### Abstract

Deriving macroscopic phenomenological laws of irreversible thermodynamics from simple microscopic models is one of the tasks of non-equilibrium statistical mechanics. We consider stationary energy transport in crystals with reference to simple mathematical models consisting of coupled oscillators on a lattice. The role of lattice dimensionality on the breakdown of the Fourier's law is discussed and some universal quantitative aspects are emphasized: the divergence of the finite-size thermal conductivity is characterized by universal laws in one and two dimensions. Equilibrium and non-equilibrium molecular dynamics methods are presented along with a critical survey of previous numerical results. Analytical results for the non-equilibrium dynamics can be obtained in the harmonic chain where the role of disorder and localization can be also understood. The traditional kinetic approach, based on the Boltzmann-Peierls equation is also briefly sketched with reference to one-dimensional chains. Simple toy models can be defined in which the conductivity is finite. Anomalous transport in integrable non-linear systems is briefly discussed. Finally, possible future research themes are outlined. (C) 2002 Elsevier Science B.V. All rights reserved.

Original language | English |
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Pages (from-to) | 1-80 |

Number of pages | 80 |

Journal | Physics Reports |

Volume | 377 |

Issue number | 1 |

DOIs | |

Publication status | Published - Apr 2003 |

### Keywords

- thermal conductivity
- classical lattices
- nonequilibrium molecular-dynamics
- disordered harmonic chain
- diatomic toda lattice
- heat-conduction
- energy-transport
- statistical-mechanics
- Hamiltonian-systems
- anharmonic chains
- Fouriers Law
- XY-model

## Cite this

*Physics Reports*,

*377*(1), 1-80. https://doi.org/10.1016/S0370-1573(02)00558-6