Solitons and Heat Transfer in Nonlinear Lattices with Cubic On-Site and Quartic Interaction Potentials


This paper deals with the transfer of soliton-like heat waves in nonlinear lattices with cubic on-site potential and quartic interparticle interaction potential. A model Hamiltonian was proposed using the second quantized operators and the same was averaged using a suitable wavefunction. The equations were derived numerically in the discrete form for the field amplitude. Moreover the resulting equations were analyzed analytically using the continuous approximation technique and the properties of heat transfer were examined theoretically.

Share and Cite:

Perseus, R. and Latha, M. (2014) Solitons and Heat Transfer in Nonlinear Lattices with Cubic On-Site and Quartic Interaction Potentials. Open Access Library Journal, 1, 1-10. doi: 10.4236/oalib.1100822.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Saito, K. (2003) Strong Evidence of Normal Heat Conduction in a One-Dimensional Quantum System. Europhysics Letters, 61, 34.
[2] Zurcher, U. and Talkner, P. (1900) Quantum-Mechanical Harmonic Chain Attached to Heat Baths. II. Nonequilibrium Properties. Physical Review A, 42, 3278.
[3] Roy, D. and Dhar, A. (2008) Role of Pinning Potentials in Heat Transport through Disordered Harmonic Chains. Physical Review E, 78, Article ID: 051112.
[4] O’Connor, A.J. and Lebowitz, J.L. (1974) Heat Conduction and Sound Transmission in Isotopically Disordered Harmonic Crystals. Journal of Mathematical Physics, 15, 692.
[5] Eckmann, J.P., Pillet, C.A. and Rey-Bellet L. (1999) Non-Equilibrium Statistical Mechanics of Anharmonic Chain Coupled to Two Heat Baths at Different Temperatures. Communications in Mathematical Physics, 201, 657.
[6] Verheggen, T. (1979) Transmission Coefficient and Heat Conduction of a Harmonic Chain with Random Masses: Asymptotic Estimates on Products of Random Matrices. Communications in Mathematical Physics, 68, 69.
[7] Dhar, A. (2001) Comment on “Can Disorder Induce a Finite Thermal Conductivity in 1D Lattices?” Physical Review Letters, 87, Article ID: 069401.
[8] Mejia-Monasterio, C., Larralde, H. and Leyvraz, F. (2001) Coupled Normal Heat and Matter Transport in a Simple Model System. Physical Review Letters, 86, 5417.s
[9] Li, B., Wang, J. and Casati, G. (2003) Heat Conductivity in Linear Mixing Systems. Physical Review E, 67, 021204.
[10] Dhar, A. and Roy, D. (2006) Heat Transport in Harmonic Lattices. Journal of Statistical Physics, 125, 801-820.
[11] Li, B.W., Wang, L. and Hu, B. (2002) Finite Thermal Conductivity in 1D Models Having Zero Lyapunov Exponents. Physical Review Letters, 88, Article ID: 223901.
[12] Nakazawa, H. (1968) Energy Flow in Harmonic Linear Chain. Progress of Theoretical Physics, 39, 236-238.
[13] Li, B.W., Zhao, H. and Hu, B. (2001) Can Disorder Induce a Finite Thermal Conductivity in 1D Lattices? Physical Review Letters, 86, 63.
[14] Mountain, R.D. and MacDonald, R.A. (1983) Thermal Conductivity of Crystals: A Molecular-Dynamics Study of Heat Flow in a Two-Dimensional Crystal. Physical Review B, 28, 3022.
[15] Jackson, E.A. and Mistriotis, A.D. (1989) Thermal Conductivity of One- and Two-Dimensional Lattices. Journal of Physics: Condensed Matter, 1, 1223.
[16] Dhar, A. (2001) Heat Conduction in the Disordered Harmonic Chain Revisited. Physical Review Letters, 86, 5882.
[17] Lepri, S., Livi, R. and Politi, A. (1997) Heat Conduction in Chains of Nonlinear Oscillators. Physical Review Letters, 78, 1896.
[18] Rubin, R.J. and Greer, W.L. (1971) Abnormal Lattice Thermal Conductivity of a One-Dimensional, Harmonic, Isotopically Disordered Crystal. Journal of Mathematical Physics, 12, 1686.
[19] Shiba, H. and Ito, N. (2008) Anomalous Heat Conduction in Three-Dimensional Nonlinear Lattices. Journal of the Physical Society of Japan, 77, Article ID: 054006.
[20] Bourbonnais, R. and Maynard, R. (1990) Energy Transport in One- and Two-Dimensional Anharmonic Lattices with Isotopic Disorder. Physical Review Letters, 64, 1397.
[21] Terraneo, M., Peyrard, M. and Casati, G. (2002) Controlling the Energy Flow in Nonlinear Lattices: A Model for a Thermal Rectifier. Physical Review Letters, 88, Article ID: 094302.
[22] Li, B.W., Wang, L. and Casati, G. (2006) Negative Differential Thermal Resistance and Thermal Transistor. Applied Physics Letters, 88, Article ID: 143501.
[23] Gaul, C. and Büttner, H. (2007) Quantum Mechanical Heat Transport in Disordered Harmonic Chains. Physical Review E, 76, Article ID: 011111.
[24] Segal, D. and Nitzan, A. (2005) Spin-Boson Thermal Rectifier. Physical Review Letters, 94, Article ID: 034301.
[25] Li, B.W., Wang, L. and Casati, G. (2004) Thermal Diode: Rectification of Heat Flux. Physical Review Letters, 93, Article ID: 184301.
[26] Toda, M. (1979) Solitons and Heat Conduction. Physica Scripta, 20, 424.
[27] Dhar, A. (2008) Heat Transport in Low-Dimensional Systems. Advances in Physics, 57, 457-537.
[28] Wang, J.S. (2007) Quantum Thermal Transport from Classical Molecular Dynamics. Physical Review Letters, 99, Article ID: 160601.
[29] Stock, G. (2009) Classical Simulation of Quantum Energy Flow in Biomolecules. Physical Review Letters, 102, Article ID: 118301.
[30] Wu, L.A. and Segal, D. (2011) Quantum Heat Transfer: A Born-Oppenheimer Method. Physical Review E, 83, Article ID: 051114.
[31] Imai, H., Wada, H. and Shiga, M. (1995) Effects of Spin Fluctuations on the Specific-Heat in YMN2 and Y0.97SC-0.03MN2. Journal of the Physical Society of Japan, 64, 2198.
[32] Theodorakopoulos, N. and Bacalis, N.C. (1992) Thermal Solitons in the Toda Chain. Physical Review B, 46, 10706.
[33] Takayama, H. and Ishikawa, M. (1986) Classical Thermodynamics of the Toda Lattice as a Classical Limit of the Two-Component Bethe Ansatz Scheme. Progress of Theoretical Physics, 76, 820.
[34] Theodorakopoulos, N. (1984) Ideal-Gas Approach to the Statistical Mechanics of Integrable Systems: The Sine-Gordon Case. Physical Review B, 30, 4071.
[35] Majernik, V. and Majernikova, E. (1995) The Possibility of Thermal Solitons. International Journal of Heat and Mass Transfer, 38, 2701-2703.
[36] Li, N.B., Zhan, F., Hanggi, P. and Li, B.W. (2009) Shuttling Heat across One-Dimensional Homogenous Nonlinear Lattices with a Brownian Heat Motor. Physical Review E, 80, Article ID: 011125.
[37] Dhar, A. (2008) Heat Transport in Low-Dimensional Systems. Advances in Physics, 57, 5.
[38] Stefano, L., Livi, R. and Politi, A. (2003) Thermal Conduction in Classical Low-Dimensional Lattices. Arxiv: Condmat/0112193v2.
[39] Radcliffe, J. M. (1971) Some Properties of Coherent Spin States. Journal of Physics A: General Physics, 4, 313.
[40] Baldwin, D., Goklas, U. and Hereman, W. (2004) Symbolic Computation of Hyperbolic Tangent Solutions for Nonlinear Differential-Difference Equations. Computer Physics Communications, 162, 203-217.

Copyright © 2022 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.