a monotonic function and the entropy production is nonnegative. An approximative formula for the asymptotic behavior of the entropy for the ballistic heat equation is obtained. A comparison with the ordinary Fourier-based heat equation and hyperbolic heat equation is made. A crucial difference in asymptotic behaviour of the entropy for ballistic and classical heat conduction equation is shown. It is shown that mathematical time reversibility of partial differential ballistic heat equation is not consistent with its physical irreversibility. The processes described by the ballistic heat equation are irreversible because of the entropy increase.