Skip to main content
SpringerLink
Log in

The Ballistic Heat Equation for a One-Dimensional Harmonic Crystal

  • Chapter
  • First Online:
Dynamical Processes in Generalized Continua and Structures

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 103))

Abstract

The analytical model of unsteady ballistic heat transfer in a one-dimensional harmonic crystal is analyzed. A nonlocal temperature is introduced as a generalization of the kinetic temperature. A closed equation determining unsteady thermal processes in terms of the nonlocal temperature is derived. For an instantaneous heat perturbation a time-reversible equation for the kinetic temperature is derived and solved. This equation can be referred as the ballistic heat conduction equation, it is somewhat similar to the hyperbolic heat conduction equation, but it has important differences. The resulting constitutive law for the heat flux in the considered system is obtained. This law significantly differs from Fourier’s law and it predicts a finite velocity of the heat front and independence of the heat flux on the crystal length. The analytical results are confirmed by computer simulations. Further developments of the presented approach are referred.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Thermal transport in low dimensions: from statistical physics to nanoscale heat transfer. In: Lepri S. (ed.) Lecture Notes in Physics, vol. 921, 418 p. Springer, Switzerland (2016)

    Google Scholar 

  2. Hoover, W.G., Hoover, C.G.: Simulation and control of chaotic nonequilibrium systems. In: Advanced Series in Nonlinear Dynamics, vol. 27, 324 p. World Scientific (2015)

    Google Scholar 

  3. Charlotte, M., Truskinovsky, L.: Lattice dynamics from a continuum viewpoint. J. Mech. Phys. Solids 60, 1508–1544 (2012)

    Article  Google Scholar 

  4. Bonetto, F., Lebowitz, J.L., Rey-Bellet, L.: Fourier’s law: a challenge to theorists. In: Fokas, A., et al. (eds.) Mathematical Physics 2000, pp. 128–150. Imperial College Press, London (2000)

    Chapter  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Dhar, A.: Heat transport in low-dimensional systems. Adv. Phys. 57, 457–537 (2008)

    Article  Google Scholar 

  7. Aoki, K., Kusnezov, D.: Bulk properties of anharmonic chains in strong thermal gradients: non-equilibrium \(\phi ^4\) theory. Phys. Lett. A 265, 250–256 (2000)

    Article  MathSciNet  Google Scholar 

  8. Gendelman, O.V., Savin, A.V.: Normal heat conductivity of the one-dimensional lattice with periodic potential. Phys. Rev. Lett. 84, 2381–2384 (2000)

    Article  Google Scholar 

  9. Giardina, C., Livi, R., Politi, A., Vassalli, M.: Finite thermal conductivity in 1D lattices. Phys. Rev. Lett. 84, 2144–2147 (2000)

    Article  Google Scholar 

  10. Gendelman, O.V., Savin, A.V.: Normal heat conductivity in chains capable of dissociation. Europhys. Lett. 106, 34004 (2014)

    Article  Google Scholar 

  11. Bonetto, F., Lebowitz, J.L., Lukkarinen, J.: Fourier’s law for a harmonic crystal with self-consistent stochastic reservoirs. J. Stat. Phys. 116, 783–813 (2004)

    Article  MathSciNet  Google Scholar 

  12. Le-Zakharov, A.A., Krivtsov, A.M.: Molecular dynamics investigation of heat conduction in crystals with defects. Dokl. Phys. 53, 261–264 (2008)

    Article  Google Scholar 

  13. Chang, C.W., Okawa, D., Garcia, H., Majumdar, A., Zettl, A.: Breakdown of fourier’s law in nanotube thermal conductors. Phys. Rev. Lett. 101, 075903 (2008)

    Article  Google Scholar 

  14. Xu, X., Pereira, L.F., Wang, Y., Wu, J., Zhang, K., Zhao, X., Bae, S., Bui, C.T., Xie, R., Thong, J.T., Hong, B.H., Loh, K.P., Donadio, D., Li, B., Ozyilmaz, B.: Length-dependent thermal conductivity in suspended single-layer graphene. Nat. Commun. 5, 3689 (2014)

    Article  Google Scholar 

  15. Hsiao, T.K., Huang, B.W., Chang, H.K., Liou, S.C., Chu, M.W., Lee, S.C., Chang, C.W.: Micron-scale ballistic thermal conduction and suppressed thermal conductivity in heterogeneously interfaced nanowires. Phys. Rev. B 91, 035406 (2015)

    Article  Google Scholar 

  16. Kannan, V., Dhar, A., Lebowitz, J.L.: Nonequilibrium stationary state of a harmonic crystal with alternating masses. Phys. Rev. E 85, 041118 (2012)

    Article  Google Scholar 

  17. Dhar, A., Dandekar, R.: Heat transport and current fluctuations in harmonic crystals. Physica A 418, 49–64 (2015)

    Article  MathSciNet  Google Scholar 

  18. Rieder, Z., Lebowitz, J.L., Lieb, E.: Properties of a harmonic crystal in a stationary nonequilibrium state. J. Math. Phys. 8, 1073–1078 (1967)

    Article  Google Scholar 

  19. Rubin, R.J.: Momentum autocorrelation functions and energy transport in harmonic crystals containing isotopic defects. Phys. Rev. 131, 964–989 (1963)

    Article  Google Scholar 

  20. Gendelman, O.V., Shvartsman, R., Madar, B., Savin, A.V.: Nonstationary heat conduction in one-dimensional models with substrate potential. Phys. Rev. E 85, 011105 (2012)

    Article  Google Scholar 

  21. Gusev, A.A., Lurie, S.A.: Wave-relaxation duality of heat propagation in Fermi-Pasta-Ulam chains. Mod. Phys. Lett. B 26, 1250145 (2012)

    Article  Google Scholar 

  22. Guzev, M.A.: The Fourier law for a one-dimensional crystal. Far East. Math. J. (1), 34–39 (2018)

    Google Scholar 

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

    Article  Google Scholar 

  24. Chandrasekharaiah, D.S.: Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev 39, 355–376 (1986)

    Article  Google Scholar 

  25. Tzou, D.Y.: Macro- to Microscale Heat Transfer: The Lagging Behavior, 566 p. Wiley (2015)

    Google Scholar 

  26. Poletkin, K.V., Gurzadyan, G.G., Shang, J., Kulish, V.: Ultrafast heat transfer on nanoscale in thin gold films. Appl. Phys. B 107, 137–143 (2012)

    Article  Google Scholar 

  27. Indeitsev, D.A., Osipova, E.V.: A two-temperature model of optical excitation of acoustic waves in conductors. Dokl. Phys. 62(3), 136–140 (2017)

    Article  Google Scholar 

  28. Lepri, S., Mejia-Monasterio, C., Politi, A.: A stochastic model of anomalous heat transport: analytical solution of the steady state. J. Phys. A Math. Theor. 42, 025001, 15 p. (2009)

    Google Scholar 

  29. Lepri, S., Mejia-Monasterio, C., Politi, A.: Nonequilibrium dynamics of a stochastic model of anomalous heat transport. J. Phys. A: Math. Theor. 43, 065002, 22 p. (2010)

    Google Scholar 

  30. Delfini, L., Lepri, S., Livi, R., Mejia-Monasterio, C., Politi, A.: Nonequilibrium dynamics of a stochastic model of anomalous heat transport: numerical analysis. J. Phys. A Math. Theor. 43, 145001, 16 p. (2009)

    Google Scholar 

  31. Krivtsov, A.M.: Dynamics of energy characteristics in one-dimensional crystal. In: Procedings of XXXIV Summer School “Advanced Problems in Mechanics”, St.-Petersburg, Russia, 2007, pp. 261–273. ISBN 5-98883-009-9

    Google Scholar 

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

    Article  Google Scholar 

  33. Krivtsov, A.M.: On unsteady heat conduction in a harmonic crystal (2015). ArXiv:1509.02506

  34. Born, M., Huang, K.: Dynamical Theory of Crystal Lattices, 432 p. Clarendon Press, Oxford (1954)

    Google Scholar 

  35. Gavrilov, S.N., Krivtsov, A.M., Tsvetkov, D.V.: Heat transfer in a one-dimensional harmonic crystal in a viscous environment subjected to an external heat supply. In: Continuum Mechanics and Thermodynamics (2018)

    Google Scholar 

  36. Manvi, R., Duvall, G.E., Lowell, S.C.: Finite amplitude longitudinal waves in lattices. Int. J. Mech. Sci. 11, 1 (1969)

    Article  Google Scholar 

  37. Holian, B.L., Straub, G.K.: Molecular dynamics of shock waves in one-dimensional chains. Phys. Rev. B 18, 1593 (1978)

    Article  Google Scholar 

  38. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1046 p. U. S. Government Printing Office (1972)

    Google Scholar 

  39. Polyanin, A.D., Nazaikinskii, V.E.: Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd edn, 1632 p. CRC Press, Boca Raton-London (2016)

    Google Scholar 

  40. Babenkov, M.B., Krivtsov, A.M., Tsvetkov, D.V.: Unsteady heat conduction processes in a harmonic crystal with a substrate potential. In: Proceedings of XXIV ICTAM, 21–26 August 2016, Montreal, Canada, 2440–2441 (2016)

    Google Scholar 

  41. Shoby, K., Yoshida, T., Mori, H.: Dynamic properties of one-dimensional harmonic liquids. II, Prog. Theoret. Phys. 66, 1160–1168 (1981)

    Google Scholar 

  42. Kuzkin, V.A., Krivtsov, A.M.: Fast and slow thermal processes in harmonic scalar lattices. J. Phys. Condens. Matter 29, 505401, 14 p. (2017)

    Google Scholar 

  43. Kuzkin, V.A.: Fast and slow thermal processes in harmonic crystals with polyatomic lattice. Arxiv preprint(2018). ArXiv:1808.00504

  44. Hoover, W.G.: Molecular dynamics. In: Lecture Notes in Physics, vol. 258, 324 p. Springer-Verlag (1986)

    Google Scholar 

  45. Holian, B.L., Hoover, W.G., Posch, H.A.: Resolution of Loschmidt’s paradox: the origin of irreversible behavior in reversible atomistic dynamics. Phys. Rev. Lett. 59, 10–13 (1987)

    Article  MathSciNet  Google Scholar 

  46. Gendelman, O.V., Savin, A.V.: Nonstationary heat conduction in one-dimensional chains with conserved momentum. Phys. Rev. E 81, 020103 (2010)

    Article  Google Scholar 

  47. Dhar, A.: Heat conduction in the disordered harmonic chain revisited. Phys. Rev. Lett. 86, 5882–5885 (2001)

    Article  Google Scholar 

  48. Hoover, W.G., Hoover, C.G.: Hamiltonian thermostats fail to promote heat flow. Commun. Nonlinear Sci. Numer. Simulat. 18, 3365–3372 (2013)

    Article  MathSciNet  Google Scholar 

  49. Kachman, T.: Master-degree thesis under supervision of O. V. Gendelman, 187 p., Technion, Izrael (2011)

    Google Scholar 

  50. Tzou, D.Y.: The generalized lagging responce in small scale and high rate heating. Int. J. Mass Transf. 38, 3231–3240 (1995)

    Article  Google Scholar 

  51. Rubin, R.J., Greer, W.L.: Abnormal lattice thermal conductivity of a one-dimensional, harmonic, isotopically disordered crystal. J. Math. Phys. 12, 1686–1701 (1971)

    Article  Google Scholar 

  52. Babenkov, M.B., Ivanova, E.A.: Analysis of the wave propagation processes in heat transfer problems of the hyperbolic type. Contin. Mech. Thermodyn. 26, 483–502 (2014)

    Article  MathSciNet  Google Scholar 

  53. 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)

    Article  Google Scholar 

  54. Krivtsov, A.M., Kuzkin, V.A.: Discrete and continuum thermomechanics. In: Encyclopedia of Continuum Mechanics, 16 p (2018)

    Google Scholar 

  55. Goldstein, R.V., Morozov, N.F.: Mechanics of deformation and fracture of nanomaterials and nanotechnology. Phys. Mesomech. 10, 235–246 (2007)

    Article  Google Scholar 

  56. Shtukin, L.V., Berinskii, I.E., Indeitsev, D.A., Morozov, N.F., Skubov, D.Y.: Electromechanical models of nanoresonators. Phys. Mesomech. 19(3), 248–254 (2016)

    Article  Google Scholar 

Download references

Acknowledgements

The author is grateful to O. V. Gendelman, W. G. Hoover, D. A. Indeitsev, M. L. Kachanov, V. A. Kuzkin, S. A. Lurie, N. F. Morozov, and V. F. Zhuravlev for helpful and stimulating discussions; to M. B. Babenkov and D. V. Tsvetkov for additional analysis and computations. This work is supported by the Russian Science Foundation (Grant No. 18-11-00201).

Author information

Authors and Affiliations

  1. Department of Theoretical Mechanics, Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia

    Anton Krivtsov

  2. Laboratory of Discrete Models in Mechanics, Institute for Problems in Mechanical Engineering RAS, St. Petersburg, Russia

    Anton Krivtsov

Authors
  1. Anton Krivtsov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anton Krivtsov .

Editor information

Editors and Affiliations

  1. Lehrstuhl Technische Mechanik, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Sachsen-Anhalt, Germany

    Holm Altenbach

  2. Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia

    Alexander Belyaev

  3. Faculty of Civil and Environmental Engineering, Gdańsk University of Technology, Gdańsk, Poland

    Victor A. Eremeyev

  4. Department of Theoretical Mechanics, Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia

    Anton Krivtsov

  5. Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia

    Alexey V. Porubov

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Krivtsov, A. (2019). The Ballistic Heat Equation for a One-Dimensional Harmonic Crystal. In: Altenbach, H., Belyaev, A., Eremeyev, V., Krivtsov, A., Porubov, A. (eds) Dynamical Processes in Generalized Continua and Structures. Advanced Structured Materials, vol 103. Springer, Cham. https://doi.org/10.1007/978-3-030-11665-1_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-11665-1_19

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-11664-4

  • Online ISBN: 978-3-030-11665-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation