Skip to main content
SpringerLink
Log in

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

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    Google Scholar 

  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. 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)

    Google Scholar 

  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)

    ADS  Google Scholar 

  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. 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)

    Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  9. Hoover, W., Hoover, C.: Simulation and Control of Chaotic Nonequilibrium Systems. World Scientific, Singapore (2015)

    MATH  Google Scholar 

  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)

    Google Scholar 

  11. Krivtsov, A.: From nonlinear oscillations to equation of state in simple discrete systems. Chaos Solitons Fractals 17(1), 79–87 (2003)

    ADS  MATH  Google Scholar 

  12. Berinskii, I.: Elastic networks to model auxetic properties of cellular materials. Int. J. Mech. Sci. 115, 481–488 (2016)

    Google Scholar 

  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)

    ADS  Google Scholar 

  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. Berinskii, I., Krivtsov, A.: A hyperboloid structure as a mechanical model of the carbon bond. Int. J. Solids Struct. 96, 145–152 (2016)

    Google Scholar 

  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. Lepri, S., Livi, R., Politi, A.: On the anomalous thermal conductivity of one-dimensional lattices. Europhys. Lett. 43(3), 271 (1998)

    ADS  Google Scholar 

  18. Dhar, A.: Heat transport in low-dimensional systems. Adv. Phys. 57(5), 457–537 (2008)

    ADS  Google Scholar 

  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)

    ADS  Google Scholar 

  20. Allen, K., Ford, J.: Energy transport for a three-dimensional harmonic crystal. Phys. Rev. 187(3), 1132 (1969)

    ADS  Google Scholar 

  21. Nakazawa, H.: On the lattice thermal conduction. Progr. Theoret. Phys. Suppl. 45, 231–262 (1970)

    ADS  Google Scholar 

  22. Lee, L., Dhar, A.: Heat conduction in a two-dimensional harmonic crystal with disorder. Phys. Rev. Lett. 95(9), 094,302 (2005)

    Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  25. Bernardin, C., Kannan, V., Lebowitz, J., Lukkarinen, J.: Harmonic systems with bulk noises. J. Stat. Phys. 146(4), 800–831 (2012)

    ADS  MathSciNet  MATH  Google Scholar 

  26. Freitas, N., Paz, J.: Analytic solution for heat flow through a general harmonic network. Phys. Rev. E 90(4), 042,128 (2014)

    Google Scholar 

  27. Freitas, N., Paz, J.: Erratum: analytic solution for heat flow through a general harmonic network. Phys. Rev. E 90(6), 069,903 (2014)

    Google Scholar 

  28. Hoover, W., Hoover, C.: Hamiltonian thermostats fail to promote heat flow. Commun. Nonlinear Sci. Numer. Simul. 18(12), 3365–3372 (2013)

    ADS  MathSciNet  MATH  Google Scholar 

  29. Lukkarinen, J., Marcozzi, M., Nota, A.: Harmonic chain with velocity flips: thermalization and kinetic theory. J. Stat. Phys. 165(5), 809–844 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  30. Le-Zakharov, A., Krivtsov, A.: Molecular dynamics investigation of heat conduction in crystals with defects. Doklady Phys. 53(5), 261–264 (2008)

    ADS  MATH  Google Scholar 

  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)

    Google Scholar 

  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)

    ADS  Google Scholar 

  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)

    ADS  Google Scholar 

  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)

    ADS  Google Scholar 

  35. Gendelman, O., Savin, A.: Nonstationary heat conduction in one-dimensional chains with conserved momentum. Phys. Rev. E 81(2), 020,103 (2010)

    Google Scholar 

  36. Guzev, M.: The exact formula for the temperature of a one-dimensional crystal. Far East. Math. J. 18(1), 39–47 (2018)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  38. Krivtsov, A.: Heat transfer in infinite harmonic one-dimensional crystals. Doklady Phys. 60(9), 407–411 (2015)

    ADS  Google Scholar 

  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

    Chapter  Google Scholar 

  40. Chandrasekharalah, D.: Thermoelasticity with second sound: a review. Appl. Mech. Rev. 39(3), 355 (1986)

    ADS  Google Scholar 

  41. Tzou, D.: Macro-to Microscale Heat Transfer: The Lagging Behavior. Wiley, New York (2014)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  48. Kuzkin, V., Krivtsov, A.: An analytical description of transient thermal processes in harmonic crystals. Phys. Solid State 59(5), 1051–1062 (2017)

    ADS  Google Scholar 

  49. Kuzkin, V., Krivtsov, A.: High-frequency thermal processes in harmonic crystals. Doklady Phys. 62(2), 85–89 (2017)

    ADS  Google Scholar 

  50. Kuzkin, V., Krivtsov, A.: Fast and slow thermal processes in harmonic scalar lattices. J. Phys. Condens Matter 29(50), 505,401 (2017)

    Google Scholar 

  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

    Article  Google Scholar 

  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. 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

    Article  ADS  MathSciNet  Google Scholar 

  54. Mielke, A.: Macroscopic behavior of microscopic oscillations in harmonic lattices via Wigner–Husimi transforms. Arch. Rational Mech. Anal. 181(3), 401–448 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  56. Savin, A., Zolotarevskiy, V., Gendelman, O.: Normal heat conductivity in two-dimensional scalar lattices. Europhys. Lett. 113(2), 24,003 (2016)

    Google Scholar 

  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)

    Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  59. Kloeden, P., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1999)

    MATH  Google Scholar 

  60. Stepanov, S.: Stochastic World. Springer, Berlin (2013)

    MATH  Google Scholar 

  61. Langevin, P.: Sur la théorie du mouvement brownien. Comptes Rendus de L’Academie des Sci. 146(530–533), 530 (1908)

    MATH  Google Scholar 

  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)

    ADS  Google Scholar 

  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)

  64. Wang, M., Uhlenbeck, G.: On the theory of the Brownian motion II. Rev. Modern Phys. 17(2–3), 323 (1945)

    ADS  MathSciNet  MATH  Google Scholar 

  65. Vladimirov, V.: Equations of Mathematical Physics. Marcel Dekker, New York (1971)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  67. Nayfeh, A.: Perturbation Methods. Wiley, New York (2008)

    Google Scholar 

  68. Kevorkian, J., Cole, J.: Multiple Scale and Singular Perturbation Methods. Springer, Berlin (2012)

    MATH  Google Scholar 

  69. Fedoryuk, M.: The Saddle-Point Method. Nauka, Moscow (1977). In Russian

    MATH  Google Scholar 

  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. 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)

    MATH  Google Scholar 

  72. Goldstein, R., Morozov, N.: Mechanics of deformation and fracture of nanomaterials and nanotechnology. Phys. Mesomech. 10(5–6), 235–246 (2007)

    Google Scholar 

  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)

    ADS  Google Scholar 

  74. Indeitsev, D., Osipova, E.: A two-temperature model of optical excitation of acoustic waves in conductors. Doklady Phys. 62(3), 136–140 (2017)

    ADS  Google Scholar 

  75. Gel’fand, I., Shilov, G.: Generalized Functions. Volume I: Properties and Operations. Academic Press, New York (1964)

    MATH  Google Scholar 

  76. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Dover, New York (1972)

    MATH  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Institute for Problems in Mechanical Engineering RAS, V.O., Bolshoy pr. 61, St. Petersburg, Russia, 199178

    Serge N. Gavrilov & Anton M. Krivtsov

  2. Peter the Great St. Petersburg Polytechnic University (SPbPU), Polytechnicheskaya str. 29, St. Petersburg, Russia, 195251

    Serge N. Gavrilov & Anton M. Krivtsov

Authors
  1. Serge N. Gavrilov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anton M. Krivtsov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Serge N. Gavrilov.

Additional information

Communicated by Andreas Öchsner.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by Russian Science Foundation (Grant No. 18-11-00201).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gavrilov, S.N., Krivtsov, A.M. 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. Continuum Mech. Thermodyn. 32, 41–61 (2020). https://doi.org/10.1007/s00161-019-00782-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-019-00782-2

Keywords

Search

Navigation