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

Abstract

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.

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References

  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.

    Article  Google 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.

    Article  Google 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.

    ADS  Article  Google 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.

    Google 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.

    ADS  Article  MATH  Google 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.

  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.

    Article  Google 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.

    ADS  Article  MATH  Google 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.

    Article  Google 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.

    ADS  Article  Google 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.

    Article  Google 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.

    ADS  MathSciNet  Article  Google Scholar 

  14. 14.

    Dhar, A., Heat Transport in Low-Dimensional Systems, Adv. Phys., 2008, vol. 57, pp. 457–537.

    ADS  Article  Google 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.

    ADS  MathSciNet  Article  MATH  Google 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.

    ADS  Article  Google 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.

    ADS  Article  Google 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.

    ADS  Article  Google 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.

    ADS  MathSciNet  Article  MATH  Google 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.

    ADS  Article  MATH  Google 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.

    ADS  Article  Google 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.

    ADS  Article  Google 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.).

    ADS  MathSciNet  Article  MATH  Google 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.

    ADS  Article  Google Scholar 

  26. 26.

    Dhar, A. and Dandekar, R., Heat Transport and Current Fluctuations in Harmonic Crystals, Physica A, vol. 418, pp. 49-64.

  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.

    Google 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.

    MathSciNet  Article  MATH  Google 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.

    Google Scholar 

  32. 32.

    Krivtsov, A.M., Energy Oscillations in a One-Dimensional Crystal, Doklady Physics, 2014, vol. 59, no. 9, pp. 427–430.

    MathSciNet  Article  Google Scholar 

  33. 33.

    Krivtsov, A.M., Heat Transfer in Infinite Harmonic OneDimensional Crystals, Doklady Physics, 2015, vol. 60, no. 9, pp. 407–411.

    ADS  Article  Google 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.

    ADS  Article  Google 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.

    ADS  Article  Google Scholar 

  39. 39.

    Slepyan, L.I. and Yakovlev, Yu.S., Integral Transforms in Nonstationary Problems of Mechanics, Leningrad: Sudostroenie, 1980.

    Google 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.

    ADS  Article  Google Scholar 

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Correspondence to A. M. Krivtsov.

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Original Russian Text © M.B. Babenkov, A.M. Krivtsov, D.V. Tsvetkov, 2016, published in Fizicheskaya Mezomekhanika, 2016, Vol. 19, No. 1, pp. 60-67.

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Babenkov, M.B., Krivtsov, A.M. & Tsvetkov, D.V. Energy oscillations in a one-dimensional harmonic crystal on an elastic substrate. Phys Mesomech 19, 282–290 (2016). https://doi.org/10.1134/S1029959916030061

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Keywords

  • one-dimensional crystal
  • thermal conductivity
  • covariances
  • energy oscillations
  • elastic substrate