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Steady-state kinetic temperature distribution in a two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a point heat source

  • Serge N. GavrilovEmail author
  • Anton M. Krivtsov
Original Article

Abstract

We consider heat transfer in an infinite two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a heat source. The basic equations for the particles of the lattice are stated in the form of a system of stochastic ordinary differential equations. We perform a continualization procedure and derive an infinite system of linear partial differential equations for covariance variables. The most important results of the paper are the deterministic differential–difference equation describing non-stationary heat propagation in the lattice and the analytical formula in the integral form for its steady-state solution describing kinetic temperature distribution caused by a point heat source of a constant intensity. The comparison between numerical solution of stochastic equations and obtained analytical solution demonstrates a very good agreement everywhere except for the main diagonals of the lattice (with respect to the point source position), where the analytical solution is singular.

Keywords

Ballistic heat transfer 2D harmonic scalar lattice Kinetic temperature 

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Notes

Acknowledgements

The authors are grateful to V.A. Kuzkin, A.S. Murachev, and E.V. Shishkina for useful and stimulating discussions.

References

  1. 1.
    Chang, C., Okawa, D., Garcia, H., Majumdar, A., Zettl, A.: Breakdown of Fourier’s law in nanotube thermal conductors. Phys. Rev. Lett. 101(7), 075,903 (2008)CrossRefGoogle Scholar
  2. 2.
    Xu, X., Pereira, L., Wang, Y., Wu, J., Zhang, K., Zhao, X., Bae, S., Bui, C., Xie, R., Thong, J., Hong, B., Loh, K., Donadio, D., Li, B., Özyilmaz, B.: Length-dependent thermal conductivity in suspended single-layer graphene. Nat. Commun. 5, 15 (2014)Google Scholar
  3. 3.
    Hsiao, T., Huang, B., Chang, H., Liou, S., Chu, M., Lee, S., Chang, C.: Micron-scale ballistic thermal conduction and suppressed thermal conductivity in heterogeneously interfaced nanowires. Phys. Rev. B 91(3), 035,406 (2015)CrossRefGoogle Scholar
  4. 4.
    Cahill, D., Ford, W., Goodson, K., Mahan, G., Majumdar, A., Maris, H., Merlin, R., Phillpot, S.: Nanoscale thermal transport. J. Appl. Phys. 93(2), 793–818 (2003)ADSCrossRefGoogle Scholar
  5. 5.
    Liu, S., Xu, X., Xie, R., Zhang, G., Li, B.: Anomalous heat conduction and anomalous diffusion in low dimensional nanoscale systems. Eur. Phys. J. B 85(337), 075204 (2012)Google Scholar
  6. 6.
    Chang, C.: Experimental probing of non-Fourier thermal conductors. In: Lepri, S. (ed.) Thermal Transport in Low Dimensions: From Statistical Physics to Nanoscale Heat Transfer. Lecture Notes in Physics, vol. 921, pp. 305–338. Springer, Berlin (2016)CrossRefGoogle Scholar
  7. 7.
    Lepri, S., Livi, R., Politi, A.: Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377(1), 1–80 (2003)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Spohn, H.: Fluctuating hydrodynamics approach to equilibrium time correlations for an harmonic chains. In: Lepri, S. (ed.) Thermal Transport in Low Dimensions: From Statistical Physics to Nanoscale Heat Transfer. Lecture Notes in Physics, pp. 107–158. Springer, Berlin (2016)CrossRefGoogle Scholar
  9. 9.
    Hoover, W., Hoover, C.: Simulation and Control of Chaotic Nonequilibrium Systems. World Scientific, Singapore (2015)zbMATHCrossRefGoogle Scholar
  10. 10.
    Daly, B., Maris, H., Imamura, K., Tamura, S.: Molecular dynamics calculation of the thermal conductivity of superlattices. Phys. Rev. B 66(2), 024,301 (2002)CrossRefGoogle Scholar
  11. 11.
    Krivtsov, A.: From nonlinear oscillations to equation of state in simple discrete systems. Chaos Solitons Fractals 17(1), 79–87 (2003)ADSzbMATHCrossRefGoogle Scholar
  12. 12.
    Berinskii, I.: Elastic networks to model auxetic properties of cellular materials. Int. J. Mech. Sci. 115, 481–488 (2016)CrossRefGoogle Scholar
  13. 13.
    Kuzkin, V., Krivtsov, A., Podolskaya, E., Kachanov, M.: Lattice with vacancies: elastic fields and effective properties in frameworks of discrete and continuum models. Philos. Mag. 96(15), 1538–1555 (2016)ADSCrossRefGoogle Scholar
  14. 14.
    Berinskii, I., Krivtsov, A.: Linear oscillations of suspended graphene. In: Altenbach H, Mikhasev GI (eds) Shell and Membrane Theories in Mechanics and Biology, pp. 99–107. Springer, Berlin (2015)Google Scholar
  15. 15.
    Berinskii, I., Krivtsov, A.: A hyperboloid structure as a mechanical model of the carbon bond. Int. J. Solids Struct. 96, 145–152 (2016)CrossRefGoogle Scholar
  16. 16.
    Bonetto, F., Lebowitz, J., Rey-Bellet, L.: Fourier’s law: a challenge to theorists. In: Fokas, A., Grigoryan, A., Kibble, T., Zegarlinski, B. (eds.) Mathematical Physics 2000. World Scientific, Singapore (2000)Google Scholar
  17. 17.
    Lepri, S., Livi, R., Politi, A.: On the anomalous thermal conductivity of one-dimensional lattices. Europhys. Lett. 43(3), 271 (1998)ADSCrossRefGoogle Scholar
  18. 18.
    Dhar, A.: Heat transport in low-dimensional systems. Adv. Phys. 57(5), 457–537 (2008)ADSCrossRefGoogle Scholar
  19. 19.
    Rieder, Z., Lebowitz, J., Lieb, E.: Properties of a harmonic crystal in a stationary nonequilibrium state. J. Math. Phys. 8(5), 1073–1078 (1967)ADSCrossRefGoogle Scholar
  20. 20.
    Allen, K., Ford, J.: Energy transport for a three-dimensional harmonic crystal. Phys. Rev. 187(3), 1132 (1969)ADSCrossRefGoogle Scholar
  21. 21.
    Nakazawa, H.: On the lattice thermal conduction. Progr. Theoret. Phys. Suppl. 45, 231–262 (1970)ADSCrossRefGoogle Scholar
  22. 22.
    Lee, L., Dhar, A.: Heat conduction in a two-dimensional harmonic crystal with disorder. Phys. Rev. Lett. 95(9), 094,302 (2005)CrossRefGoogle Scholar
  23. 23.
    Kundu, A., Chaudhuri, A., Roy, D., Dhar, A., Lebowitz, J., Spohn, H.: Heat conduction and phonon localization in disordered harmonic crystals. Europhys. Lett. 90(4), 40,001 (2010)CrossRefGoogle Scholar
  24. 24.
    Dhar, A., Saito, K.: Heat transport in harmonic systems. In: Lepri, S. (ed.) Thermal Transport in Low Dimensions: From Statistical Physics to Nanoscale Heat Transfer. Lecture Notes in Physics, vol. 921, pp. 39–106. Springer, Berlin (2016)CrossRefGoogle Scholar
  25. 25.
    Bernardin, C., Kannan, V., Lebowitz, J., Lukkarinen, J.: Harmonic systems with bulk noises. J. Stat. Phys. 146(4), 800–831 (2012)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Freitas, N., Paz, J.: Analytic solution for heat flow through a general harmonic network. Phys. Rev. E 90(4), 042,128 (2014)CrossRefGoogle Scholar
  27. 27.
    Freitas, N., Paz, J.: Erratum: analytic solution for heat flow through a general harmonic network. Phys. Rev. E 90(6), 069,903 (2014)CrossRefGoogle Scholar
  28. 28.
    Hoover, W., Hoover, C.: Hamiltonian thermostats fail to promote heat flow. Commun. Nonlinear Sci. Numer. Simul. 18(12), 3365–3372 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Lukkarinen, J., Marcozzi, M., Nota, A.: Harmonic chain with velocity flips: thermalization and kinetic theory. J. Stat. Phys. 165(5), 809–844 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Le-Zakharov, A., Krivtsov, A.: Molecular dynamics investigation of heat conduction in crystals with defects. Doklady Phys. 53(5), 261–264 (2008)ADSzbMATHCrossRefGoogle Scholar
  31. 31.
    Gendelman, O., Shvartsman, R., Madar, B., Savin, A.: Nonstationary heat conduction in one-dimensional models with substrate potential. Phys. Rev. E 85(1), 011,105 (2012)CrossRefGoogle Scholar
  32. 32.
    Tsai, D., MacDonald, R.: Molecular-dynamical study of second sound in a solid excited by a strong heat pulse. Phys. Rev. B 14(10), 4714 (1976)ADSCrossRefGoogle Scholar
  33. 33.
    Ladd, A., Moran, B., Hoover, W.: Lattice thermal conductivity: a comparison of molecular dynamics and anharmonic lattice dynamics. Phys. Rev. B 34(8), 5058 (1986)ADSCrossRefGoogle Scholar
  34. 34.
    Volz, S., Saulnier, J.B., Lallemand, M., Perrin, B., Depondt, P., Mareschal, M.: Transient Fourier-law deviation by molecular dynamics in solid argon. Phys. Rev. B 54(1), 340 (1996)ADSCrossRefGoogle Scholar
  35. 35.
    Gendelman, O., Savin, A.: Nonstationary heat conduction in one-dimensional chains with conserved momentum. Phys. Rev. E 81(2), 020,103 (2010)CrossRefGoogle Scholar
  36. 36.
    Guzev, M.: The exact formula for the temperature of a one-dimensional crystal. Far East. Math. J. 18(1), 39–47 (2018)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Krivtsov, A.: Energy oscillations in a one-dimensional crystal. Doklady Phys. 59(9), 427–430 (2014)CrossRefGoogle Scholar
  38. 38.
    Krivtsov, A.: Heat transfer in infinite harmonic one-dimensional crystals. Doklady Phys. 60(9), 407–411 (2015)ADSCrossRefGoogle Scholar
  39. 39.
    Krivtsov, A.: The ballistic heat equation for a one-dimensional harmonic crystal. In: Altenbach, H., et al. (eds.) Dynamical Processes in Generalized Continua and Structures, Advanced Structured Materials 103, pp. 345–358. Springer, Berlin (2019).  https://doi.org/10.1007/978-3-030-11665-1_19 CrossRefGoogle Scholar
  40. 40.
    Chandrasekharalah, D.: Thermoelasticity with second sound: a review. Appl. Mech. Rev. 39(3), 355 (1986)ADSCrossRefGoogle Scholar
  41. 41.
    Tzou, D.: Macro-to Microscale Heat Transfer: The Lagging Behavior. Wiley, New York (2014)CrossRefGoogle Scholar
  42. 42.
    Cattaneo, C.: Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée. Comptes Rendus de L’Academie des Sci. 247(4), 431–433 (1958)zbMATHMathSciNetGoogle Scholar
  43. 43.
    Vernotte, P.: Les paradoxes de la théorie continue de léquation de la chaleur. Comptes Rendus de L’Academie des Sci. 246(22), 3154–3155 (1958)zbMATHMathSciNetGoogle Scholar
  44. 44.
    Sokolov, A., Krivtsov, A., Müller, W.: Localized heat perturbation in harmonic 1D crystals: solutions for the equation of anomalous heat conduction. Phys. Mesomech. 20(3), 305–310 (2017)CrossRefGoogle Scholar
  45. 45.
    Krivtsov, A., Sokolov, A., Müller, W., Freidin, A.: One-dimensional heat conduction and entropy production. In: dell’Isola, F., Eremeyev, V., Porubov, A. (eds.) Advances in Mechanics of Microstructured Media and Structures, pp. 197–213. Springer, Berlin (2018)CrossRefGoogle Scholar
  46. 46.
    Sokolov, A., Krivtsov, A., Müller, W., Vilchevskaya, E.: Change of entropy for the one-dimensional ballistic heat equation: sinusoidal initial perturbation. Phys. Rev. E 99, 042,107 (2019).  https://doi.org/10.1103/PhysRevE.99.042107 MathSciNetCrossRefGoogle Scholar
  47. 47.
    Babenkov, M., Krivtsov, A., Tsvetkov, D.: Energy oscillations in a one-dimensional harmonic crystal on an elastic substrate. Phys. Mesomech. 19(3), 282–290 (2016)CrossRefGoogle Scholar
  48. 48.
    Kuzkin, V., Krivtsov, A.: An analytical description of transient thermal processes in harmonic crystals. Phys. Solid State 59(5), 1051–1062 (2017)ADSCrossRefGoogle Scholar
  49. 49.
    Kuzkin, V., Krivtsov, A.: High-frequency thermal processes in harmonic crystals. Doklady Phys. 62(2), 85–89 (2017)ADSCrossRefGoogle Scholar
  50. 50.
    Kuzkin, V., Krivtsov, A.: Fast and slow thermal processes in harmonic scalar lattices. J. Phys. Condens Matter 29(50), 505,401 (2017)CrossRefGoogle Scholar
  51. 51.
    Murachev, A., Krivtsov, A., Tsvetkov, D.: Thermal echo in a finite one-dimensional harmonic crystal. J. Phys. Condens. Matter 31(9), 095,702 (2019).  https://doi.org/10.1088/1361-648X/aaf3c6 CrossRefGoogle Scholar
  52. 52.
    Podolskaya, E., Krivtsov, A., Tsvetkov, D.: Anomalous heat transfer in one-dimensional diatomic harmonic crystal. Mater. Phys. Mech. 40, 172–180 (2018)Google Scholar
  53. 53.
    Gavrilov, S., Krivtsov, A., Tsvetkov, D.: Heat transfer in a one-dimensional harmonic crystal in a viscous environment subjected to an external heat supply. Continuum Mech. Thermodyn. 31, 255–272 (2019).  https://doi.org/10.1007/s00161-018-0681-3 ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    Mielke, A.: Macroscopic behavior of microscopic oscillations in harmonic lattices via Wigner–Husimi transforms. Arch. Rational Mech. Anal. 181(3), 401–448 (2006)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Harris, L., Lukkarinen, J., Teufel, S., Theil, F.: Energy transport by acoustic modes of harmonic lattices. SIAM J. Math. Anal. 40(4), 1392–1418 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Savin, A., Zolotarevskiy, V., Gendelman, O.: Normal heat conductivity in two-dimensional scalar lattices. Europhys. Lett. 113(2), 24,003 (2016)CrossRefGoogle Scholar
  57. 57.
    Nishiguchi, N., Kawada, Y., Sakuma, T.: Thermal conductivity in two-dimensional monatomic non-linear lattices. J. Phys. Condens. Matter 4(50), 10,227 (1992)CrossRefGoogle Scholar
  58. 58.
    Kuzkin, V.A.: Thermal equilibration in infinite harmonic crystals. Continuum Mech. Thermodyn. (2019).  https://doi.org/10.1007/s00161-019-00758-2
  59. 59.
    Kloeden, P., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1999)zbMATHGoogle Scholar
  60. 60.
    Stepanov, S.: Stochastic World. Springer, Berlin (2013)zbMATHCrossRefGoogle Scholar
  61. 61.
    Langevin, P.: Sur la théorie du mouvement brownien. Comptes Rendus de L’Academie des Sci. 146(530–533), 530 (1908)zbMATHGoogle Scholar
  62. 62.
    Lemons, D., Gythiel, A.: Paul Langevin’s 1908 paper "On the theory of Brownian motion" [“Sur la théorie du mouvement brownien”]. CR Acad. Sci.(Paris) 146, 530–533 (1908)]. Am. J. Phys. 65(11), 1079–1081 (1997)ADSCrossRefGoogle Scholar
  63. 63.
    Krivtsov, A.: Dynamics of heat processes in one-dimensional harmonic crystals. In: Problems of Mathematical Physics and Applied Mathematics: Proceedings of the Seminar in Honor of Prof. E.A. Tropp’s 75th Anniversary, pp. 63–81. Ioffe Institute, St. Petersburg (2016) (in Russian)Google Scholar
  64. 64.
    Wang, M., Uhlenbeck, G.: On the theory of the Brownian motion II. Rev. Modern Phys. 17(2–3), 323 (1945)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Vladimirov, V.: Equations of Mathematical Physics. Marcel Dekker, New York (1971)zbMATHGoogle Scholar
  66. 66.
    Lepri, S., Mejía-Monasterio, C., Politi, A.: Nonequilibrium dynamics of a stochastic model of anomalous heat transport. J. Phys. A 43(6), 065,002 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Nayfeh, A.: Perturbation Methods. Wiley, New York (2008)Google Scholar
  68. 68.
    Kevorkian, J., Cole, J.: Multiple Scale and Singular Perturbation Methods. Springer, Berlin (2012)zbMATHGoogle Scholar
  69. 69.
    Fedoryuk, M.: The Saddle-Point Method. Nauka, Moscow (1977). In RussianzbMATHGoogle Scholar
  70. 70.
    Jones, E., Oliphant, T., Peterson, P., et al.: SciPy: Open source scientific tools for Python. http://www.scipy.org/. Accessed 10 Jan 2019
  71. 71.
    Giannoulis, J., Herrmann, M., Mielke, A.: Continuum descriptions for the dynamics in discrete lattices: derivation and justification. In: Mielke, A. (ed.) Analysis, Modeling and Simulation of Multiscale Problems, pp. 435–466. Springer, Berlin (2006)zbMATHCrossRefGoogle Scholar
  72. 72.
    Goldstein, R., Morozov, N.: Mechanics of deformation and fracture of nanomaterials and nanotechnology. Phys. Mesomech. 10(5–6), 235–246 (2007)CrossRefGoogle Scholar
  73. 73.
    Hwang, G., Kwon, O.: Measuring the size dependence of thermal conductivity of suspended graphene disks using null-point scanning thermal microscopy. Nanoscale 8(9), 5280–5290 (2016)ADSCrossRefGoogle Scholar
  74. 74.
    Indeitsev, D., Osipova, E.: A two-temperature model of optical excitation of acoustic waves in conductors. Doklady Phys. 62(3), 136–140 (2017)ADSCrossRefGoogle Scholar
  75. 75.
    Gel’fand, I., Shilov, G.: Generalized Functions. Volume I: Properties and Operations. Academic Press, New York (1964)zbMATHGoogle Scholar
  76. 76.
    Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Dover, New York (1972)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Institute for Problems in Mechanical Engineering RASSt. PetersburgRussia
  2. 2.Peter the Great St. Petersburg Polytechnic University (SPbPU)St. PetersburgRussia

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