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 in the
breakdown of 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 non-equilibrium dynamics can be obtained in the
harmonic chain, where the role of disorder and localization can also be
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 nonlinear
systems is briefly discussed. Finally, possible future research themes
are outlined.