Skip to main content

The Ballistic Heat Equation for a One-Dimensional Harmonic Crystal

Part of the Advanced Structured Materials book series (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, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-11665-1_19
  • Chapter length: 14 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   139.00
Price excludes VAT (USA)
  • ISBN: 978-3-030-11665-1
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Hardcover Book
USD   179.99
Price excludes VAT (USA)
Fig. 1
Fig. 2
Fig. 3

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)

    CrossRef  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)

    CrossRef  Google Scholar 

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

    MathSciNet  CrossRef  Google Scholar 

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

    CrossRef  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)

    MathSciNet  CrossRef  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)

    CrossRef  Google Scholar 

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

    CrossRef  Google Scholar 

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

    CrossRef  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)

    MathSciNet  CrossRef  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)

    CrossRef  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)

    CrossRef  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)

    CrossRef  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)

    CrossRef  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)

    CrossRef  Google Scholar 

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

    MathSciNet  CrossRef  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)

    CrossRef  Google Scholar 

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

    CrossRef  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)

    CrossRef  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)

    CrossRef  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)

    CrossRef  Google Scholar 

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

    CrossRef  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)

    CrossRef  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)

    CrossRef  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)

    CrossRef  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)

    CrossRef  Google Scholar 

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

    CrossRef  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)

    MathSciNet  CrossRef  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)

    CrossRef  Google Scholar 

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

    CrossRef  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)

    MathSciNet  CrossRef  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)

    CrossRef  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)

    CrossRef  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)

    MathSciNet  CrossRef  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)

    CrossRef  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)

    CrossRef  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)

    CrossRef  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

Authors

Corresponding author

Correspondence to Anton Krivtsov .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

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)