Energy transport in weakly anharmonic chains. (English) Zbl 1135.82326
Summary: 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.
MSC:
| 82C70 | Transport processes in time-dependent statistical mechanics |
| 82C20 | Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics |
Keywords:
Fourier’s law; phonon Boltzmann equation; molecular dynamics, one-dimensional lattice dynamicsReferences:
| [1] | Fermi, E.; Pasta, J.; Ulam, S.; Newell, A. C., Studies in nonlinear problems, I, Nonlinear Wave Motion, 143-156 (1974), Providence, RI: American Mathematical Society, Providence, RI · Zbl 0353.70028 |
| [2] | Focus Issue: The “Fermi-Pasta-Ulam” problem—the first 50 years, Chaos15(1): (2005). |
| [3] | 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). · Zbl 1074.82530 |
| [4] | Lepri, S.; Livi, R.; Politi, A., Thermal conduction in classical low-dimensional lattices, Phys. Rep., 377, 1-80 (2003) · doi:10.1016/S0370-1573(02)00558-6 |
| [5] | Rieder, Z.; Lebowitz, J. L.; Lieb, E., Properties of a harmonic crystal in a stationary nonequilibrium state, J. Math. Phys., 8, 1073-1078 (1967) · doi:10.1063/1.1705319 |
| [6] | Rubin, R. J.; Greer, W. L., Abnormal lattice thermal conductivity of a one-dimensional, harmonic, isotopically disordered crystal, J. Math. Phys., 12, 1686-1701 (1971) · doi:10.1063/1.1665793 |
| [7] | Casher, A.; Lebowitz, J. L., Heat flow in regular disordered harmonic chains, J. Math. Phys., 12, 1701-1711 (1971) · doi:10.1063/1.1665794 |
| [8] | Keller, J. B.; Papanicolaou, G. C.; Weilenmann, J., Heat conduction in a one-dimensional random medium, Commun. Pure Appl. Math., 32, 583-592 (1978) · Zbl 0362.35037 |
| [9] | Dhar, A., Heat conduction in the disordered harmonic chain revisited, Phys. Rev. Lett., 86, 5882-5885 (2001) · doi:10.1103/PhysRevLett.86.5882 |
| [10] | Lepri, S.; Livi, R.; Politi, A., On the anomalous thermal conductivity of one-dimensional lattices, Europhys. Lett., 43, 271-276 (1998) · doi:10.1209/epl/i1998-00352-3 |
| [11] | Narayan, O.; Ramaswamy, S., Anomalous heat conduction in one-dimensional momentum-conserving systems, Phys. Rev. Lett., 89, 200601 (2002) · doi:10.1103/PhysRevLett.89.200601 |
| [12] | Ziman, J. M., Electrons and Phonons (1960), Oxford: Clarendon Press, Oxford · Zbl 0983.82002 |
| [13] | Gurevich, V. L., Transport in Phonon Systems (1986), Amsterdam: North-Holland, Amsterdam |
| [14] | H. Spohn, The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics. J. Stat. Phys. Online First (2006), URL doi:10.1007/ . · Zbl 1106.82033 |
| [15] | Tritt, T. M., Thermal Conductivity: Theory, Properties, and Applications (Physics of Solids and Liquids (2005), Berlin: Springer, Berlin |
| [16] | Lefevere, R.; Schenkel, A., Normal heat conductivity in a strongly pinned chain of anharmonic oscillators, J. Stat. Mech., 2006, 2, L02001 (2006) · doi:10.1088/1742-5468/2006/02/L02001 |
| [17] | Pereverzev, A., Fermi-Pasta-Ulam β lattice: Peierls equation and anomalous heat conductivity, Phys. Rev. E, 68, 056124 (2003) · doi:10.1103/PhysRevE.68.056124 |
| [18] | Aoki, K.; Kusnezov, D., Nonequilibrium statistical mechanics of classical lattice φ4 field theory, Ann. Phys., 295, 50-80 (2002) · Zbl 1008.81070 · doi:10.1006/aphy.2001.6207 |
| [19] | J. Lukkarinen and H. Spohn, in preparation. |
| [20] | Nosé, S., A unified formulation of the constant temperature molecular dynamics methods, J. Chem. Phys., 81, 511-519 (1984) · doi:10.1063/1.447334 |
| [21] | Hoover, W. G., Canonical dynamics: Equilibrium phase-space distributions, Phys. Rev. A, 31, 1695-1697 (1985) · doi:10.1103/PhysRevA.31.1695 |
| [22] | Aoki, K.; Kusnezov, D., Bulk properties of anharmonic chains in strong thermal gradients: Non-equilibrium φ4 field theory, Phys. Lett. A, 265, 250-256 (2000) · Zbl 0940.82043 · doi:10.1016/S0375-9601(99)00899-3 |
| [23] | Hu, B.; Li, B.; Zhao, H., Heat conduction in one-dimensional nonintegrable systems, Phys. Rev. E, 61, 3828-3831 (2000) · doi:10.1103/PhysRevE.61.3828 |
| [24] | Aoki, K.; Kusnezov, D., Violations of local equilibrium linear response in classical lattice systems, Phys. Lett. A, 309, 377-381 (2003) · Zbl 1011.82016 · doi:10.1016/S0375-9601(03)00293-7 |
| [25] | McGaughey, A. J. H.; Kaviany, M., Thermal conductivity decomposition analysis using molecular dynamics simulations. Part I. Lennard-Jones argon, Int. J. Heat Mass Transfer, 47, 1783-1798 (2004) · Zbl 1056.80003 · doi:10.1016/j.ijheatmasstransfer.2003.11.002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
