Abstract
We investigate the energy transport in a one-dimensional lattice of oscillators with a harmonic nearest neighbor coupling and a harmonic plus quartic on-site potential. As numerically observed for particular coupling parameters before, and confirmed by our study, such chains satisfy Fourier’s law: a chain of length N coupled to thermal reservoirs at both ends has an average steady state energy current proportional to 1/N. On the theoretical level we employ the Peierls transport equation for phonons and note that beyond a mere exchange of labels it admits nondegenerate phonon collisions. These collisions are responsible for a finite heat conductivity. The predictions of kinetic theory are compared with molecular dynamics simulations. In the range of weak anharmonicity, respectively low temperatures, reasonable agreement is observed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Fermi, J. Pasta and S. Ulam, Studies in nonlinear problems, I, in A. C.Newell, ed., Nonlinear Wave Motion, Providence, RI: American Mathematical Society, pp. 143–156 (1974). Originally published as Los Alamos Report LA-1940 in 1955.
Focus Issue: The “Fermi-Pasta-Ulam” problem—the first 50 years, Chaos 15(1): (2005).
F. Bonetto, J. L. Lebowitz and L. Rey-Bellet, Fourier’s law: A challenge to theorists, in A. Fokas, A. Grigoryan, T. Kibble and B. Zegarlinski, eds., Mathematical Physics,London:Imperial College Press, pp. 128–150 (2000).
S. Lepri, R. Livi and A. Politi, Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377:1–80 (2003).
Z. Rieder, J. L. Lebowitz and E. Lieb, Properties of a harmonic crystal in a stationary nonequilibrium state. J. Math. Phys. 8:1073–1078 (1967).
R. J. Rubin and W. L. Greer, Abnormal lattice thermal conductivity of a one-dimensional, harmonic, isotopically disordered crystal. J. Math. Phys. 12:1686–1701 (1971).
A. Casher and J. L. Lebowitz, Heat flow in regular and disordered harmonic chains. J. Math. Phys. 12:1701–1711 (1971).
J. B. Keller, G. C. Papanicolaou and J. Weilenmann, Heat conduction in a one-dimensional random medium. Commun. Pure Appl. Math. 32:583–592 (1978).
A. Dhar, Heat conduction in the disordered harmonic chain revisited. Phys. Rev. Lett. 86:5882–5885 (2001).
S. Lepri, R. Livi and A. Politi, On the anomalous thermal conductivity of one-dimensional lattices. Europhys. Lett. 43:271–276 (1998).
O. Narayan and S. Ramaswamy, Anomalous heat conduction in one-dimensional momentum-conserving systems. Phys. Rev. Lett. 89:200601 (2002).
J. M. Ziman, Electrons and Phonons (Clarendon Press, Oxford, 1960).
V. L. Gurevich, Transport in Phonon Systems (North-Holland, Amsterdam, 1986).
H. Spohn, The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics. J. Stat. Phys. Online First (2006), URL http://dx.doi.org/10.1007/ s10955-005-8088-5 .
T. M. Tritt, ed., Thermal Conductivity: Theory, Properties, and Applications (Physics of Solids and Liquids) (Springer, Berlin, 2005).
R. Lefevere and A. Schenkel, Normal heat conductivity in a strongly pinned chain of anharmonic oscillators. J. Stat. Mech. 2006(2):L02001 (2006).
A. Pereverzev, Fermi-Pasta-Ulam β lattice: Peierls equation and anomalous heat conductivity. Phys. Rev. E 68:056124 (2003).
K. Aoki and D. Kusnezov, Nonequilibrium statistical mechanics of classical lattice φ4 field theory. Ann. Phys. 295:50–80 (2002).
J. Lukkarinen and H. Spohn, in preparation.
S. Nosé, A unified formulation of the constant temperature molecular dynamics methods. J. Chem. Phys. 81:511–519 (1984).
W. G. Hoover, Canonical dynamics: Equilibrium phase-space distributions. Phys. Rev. A 31:1695–1697 (1985).
K. Aoki and D. Kusnezov, Bulk properties of anharmonic chains in strong thermal gradients: Non-equilibrium φ4 field theory. Phys. Lett. A 265:250–256 (2000).
B. Hu, B. Li and H. Zhao, Heat conduction in one-dimensional nonintegrable systems. Phys. Rev. E 61:3828–3831 (2000).
K. Aoki and D. Kusnezov, Violations of local equilibrium and linear response in classical lattice systems. Phys. Lett. A 309:377–381 (2003).
A. J. H. McGaughey and M. Kaviany, Thermal conductivity decomposition and analysis using molecular dynamics simulations. Part I. Lennard-Jones argon. Int. J. Heat Mass Transfer 47:1783–1798 (2004).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Aoki, K., Lukkarinen, J. & Spohn, H. Energy Transport in Weakly Anharmonic Chains. J Stat Phys 124, 1105–1129 (2006). https://doi.org/10.1007/s10955-006-9171-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-006-9171-2