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]

A general method to determine covariant Lyapunov vectors in both discrete-and continuous-time dynamical systems is introduced. This allows us to address fundamental questions such as the degree of hyperbolicity, which can be quantified in terms of the transversality of these intrinsic vectors. For spatially extended systems, the covariant Lyapunov vectors have localization properties and spatial Fourier spectra qualitatively different from those composing the orthonormalized basis obtained in the standard procedure used to calculate the Lyapunov exponents. DOI: 10.1103/PhysRevLett.99.130601 PACS numbers: 05.70.Ln, 05.90.+m, 45.70.ÿn, 87.18.Ed Measuring Lyapunov exponents (LEs) is a central issue in the investigation of chaotic dynamical systems because they are intrinsic observables that allow us to quantify a number of different physical properties such as sensitivity to initial conditions, local entropy production and attractor dimension [1]. Moreover, in the context of spatiotemporal chaos, the very existence of a well-defined Lyapunov spectrum in the thermodynamic limit is a proof of the extensivity of chaos [2], and it has been speculated that the small exponents contain information on the ''hydrodynamic'' modes of the dynamics (e.g., see [3] and references therein).In this latter perspective, a growing interest has been devoted not only to the LEs but also to some corresponding vectors, with the motivation that they could contribute to identifying both the real-space structure of collective modes [4] and the regions characterized by stronger or weaker instabilities [5]. However, the only available approach so far is based on the vectors yielded by the standard procedure used to calculate the LEs [6]. This allows us to identify the most expanding subspaces, but has the drawback that these vectors-that we shall call GramSchmidt vectors (GSV) after the procedure used-are, by construction, orthogonal, even where stable and unstable manifolds are nearly tangent. Moreover, GSV are not invariant under time reversal, and they are not covariant; i.e., the GSV at a given phase-space point are not mapped by the linearized dynamics into the GSV of the forward images of this point.While the existence, for invertible dynamics, of a coordinate-independent, local decomposition of phase space into covariant Lyapunov directions -the so-called Oseledec splitting [1]-has been discussed by Ruelle long ago [7], it received almost no attention in the literature, because of the absence of algorithms to practically determine it. In this Letter, we propose an innovative approach based on both forward and backward iterations of the tangent dynamics, which allows determining a set of directions at each point of phase space that are invariant under time reversal and covariant with the dynamics. We argue that, for any invertible dynamical system, the intrinsic tangent-space decomposition introduced by these covariant Lyapunov vectors (CLV) coincides with the Oseledec splitting.As a first important and general application of the CLV...

The divergence of the thermal conductivity in the thermodynamic limit is thoroughly investigated. The divergence law is consistently determined with two different numerical approaches based on equilibrium and non-equilibrium simulations. A possible explanation in the framework of linear-response theory is also presented, which traces back the physical origin of this anomaly to the slow diffusion of the energy of long-wavelength Fourier modes. Finally, the results of dynamical simulations are compared with the predictions of mode-coupling theory.An extremely idealized, but physically meaningful, model of an insulating solid is a set of N atoms of mass m, arranged on a 1-d lattice with spacing a and coupled by nonlinear forces. Denoting with q l the displacement of the l-th particle from its equilibrium position la, the corresponding Hamiltonian readswhere p l = mq l . If the chain is put in contact at its boundaries with two heat baths at different temperatures T 0 and T 0 + ∆T , a nonequilibrium stationary state arises, characterized by a non vanishing average heat flux J. The microscopic definition of J is [1,2]where f l = −V ′ (q l − q l−1 ) is the interaction force and j l represents the local flux at site l (i.e. the sum of the fluxes of potential energy from its neighbours). The lattice thermal conductivity κ can be defined through the Fourier law,where x = la is the coordinate along the chain and · denotes a time average in the stationary regime.

We discuss the thermal conductivity of a chain of coupled rotators, showing that it is the first example of a 1d nonlinear lattice exhibiting normal transport properties in the absence of an on-site potential. Numerical estimates obtained by simulating a chain in contact with two thermal baths at different temperatures are found to be consistent with those ones based on linear response theory. The dynamics of the Fourier modes provides direct evidence of energy diffusion. The finiteness of the conductivity is traced back to the occurrence of phase-jumps. Our conclusions are confirmed by the analysis of two variants of this model.

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