Thermal equilibration in infinite harmonic crystals

Abstract

We study transient thermal processes in infinite harmonic crystals having a unit cell with an arbitrary number of particles. Initially, particles have zero displacements and random velocities such that spatial distribution of temperature is uniform. Initial kinetic and potential energies are different and therefore the system is far from thermal equilibrium. Time evolution of kinetic temperatures, corresponding to different degrees of freedom of the unit cell, is investigated. It is shown that the temperatures oscillate in time and tend to generally different equilibrium values. The oscillations are caused by two physical processes: equilibration of kinetic and potential energies and redistribution of temperature among degrees of freedom of the unit cell. An exact formula describing these oscillations is obtained. At large times, a crystal approaches thermal equilibrium, i.e., a state in which the temperatures are constant in time. A relation, referred to as the non-equipartition theorem, between equilibrium values of the temperatures and initial conditions is derived. For illustration, transient thermal processes in a diatomic chain and graphene lattice are considered. Analytical results are supported by numerical solution of lattice dynamics equations.

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References

  1. 1.

    Allen, M.P., Tildesley, D.J.: Computer Simulation of Liquids, p. 385. Clarendon Press, Oxford (1987)

    Google Scholar 

  2. 2.

    Babenkov, M.B., Krivtsov, A.M., Tsvetkov, D.V.: Energy oscillations in 1D harmonic crystal on elastic foundation. Phys. Mesomech. 19(1), 60–67 (2016)

    Google Scholar 

  3. 3.

    Balandin, A.A.: Thermal properties of graphene and nanostructured carbon materials. Nat. Mater. 10, 569 (2011)

    ADS  Article  Google Scholar 

  4. 4.

    Barani, E., Lobzenko, I.P., Korznikova, E.A., Soboleva, E.G., Dmitriev, S.V., Zhou, K., Marjaneh, A.M.: Transverse discrete breathers in unstrained graphene. Eur. Phys. J. B 90(3), 1 (2017)

    Article  Google Scholar 

  5. 5.

    Benettin, G., Lo Vecchio, G., Tenenbaum, A.: Stochastic transition in two-dimensional Lennard–Jones systems. Phys. Rev. A 22, 1709 (1980)

    ADS  MathSciNet  Article  Google Scholar 

  6. 6.

    Berinskii, I.E., Krivtsov, A.M.: Linear oscillations of suspended graphene. In: Altenbach, H., Mikhasev, G. (eds.) Shell and Membrane Theories in Mechanics and Biology. Advanced Structured Materials, vol. 45. Springer, Berlin (2015)

    Google Scholar 

  7. 7.

    Boldrighini, C., Pellegrinotti, A., Triolo, L.: Convergence to stationary states for infinite harmonic systems. J. Stat. Phys. 30(1), 123–155 (1983)

    ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    Casas-Vazquez, J., Jou, D.: Temperature in non-equilibrium states: a review of open problems and current proposals. Rep. Prog. Phys. 66, 1937–2023 (2003)

    ADS  Article  Google Scholar 

  9. 9.

    Casher, A., Lebowitz, J.L.: Heat flow in regular and disordered harmonic chains. J. Math. Phys. 12, 1701 (1971)

    ADS  Article  Google Scholar 

  10. 10.

    Chang, A.Y., Cho, Y.-J., Chen, K.-C., Chen, C.-W., Kinaci, A., Diroll, B.T., Wagner, M.J., Chan, M.K.Y., Lin, H.-W., Schaller, R.D.: Slow organic-to-inorganic sub-lattice thermalization in methylammonium lead halide perovskites observed by ultrafast photoluminescence. Adv. Energy Mater. 6, 1600422 (2016)

    Article  Google Scholar 

  11. 11.

    Dove, M.T.: Introduction to Lattice Dynamics. Cambridge University Press, London (1993)

    Google Scholar 

  12. 12.

    Dobrushin, R.L., Pellegrinotti, A., Suhov, Y.M., Triolo, L.: One-dimensional harmonic lattice caricature of hydrodynamics. J. Stat. Phys. 43, 3 (1986)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Dudnikova, T.V., Komech, A.I., Spohn, H.: On the convergence to statistical equilibrium for harmonic crystals. J. Math. Phys. 44, 2596 (2003)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Dudnikova, T.V., Komech, A.I.: On the convergence to a statistical equilibrium in the crystal coupled to a scalar field. Russ. J. Math. Phys. 12(3), 301–325 (2005)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Fedoryuk, M.V.: The stationary phase method and pseudodifferential operators. Russ. Math. Surv. 6(1), 65–115 (1971)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Guo, P., Gong, J., Sadasivam, S., Xia, Y., Song, T.-B., Diroll, B.T., Stoumpos, C.C., Ketterson, J.B., Kanatzidis, M.G., Chan, M.K.Y., Darancet, P., Xu, T., Schaller, R.D.: Slow thermal equilibration in methylammonium lead iodide revealed by transient mid-infrared spectroscopy. Nat. Commun. 9, 2792 (2018)

    ADS  Article  Google Scholar 

  17. 17.

    Guzev, M.A.: The exact formula for the temperature of a one-dimensional crystal. Dal’nevost. Mat. Zh. 18, 39 (2018)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    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. Cont. Mech. Thermodyn. 31(1), 255–272 (2019)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Harris, L., Lukkarinen, J., Teufel, S., Theil, F.: Energy transport by acoustic modes of harmonic lattices. SIAM J. Math. Anal. 40(4), 1392 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Hizhnyakov, V., Klopov, M., Shelkan, A.: Transverse intrinsic localized modes in monoatomic chain and in graphene. Phys. Lett. A 380(9–10), 1075–1081 (2016)

    ADS  MATH  Article  Google Scholar 

  21. 21.

    van Hemmen, J.L.: A generalized equipartition theorem. Phys. Lett. 79A, 1 (1980)

    MathSciNet  Google Scholar 

  22. 22.

    Hemmer, P.C.: Dynamic and Stochastic Types of Motion in the Linear Chain. Norges tekniske hoiskole, Trondheim (1959)

    Google Scholar 

  23. 23.

    Holian, B.L., Hoover, W.G., Moran, B., Straub, G.K.: Shock-wave structure via nonequilibrium molecular dynamics and Navier–Stokes continuum mechanics. Phys. Rev. A 22, 2798 (1980)

    ADS  Article  Google Scholar 

  24. 24.

    Holian, B.L., Mareschal, M.: Heat-flow equation motivated by the ideal-gas shock wave. Phys. Rev. E 82, 026707 (2010)

    ADS  Article  Google Scholar 

  25. 25.

    Hoover, W.G.: Computational Statistical Mechanics, p. 330. Elsevier, New York (1991)

    Google Scholar 

  26. 26.

    Hoover, W.G., Hoover, C.G., Travis, K.P.: Shock-wave compression and Joule–Thomson expansion. Phys. Rev. Lett. 112, 144504 (2014)

    ADS  Article  Google Scholar 

  27. 27.

    Huerta, M.A., Robertson, H.S.: Entropy, information theory, and the approach to equilibrium of coupled harmonic oscillator systems. J. Stat. Phys. 1(3), 393–414 (1969)

    ADS  Article  Google Scholar 

  28. 28.

    Huerta, M.A., Robertson, H.S., Nearing, J.C.: Exact equilibration of harmonically bound oscillator chains. J. Math. Phys. 12, 2305 (1971)

    ADS  MathSciNet  Article  Google Scholar 

  29. 29.

    Indeitsev, D.A., Naumov, V.N., Semenov, B.N., Belyaev, A.K.: Thermoelastic waves in a continuum with complex structure. Z. Angew. Math. Mech. 89, 279 (2009)

    MATH  Article  Google Scholar 

  30. 30.

    Inogamov, N.A., Petrov, Y.V., Zhakhovsky, V.V., Khokhlov, V.A., Demaske, B.J., Ashitkov, S.I., Khishchenko, K.V., Migdal, K.P., Agranat, M.B., Anisimov, S.I., Fortov, V.E., Oleynik, I.I.: Two-temperature thermodynamic and kinetic properties of transition metals irradiated by femtosecond lasers. AIP Conf. Proc. 1464, 593 (2012)

    ADS  Article  Google Scholar 

  31. 31.

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

    ADS  Article  Google Scholar 

  32. 32.

    Kato, A., Jou, D.: Breaking of equipartition in one-dimensional heat-conducting systems. Phys. Rev. E 64, 052201 (2001)

    ADS  Article  Google Scholar 

  33. 33.

    Khadeeva, L.Z., Dmitriev, S.V., Kivshar, YuS: Discrete breathers in deformed graphene. JETP Lett. 94, 539 (2011)

    ADS  Article  Google Scholar 

  34. 34.

    Klein, G., Prigogine, I.: Sur la mecanique statistique des phenomenes irreversibles III. Physica 19, 1053 (1953)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Kittel, C.: Introduction to Solid State Physics, vol. 8. Wiley, New York (1976)

    Google Scholar 

  36. 36.

    Krivtsov, A.M.: Dynamics of energy characteristics in one-dimensional crystal. In: Proceedings of XXXIV Summer School “Advanced Problems in Mechanics”, St.-Petersburg, pp. 261–273 (2007)

  37. 37.

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

    Article  Google Scholar 

  38. 38.

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

    ADS  Article  Google Scholar 

  39. 39.

    Krivtsov, A.M.: The ballistic heat equation for a one-dimensional harmonic crystal. In: Altenbach, H., Belyaev, A., Eremeyev, V.A., Krivtsov, A., Porubov, A.V. (eds.) Dynamical Processes in Generalized Continua and Structures. Springer, Berlin (2019)

    Google Scholar 

  40. 40.

    Krivtsov, A.M., Sokolov, A.A., Müller, W.H., Freidin, A.B.: One-dimensional heat conduction and entropy production. Adv. Struct. Mater. 87, 197–213 (2018)

    MathSciNet  Article  Google Scholar 

  41. 41.

    Kosevich, A.M.: The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices. Wiley, New York (2006)

    Google Scholar 

  42. 42.

    Kuzkin, V.A., Krivtsov, A.M.: High-frequency thermal processes in harmonic crystals. Dokl. Phys. 62(2), 85 (2017)

    ADS  Article  Google Scholar 

  43. 43.

    Kuzkin, V.A., Krivtsov, A.M.: An analytical description of transient thermal processes in harmonic crystals. Phys. Solid State 59(5), 1051 (2017)

    ADS  Article  Google Scholar 

  44. 44.

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

    Article  Google Scholar 

  45. 45.

    Kuzkin, V.A.: arXiv:1808.07255 [cond-mat.stat-mech]

  46. 46.

    Lanford, O.E., Lebowitz, J.L.: Time evolution and ergodic properties of harmonic systems. In: Lecture Notes in Physics, vol. 38, pp. 144–177. Springer, Berlin (1975)

  47. 47.

    Linn, S.L., Robertson, H.S.: Thermal energy transport in harmonic systems. J. Phys. Chem. Sol. 45(2), 133 (1984)

    ADS  Article  Google Scholar 

  48. 48.

    der Linde, D., Sokolowski-Tinten, K., Bialkowski, J.: Laser–solid interaction in the femtosecond time regime. App. Surf. Sci. 109–110, 1 (1997)

    Article  Google Scholar 

  49. 49.

    Lepri, S., Mejia-Monasterio, C., Politi, A.: A stochastic model of anomalous heat transport: analytical solution of the steady state. J. Phys. A 42, 2–025001 (2008)

    MathSciNet  MATH  Google Scholar 

  50. 50.

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

    ADS  MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    Marcelli, G., Tenenbaum, A.: Quantumlike short-time behavior of a classical crystal. Phys. Rev. E 68, 041112 (2003)

    ADS  Article  Google Scholar 

  52. 52.

    Mielke, A.: Macroscopic behavior of microscopic oscillations in harmonic lattices via Wigner–Husimi transforms. Arch. Ration. Mech. Anal. 181, 401 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  53. 53.

    Mishuris, G.S., Movchan, A.B., Slepyan, L.I.: Localised knife waves in a structured interface. J. Mech. Phys. Solids 57, 1958 (2009)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Murachev, A.S., Krivtsov, A.M., Tsvetkov, D.V.: Thermal echo in a finite one-dimensional harmonic crystal. J. Phys. Condens. Mater. 31(9), 1 (2019)

    Article  Google Scholar 

  55. 55.

    Prigogine, I., Henin, F.: On the general theory of the approach to equilibrium. I. Interacting normal modes. J. Math. Phys. 1, 349 (1960)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  56. 56.

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

    ADS  Article  Google Scholar 

  57. 57.

    Simon, S.H.: The Oxford Solid State Basics. Oxford University Press, Oxford (2013)

    Google Scholar 

  58. 58.

    Schrödinger, E.: Zur dynamik elastisch gekoppelter punktsysteme. Annalen der Physik 44, 916 (1914)

    MATH  ADS  Article  Google Scholar 

  59. 59.

    Slepyan, L.I.: On the energy partition in oscillations and waves. Proc. R. Soc. A 471, 20140838 (2015)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  60. 60.

    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 

  61. 61.

    Spohn, H., Lebowitz, J.L.: Stationary non-equilibrium states of infinite harmonic systems. Commun. Math. Phys. 54, 97 (1977)

    ADS  MathSciNet  Article  Google Scholar 

  62. 62.

    Tsaplin, V.A., Kuzkin, V.A.: Temperature oscillations in harmonic triangular lattice with random initial velocities. Lett. Mater. 8(1), 16–20 (2018)

    Article  Google Scholar 

  63. 63.

    Titulaer, U.M.: Ergodic features of harmonic-oscillator systems. III. Asymptotic dynamics of large systems. Physica 70, 257 (1973)

    ADS  MathSciNet  Article  Google Scholar 

  64. 64.

    Uribe, F.J., Velasco, R.M., Garcia-Colin, L.S.: Two kinetic temperature description for shock waves. Phys. Rev. E 58, 3209 (1998)

    ADS  Article  Google Scholar 

  65. 65.

    Xiong, D., Zhang, Y., Zhao, H.: Heat transport enhanced by optical phonons in one-dimensional anharmonic lattices with alternating bonds. Phys. Rev. E 88, 052128 (2013)

    ADS  Article  Google Scholar 

  66. 66.

    Ziman, J.M.: Electrons and Phonons. The Theory of Transport Phenomena in Solids, p. 554. Oxford University Press, New York (1960)

    Google Scholar 

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Acknowledgements

The author is deeply grateful to A.M. Krivtsov, S.V. Dmitriev, M.A. Guzev, D.A. Indeitsev, E.A. Ivanova, S.N. Gavrilov, I.E. Berinskii, and A.S. Murachev for useful discussions. The work was financially supported by the Russian Science Foundation under Grant No. 17-71-10213.

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Kuzkin, V.A. Thermal equilibration in infinite harmonic crystals. Continuum Mech. Thermodyn. 31, 1401–1423 (2019). https://doi.org/10.1007/s00161-019-00758-2

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Keywords

  • Thermal equilibrium
  • Stationary state
  • Approach to equilibrium
  • Polyatomic crystal
  • Complex lattice
  • Kinetic temperature
  • Harmonic crystal
  • Transient processes
  • Equipartition theorem
  • Non-equipartition theorem
  • Temperature matrix