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 nonlinear systems is briefly discussed. Finally, possible future research themes are outlined.Comment: 90 pages, revised versio

In one and two dimensions, transport coefficients may diverge in the thermodynamic limit due to long-time correlation of the corresponding currents. The effective asymptotic behaviour is addressed with reference to the problem of heat transport in 1d crystals, modeled by chains of classical nonlinear oscillators. Extensive accurate equilibrium and nonequilibrium numerical simulations confirm that the finite-size thermal conductivity diverges with the system size L as κ ∝ L α . However, the exponent α deviates systematically from the theoretical prediction α = 1/3 proposed in a recent paper [O. Narayan, S. Ramaswamy, Phys. Rev. Lett. 89, 200601 (2002) Strong spatial constraints can significantly alter transport properties. The ultimate reason is that the response to external forces depends on statistical fluctuations which, in turn, crucially depend on the system dimensionality d. A relevant example is the anomalous behaviour of heat conductivity for d ≤ 2. After the publication of the first convincing numerical evidence of a diverging thermal conductivity in anharmonic chains [1], this issue attracted a renovated interest within the theoretical community. A fairly complete overview is given in Ref. [2], where the effects of lattice dimensionality on the breakdown of Fourier's law are discussed as well. Anomalous behaviour means both a nonintegrable algebraic decay of equilibrium correlations of the heat current J(t) (the Green-Kubo integrand) at large times t → ∞ and a divergence of the finite-size conductivity κ(L) in the L → ∞ limit. This is very much reminiscent of the problem of long-time tails in fluids [3] where, in low spatial dimension, transport coefficients may not exist at all, thus implying a breakdown of the phenomenological constitutive laws of hydrodynamics. In 1d one findswhere α > 0, −1 < δ < 0, and is the equilibrium average. For small applied gradients, linear-response theory allows establishing a connection between the two exponents. By assuming that κ(L) can be estimated by cutting-off the integral in the Green-Kubo formula at the "transit time" L/v (v being some propagation velocity of excitations), one obtains κ ∝ L −δ i.e. α = −δ. Determining the asymptotic dependence of heat conductivity is not only important for assessing the universality of this phenomenon, but may be also relevant for predicting transport properties of real materials. For instance, recent molecular dynamics results obtained with phenomenological carbon potentials indicate an unusually high conductivity of single-walled nanotubes [4]: a power-law divergence with the tube length has been observed with an exponent very close to the one obtained in simple 1d models [5].The analysis of several models [2] clarified that anomalous conductivity should occur generically whenever momentum is conserved. For lattice models, this amounts to requiring that at least one acoustic phonon branch is present in the harmonic limit. The only known exception is the coupled-rotor model, where normal transport [6] is believed to arise as a co...

We numerically study heat conduction in chains of nonlinear oscillators with time-reversible thermostats. A nontrivial temperature profile is found to set in, which obeys a simple scaling relation for increasing the number N of particles. The thermal conductivity diverges approximately as N 1͞2 , indicating that chaotic behavior is not enough to ensure the Fourier law. Finally, we show that the microscopic dynamics ensures fulfillment of a macroscopic balance equation for the entropy production. [S0031-9007(97)02611-2]