On the anomalous thermal conductivity of one-dimensional lattices

Abstract

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.

Energy Transport in Weakly Anharmonic Chains

“…The speed of phonon propagation can also be measured directly from the time and space dependences of the autocorrelation function J 0,1 (0)J j,j+1 (t) in thermal equilibrium [9,16]. The velocity of the peaks in the correlation function is equated with the average phonon velocity relevant for thermal transport.…”

“…Thus the average energy current depends on the precise spectral statistics of the thermal reservoir [6,7]. For anharmonic chains there are attempts to predict the exponent for the anomalous heat conduction through mode-coupling theory [8,9].…”

Dynamics of Oscillator Chains

“…isotropic, manifold (2.113) reduces to a constant 114) where N is the number of degrees of freedom and R is the scalar curvature. For a congruence of geodesics {γ τ (s) = γ(s, τ ) | τ ∈ R} issuing from a neighborhood I of a point of a manifold [for more details see [142]], dependent on the parameter τ , fixing a reference geodesicγ(s, τ 0 ), ifγ(s) is the vector field tangent toγ in s, and J(s) the vector field tangent in τ 0 to the curves γ s (τ ) for a fixed s, then the evolution of J contains the information on the stability (or instability) of the reference geodesicγ; if |J| grows exponentially, then the geodesic will be unstable in the Lyapunov sense, otherwise it will be stable.…”

“…The increase in computer power led to a revival of the heat conduction problem inbetween the mid-1980s and the mid-1990s, when nonequilibrium simulations of the FPU model [106,107] and of the diatomic Toda chain [108,109,110,111] of alternating light and heavy masses were performed. Subsenquently, there were systematic studies on the size dependence of the heat conductivity for the FPU chain with quartic [112,113,114] or cubic [115] nonlinear potential as well as for the diatomic Toda chain [116,117]. They indicated a divergence of the heat conductivity with N , the number of mass points, which was interpreted as due to ballistic transport of energy through the chain.…”

“…This can be understood by, for example looking at the spatio-temporal correlation function C(i, t) = j i (t)j 0 (0) of the local heat flux plotted in Fig. 2.26 (see [114]). Accordingly, one can turn the time divergence of κ as determined from the Green-Kubo formula into a divergence with N by restricting the integral in formula (2.129) to times smaller than the "transit time" N a/v s .…”