Physical Mesomechanics

, Volume 19, Issue 3, pp 282–290 | Cite as

Energy oscillations in a one-dimensional harmonic crystal on an elastic substrate

  • M. B. Babenkov
  • A. M. KrivtsovEmail author
  • D. V. Tsvetkov


A one-dimensional harmonic crystal on an elastic substrate is considered as a stochastic system into which randomness is introduced through initial conditions. The use of the particle velocity and displacement covariances reduces the stochastic problem to a closed deterministic problem for statistical characteristics of particle pairs. An equation of rapid motion that describes oscillations of potential and kinetic energy components of the system has been derived and solved. The obtained solutions are used to determine the character and to estimate the time of decay of the transient process that brings the system to thermodynamic equilibrium.


one-dimensional crystal thermal conductivity covariances energy oscillations elastic substrate 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Goldstein, R.V. and Morozov, N.F., Mechanics of Deformation and Fracture of Nanomaterials and Nanotechnologies, Phys. Mesomech., 2007, vol. 10, no. 5–6, pp. 235–246.CrossRefGoogle Scholar
  2. 2.
    Goldstein, R.V. and Morozov, N.F., Fundamental Problems of Solid Mechanics in High Technologies, Phys. Mesomech., 2012, vol. 15, no. 3–4, pp. 224–231.CrossRefGoogle Scholar
  3. 3.
    Krivtsov, A.M. and Morozov, N.F., On Mechanical Characteristics of Nanocrystals, Phys. Solid State, 2002, vol. 44, no. 12, pp. 2260–2262.ADSCrossRefGoogle Scholar
  4. 4.
    Hoover, W.G. and Hoover, C.G., Simulation and Control of Chaotic Nonequilibrium Systems: Advanced Series in Nonlinear Dynamics: V 27, Singapore: World Scientific, 2015.CrossRefzbMATHGoogle Scholar
  5. 5.
    Porubov, A.V. and Berinskii, I.E., Nonlinear Plane Waves in Materials Having Hexagonal Internal Structure, Int. J. Nonlinear Mech., 2014, vol. 67, pp. 27–33.ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Bonetto, F., Lebowitz, J.L., and Rey-Bellet, L., Fourier’s Law: A Challenge to Theorists, Mathematical Physics 2000, Fokas, A., et al., Eds., London: Imperial College Press, 2000, pp. 128–150.Google Scholar
  7. 7.
    Eremeev, V.A., Ivanova, E.A., and Morozov, N.F., Some Problems of Nanomechanics, Phys. Mesomech., 2014, vol. 17, no. 1, pp. 23–29.CrossRefGoogle Scholar
  8. 8.
    Eremeyev, V.A., Ivanova, E.A., and Indeitsev, D.A., Wave Processes in Nanostructures Formed by Nanotube Arrays or Nanosize Crystals, J. Appl. Mech. Tech. Phys., 2010, vol. 51, no. 4, pp. 569–578.ADSCrossRefzbMATHGoogle Scholar
  9. 9.
    Kuzkin, V.A., Comment on “Negative Thermal Expansion in Single-Component Systems with Isotropic Interactions”, J. Phys. Chem., 2014, vol. 118, no. 41, pp. 9793–9794.CrossRefGoogle Scholar
  10. 10.
    Kuzkin, V.A. and Krivtsov, A.M., Nonlinear Positive/Negative Thermal Expansion and Equations of State of a Chain with Longitudinal and Transverse Vibrations, Phys. Solid State. B, 2015, vol. 252, no. 7, pp. 1664–1670.ADSCrossRefGoogle Scholar
  11. 11.
    Goldstein, R.V., Gorodtsov, V.A., and Lisovenko, D.S., Mesomechanics of Multiwall Carbon Nanotubes and Nanowhiskers, Phys. Mesomech., 2009, vol. 12, no. 1, pp. 38–53.CrossRefGoogle Scholar
  12. 12.
    Podolskaya, E.A., Panchenko, A.Y., Freidin, A.B., and Krivtsov, A.M., Loss of Ellipticity and Structural Transformations in Planar Simple Crystal Lattices, Acta Mech., 2015, pp. 1–17.Google Scholar
  13. 13.
    Lepri, S., Livi, R., and Politi, A., Thermal Conduction in Classical Low-Dimensional Lattices, Phys. Rep., 2003, vol. 377, pp. 1–80.ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Dhar, A., Heat Transport in Low-Dimensional Systems, Adv. Phys., 2008, vol. 57, pp. 457–537.ADSCrossRefGoogle Scholar
  15. 15.
    Aoki, K. and Kusnezov, D., Bulk Properties of Anharmonic Chains in Strong Thermal Gradients: Non-Equilibrium Theory, Phys. Lett. A, 2000, vol. 265, pp. 250–256.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gendelman, O.V and Savin, A.V., Normal Heat Conductivity of the One-Dimensional Lattice with Periodic Potential, Phys. Rev. Lett., 2000, vol. 84, pp. 2381–2384.ADSCrossRefGoogle Scholar
  17. 17.
    Giardina, C., Livi, R., Politi, A., and Vassalli, M., Finite Thermal Conductivity in 1D Lattices, Phys. Rev. Lett., 2000, vol. 84, pp. 2144–2147.ADSCrossRefGoogle Scholar
  18. 18.
    Gendelman, O.V. and Savin, A.V., Normal Heat Conductivity in Chains Capable of Dissociation, Europhys. Lett., 2014, vol. 106, p. 34004.ADSCrossRefGoogle Scholar
  19. 19.
    Bonetto, F., Lebowitz, J.L., and Lukkarinen, J., Fourier’s Law for a Harmonic Crystal with Self-Consistent Stochastic Reservoirs, J. Stat. Phys., 2004, vol. 116, pp. 783–813.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Le-Zakharov, A.A. and Krivtsov, A.M., Molecular Dynamics Investigation of Heat Conduction in Crystals with Defects, Doklady Physics, 2008, vol. 53, no. 5, pp. 261–264.ADSCrossRefzbMATHGoogle Scholar
  21. 21.
    Chang, C.W., Okawa, D., Garcia, H., Majumdar, A., and Zettl, A., Breakdown of Fourier’s Law in Nanotube Thermal Conductors, Phys. Rev. Lett., 2008, vol. 101, p. 075903.ADSCrossRefGoogle Scholar
  22. 22.
    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., and Ozyilmaz, B., Length- Dependent Thermal Conductivity in Suspended SingleLayer Graphene, Nat. Commun., 2014, vol. 5, p. 36–89.Google Scholar
  23. 23.
    Hsiao, T.K., Huang, B.W., Chang, H.K., Liou, S.C., Chu, M.W., Lee, S.C., and Chang, C.W., Micron-Scale Ballistic Thermal Conduction and Suppressed Thermal Conductivity in Heterogeneously Interfaced Nanowires, Phys. Rev. B, 2015, vol. 91, p. 035406.ADSCrossRefGoogle Scholar
  24. 24.
    Lepri, S., Mejia-Monasterio, C., and Politi, A., Nonequilibrium Dynamics of a Stochastic Model of Anomalous Heat Transport, J. Phys. A: Math. Theor., 2010, vol. 43, p. 065002 (22 p.).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kannan, V., Dhar, A., and Lebowitz, J.L., Nonequilibrium Stationary State of a Harmonic Crystal with Alternating Masses, Phys. Rev. E, 2012, vol. 85, p. 041118.ADSCrossRefGoogle Scholar
  26. 26.
    Dhar, A. and Dandekar, R., Heat Transport and Current Fluctuations in Harmonic Crystals, Physica A, vol. 418, pp. 49-64.Google Scholar
  27. 27.
    Ivanova, E.A. and Vilchevskaya, E.N., Description of Thermal and Micro-Structural Processes in Generalized Continua: Zhilin’s Method and Its Modifications, Generalized Continua as Models for Materials with Multi-Scale Effects or under Multi-Field Actions, Altenbach, H., Forest, S., and Krivtsov, A.M., Eds., Berlin: Springer, 2013, pp.179–197.CrossRefGoogle Scholar
  28. 28.
    Ivanova, E.A., Description of Mechanism of Thermal Conduction and Internal Damping by Means of Two Component Cosserat Continuum, Acta Mech., 2014, vol. 225, no. 3, pp. 757–795.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tzou, D.Y., Macro- to Microscale Heat Transfer: The Lagging Behavior, Chichester: John Wiley & Sons, 2015.Google Scholar
  30. 30.
    Landau, L.D. and Lifshitz, E.M., Mechanics, A Course of Theoretical Physics, Volume 1, Oxford: Pergamon Press, 1969.Google Scholar
  31. 31.
    Allen, M.P. and Tildesley, A.K., Computer Simulation of Liquids, Oxford: Clarendon Press, 1987.zbMATHGoogle Scholar
  32. 32.
    Krivtsov, A.M., Energy Oscillations in a One-Dimensional Crystal, Doklady Physics, 2014, vol. 59, no. 9, pp. 427–430.MathSciNetCrossRefGoogle Scholar
  33. 33.
    Krivtsov, A.M., Heat Transfer in Infinite Harmonic OneDimensional Crystals, Doklady Physics, 2015, vol. 60, no. 9, pp. 407–411.ADSCrossRefGoogle Scholar
  34. 34.
    Krivtsov, A.M., On Unsteady Heat Conduction in a Harmonic Crystal, ArXiv: 1509.02506, 2015.Google Scholar
  35. 35.
    Krivtsov, A.M., Dynamics of Thermal Processes in OneDimensional Harmonic Crystals, Problems of Mathematical Physics and Applied Mathematics, Tropp, E.A., Ed., St. Petersburg: Ioffe Institute, 2016, pp. 63–81.Google Scholar
  36. 36.
    Krivtsov, A.M., Dynamics of Energy Characteristics in One-Dimensional Crystal, Proc. of XXXIVSummer School “AdvancedProblems in Mechanics ”, St. Petersburg, Russia, 2006, pp. 274208.Google Scholar
  37. 37.
    Poletkin, K.V., Gurzadyan, G.G., Shang, J., and Kulish, V., Ultrafast Heat Transfer on Nanoscale in Thin Gold Films, Appl. Phys. B, 2012, vol. 107, pp. 137–143.ADSCrossRefGoogle Scholar
  38. 38.
    Rieder, Z., Lebowitz, J.L., and Lieb, E., Properties of a Harmonic Crystal in a Stationary Nonequilibrium State, J. Math. Phys., 1967, vol. 8, no. 5, pp. 1073–1078.ADSCrossRefGoogle Scholar
  39. 39.
    Slepyan, L.I. and Yakovlev, Yu.S., Integral Transforms in Nonstationary Problems of Mechanics, Leningrad: Sudostroenie, 1980.zbMATHGoogle Scholar
  40. 40.
    Gendelman, O.V., Shvartsman, R., Madar, B., and Savin, A.V., Nonstationary Heat Conduction in One-Dimensional Models with Substrate Potential, Phys. Rev. E, 2012, vol. 85, no. 1, p. 011105.ADSCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  • M. B. Babenkov
    • 1
    • 2
  • A. M. Krivtsov
    • 1
    • 2
    Email author
  • D. V. Tsvetkov
    • 1
    • 2
  1. 1.Peter the Great St. Petersburg Polytechnic UniversitySt. PetersburgRussia
  2. 2.Institute of Problems of Mechanical EngineeringRussian Academy of SciencesSt. PetersburgRussia

Personalised recommendations