Skip to main content

One-Dimensional Heat Conduction and Entropy Production

Part of the Advanced Structured Materials book series (STRUCTMAT,volume 87)


This review paper analyzes the entropy concept and the second law of thermodynamics in the context of one-dimensional media. For simplicity, only thermal processes are taken into account and mechanical motions are neglected. The relation between entropy and temperature and the constraints on the direction of the heat flux are discussed. A comparison with the approach of P. A. Zhilin and the approach based on statistical mechanics is presented. The obtained conclusions are applied to three models: classical, hyperbolic and ballistic heat conduction. It is shown that the concept according to which heat flows from hot to cold is consistent only with the classical model. The peculiarities of the entropy definition and the second law of thermodynamics formulation for non-classical systems are discussed.

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-73694-5_12
  • Chapter length: 17 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
USD   139.00
Price excludes VAT (USA)
  • ISBN: 978-3-319-73694-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   179.00
Price excludes VAT (USA)
Hardcover Book
USD   179.00
Price excludes VAT (USA)


  1. 1.

    Literally it is said in [8]: “the rate of change of internal entropy...”. However, the word “internal” is used only as an antithesis to entropy coming from outside. Therefore, this notation of “internal entropy” is equivalent to the notation of “entropy” used in the current work. Moreover, one cannot divide the entropy into internal and external. Entropy supply is different—it can be associated with a transfer from outside or with an internal processes. After entropy has entered the system, it “mixes,” and it is impossible to divide it into internal and external.

  2. 2.

    Due to external heat supply (12) any value for quantity \(\dot{S}\) can be realized.

  3. 3.

    Since there are no mechanical motions, the volume remains unchanged.

  4. 4.

    This inequality is sometimes called the Planck inequality or the Clausius-Planck inequality, however, in monograph [11] these terms are not used.

  5. 5.

    In monograph [11] the term “Fourier’s inequality” is not mentioned, instead the term “zeroth law of thermodynamics” is used.

  6. 6.

    \(\omega _e=\sqrt{C/m}\): the frequency of a particle with the mass m on a spring with the stiffness C, which is \(C=\varPi ''(a)\), where \(\varPi \) is the potential of the atomic interaction, a is the lattice step.


  1. Cannon, J.R.: The One-Dimensional Heat Equation. Cambridge University Press (1984)

    Google Scholar 

  2. Lepri, S., Livi, R., Politi, A.: Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377, 1–80 (2003)

    MathSciNet  CrossRef  Google Scholar 

  3. Cattaneo, C.: Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantane. Comptes Rendus 247, 431–433 (1958)

    MATH  Google Scholar 

  4. Vernotte, P.: Les paradoxes de la theorie continue de l’équation de la chaleur. Comptes Rendus 246, 3154–3155 (1958)

    MathSciNet  MATH  Google Scholar 

  5. Krivtsov, A.M.: Heat transfer in infinite harmonic one-dimensional crystals. Dokl. Phys. 60(9), 407–411 (2015)

    CrossRef  Google Scholar 

  6. Indeitsev, D., Osipova, E.: Two-temperature model of optical excitation of acoustic waves in conductors. Dokl. Phys. 62(6), 538–541 (2017)

    Google Scholar 

  7. Krivtsov, A.M.: Deformation and fracture of solids with microstructure. Moscow, Fizmatlit. 304 p. (2007) (in Russian)

    Google Scholar 

  8. Palmov, V.: Vibrations of Elasto-Plastic Bodies. Springer, Berlin, Heidelberg (1998)

    CrossRef  MATH  Google Scholar 

  9. Jou, D., Lebon, G., Casas-Vazquez, J.: Extended Irreversible Thermodynamics. Springer, Netherlands (2010)

    Google Scholar 

  10. Müller, I., Ruggeri, T.: Extended Thermodynamics. Springer, New York (1993)

    CrossRef  MATH  Google Scholar 

  11. Zhilin, P.A.: Rational mechanics of continuous media, 584 pp. Publishing of Polytechnic University (2012)

    Google Scholar 

  12. Babenkov, M.B., Krivtsov, A.M., Tsvetkov, D.V.: Energy oscillations in a one-dimensional harmonic crystal on an elastic substrate. Phys. Mesomech. 19(3), 282–290 (2016)

    CrossRef  Google Scholar 

  13. Berinskii, I.: Elastic networks to model auxetic properties of cellular materials. Int. J. Mech. Sci. 115, 481 (2016)

    CrossRef  Google Scholar 

  14. Krivtsov, A.M.: Energy oscillations in a one-dimensional crystal. Dokl. Phys. 59(9), 427–430 (2014)

    MathSciNet  CrossRef  Google Scholar 

  15. Kuzkin, V.A., Krivtsov, A.M.: Fast and slow thermal processes in harmonic scalar lattices. J. Phys.: Condens. Matter 29(50), 505401 (2017)

    Google Scholar 

  16. Hoover, W.G., Hoover, C.G.: Simulation and Control of Chaotic Nonequilibrium Systems. Advanced Series in Nonlinear Dynamics, vol. 27, 324 p. World Scientific (2015)

    Google Scholar 

  17. Sokolov, A.A., Krivtsov, A.M., Müller, W.H.: Localized heat perturbation in harmonic 1D crystals: solutions for an equation of anomalous heat conduction. Phys. Mesomech. 20(3), 305–310 (2017)

    CrossRef  Google Scholar 

  18. Krivtsov, A.M.: A microcanonical distribution for a one-dimensional chain. The document from 12/12/2013–08/01/2014. 32 pp. (unpublished)

    Google Scholar 

Download references


The authors of this work would like to thank E. A. Ivanova and E. N. Vilchevskaya for the useful discussions.

Author information

Authors and Affiliations


Corresponding author

Correspondence to A. A. Sokolov .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Verify currency and authenticity via CrossMark

Cite this chapter

Krivtsov, A.M., Sokolov, A.A., Müller, W.H., Freidin, A.B. (2018). One-Dimensional Heat Conduction and Entropy Production. In: dell'Isola, F., Eremeyev, V., Porubov, A. (eds) Advances in Mechanics of Microstructured Media and Structures. Advanced Structured Materials, vol 87. Springer, Cham.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-73693-8

  • Online ISBN: 978-3-319-73694-5

  • eBook Packages: EngineeringEngineering (R0)