Heat jet approach for finite temperature atomic simulations of two-dimensional square lattice

Liu, Baiyili;Tang, Shaoqiang

  • Received : 2015.09.23
  • Accepted : 2015.12.23
  • Published : 2016.07.25


We propose a heat jet approach for a two-dimensional square lattice with nearest neighbouring harmonic interaction. First, we design a two-way matching boundary condition that linearly relates the displacement and velocity at atoms near the boundary, and a suitable input in terms of given incoming wave modes. Then a phonon representation for finite temperature lattice motion is adopted. The proposed approach is simple and compact. Numerical tests validate the effectiveness of the boundary condition in reflection suppression for outgoing waves. It maintains target temperature for the lattice, with expected kinetic energy distribution and heat flux. Moreover, its linear nature facilitates reliable finite temperature atomic simulations with a correct description for non-thermal motions.


heat jet approach;atomic simulations;finite temperature;square lattice


  1. Lepri, S., Livi, R. and Politi, A. (2003), "Thermal conduction in classical low-dimensional lattices", Phys. Reports, 377(1), 1-80.
  2. Andersen, H.C. (1980), "Molecular dynamics simulations at constant pressure and/or temperature", J. Chem. Phys., 72(4), 2384-2393.
  3. Berendsen, H.J., Postma, J.V., van Gunsteren, W.F., DiNola, A.R.H.J. and Haak, J.R. (1984), "Molecular dynamics with coupling to an external bath", J. Chem. Phys., 81(8), 3684-3690.
  4. Nose, S. (1984), "A unified formulation of the constant temperature molecular dynamics methods", J. Chem. Phys., 81(1), 511-519.
  5. Hoover, W.G. (1985). "Canonical dynamics: equilibrium phase-space distributions", Phys. Rev. A., 31(3), 1695-1697.
  6. Bussi, G. and Parrinello, M. (2007), "Accurate sampling using Langevin dynamics", Phys. Rev. E., 75(5), 056707.
  7. Dhar, A. (2008), "Heat transport in low-dimensional systems", Adv. Phys., 57(5), 457-537.
  8. Xiong, D., Zhang, Y. and Zhao, H. (2014), "Temperature dependence of heat conduction in the Fermi-Pasta-Ulam-beta lattice with next-nearest-neighbor coupling", Phys. Rev. E., 90(2), 022117.
  9. Hatano, T. (1999), "Heat conduction in the diatomic Toda lattice revisited", Phys. Rev. E., 59(1), R1-R4.
  10. Ai, B. and Hu, B. (2011), "Heat conduction in deformable Frenkel-Kontorova lattices: Thermal conductivity and negative differential thermal resistance", Phys. Rev. E., 83(1), 011131.
  11. Giardina, C., Livi, R., Politi, A. and Vassalli, M. (2000), "Finite thermal conductivity in 1D lattices", Phys. Rev. L., 84(10), 2144-2147.
  12. Zhong, Y., Zhang, Y., Wang, J. and Zhao, H. (2012), "Normal heat conduction in one-dimensional momentum conserving lattices with asymmetric interactions", Phys. Rev. E., 85(6), 060102.
  13. Savin, A.V. and Kosevich, Y.A. (2014), "Thermal conductivity of molecular chains with asymmetric potentials of pair interactions", Phys. Rev. E., 89(3), 032102.
  14. Dhar, A., Venkateshan, K. and Lebowitz, J.L. (2011), "Heat conduction in disordered harmonic lattices with energy-conserving noise", Phys. Rev. E., 83(2), 021108.
  15. Jackson, E.A. and Mistriotis, A.D. (1989), "Thermal conductivity of one-and two-dimensional lattices", J. Phys. Condens. Matt., 1(7), 1223-1238.
  16. Lippi, A. and Livi, R. (2000), "Heat conduction in two-dimensional nonlinear lattices", J. Stat. Phys., 100(5), 1147-1172.
  17. Yang, L., Grassberger, P. and Hu, B. (2006), "Dimensional crossover of heat conduction in low dimensions", Phys. Rev. E., 74(6), 062101.
  18. Xiong, D., Wang, J., Zhang, Y. and Zhao, H. (2010), "Heat conduction in two-dimensional disk models", Phys. Rev. E., 82(3), 030101.
  19. Nishiguchi, N., Kawada, Y. and Sakuma, T. (1992), "Thermal conductivity in two-dimensional monatomic non-linear lattices", J. Phys. Condens. Matt., 4(50), 10227-10236.
  20. Barik, D. (2006), "Heat conduction in 2D harmonic lattices with on-site potential", Europhys. Lett., 75(1), 42-48.
  21. Yang, L. (2002), "Finite heat conduction in a 2D disorder lattice", Phys. Rev. Lett., 88(9), 094301.
  22. Karpov, E.G., Park, H.S. and Liu, W.K. (2007), "A phonon heat bath approach for the atomistic and multiscale simulation of solids", Int. J. Numer. Method. Eng., 70(3), 351-378.
  23. Tang, S. and Liu, B. (2015), "Heat jet approach for atomic simulations at finite temperature", Comm. Comput. Phys., 18(5), 1445-1460.
  24. Pang, G. and Tang, S. (2011), "Time history kernel functions for square lattice", Comput. Mech., 48(6), 699-711.
  25. Wang, X. and Tang, S. (2013), "Matching boundary conditions for lattice dynamics", Int. J. Numer. Method. Eng., 93(12), 1255-1285.
  26. Tang, S. (2010), "A two-way interfacial condition for lattice simulations", Adv. Appl. Math. Mech., 2, 45-55.
  27. Born, M. and Huang, K. (1954), Dynamical theory of crystal lattices, Clarendon: Oxford.
  28. Tang, S. (2008), "A finite difference approach with velocity interfacial conditions for multiscale computations of crystalline solids", J. Comput. Phys., 227(8), 4038-4062.
  29. Tang, S., Zhang, L., Ying, Y.P. and Zhang, Y.J., "A finite difference approach for finite temperature multiscale computations", preprint.

Cited by

  1. Eliminating corner effects in square lattice simulation 2017,
  2. Heat jet approach for finite temperature atomic simulations of triangular lattice vol.59, pp.5, 2017,


Supported by : NSFC