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

Aleksei A. Sokolov1,2,*, Anton M. Krivtsov2,3,†, Wolfgang H. Müller1,‡, and Elena N. Vilchevskaya2,3,§

  • 1Continuum Mechanics and Materials Theory, Technische Universität Berlin, Einsteinufer 5, 10587 Berlin, Germany
  • 2Theoretical and Applied Mechanics, Peter the Great Saint Petersburg Polytechnic University, Politekhnicheskaja 29, 195251 Saint Petersburg, Russia
  • 3Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, Bol'shoy pr. 61, V.O., 199178 Saint Petersburg, Russia

  • *sokolovalexey1@gmail.com
  • †akrivtsov@bk.ru
  • ‡wolfgang.h.mueller@tu-berlin.de
  • §vilchevska@gmail.com
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Phys. Rev. E 99, 042107 – Published 4 April, 2019

DOI: https://doi.org/10.1103/PhysRevE.99.042107

Abstract

This work presents a thermodynamic analysis of the ballistic heat equation from two viewpoints: classical irreversible thermodynamics (CIT) and extended irreversible thermodynamics (EIT). A formula for calculating the entropy within the framework of EIT for the ballistic heat equation is derived. The 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 that the entropy production can be negative. The results obtained for EIT show that the entropy is a monotonic function and that the entropy production is nonnegative. A comparison between the entropy behaviors predicted for the ballistic, for the ordinary Fourier-based, and for the hyperbolic heat equation is made. A crucial difference of the asymptotic behavior of the entropy for the ballistic heat equation is shown. It is argued that mathematical time reversibility of the 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.

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Phys. Rev. E 99, 042107– Published 4 April, 2019
  • Received 24 August 2018
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