Heat transport in low-dimensional systems

Abstract

Recent results on theoretical studies of heat conduction in low-dimensional systems are presented. These studies are on simple, yet nontrivial, models. Most of these are classical systems, but some quantum-mechanical work is also reported. Much of the work has been on lattice models corresponding to phononic systems, and some on hard particle and hard disc systems. A recently developed approach, using generalized Langevin equations and phonon Green's functions, is explained and several applications to harmonic systems are given. For interacting systems, various analytic approaches based on the Green-Kubo formula are described, and their predictions are compared with the latest results from simulation. These results indicate that for momentum-conserving systems, transport is anomalous in one and two dimensions, and the thermal conductivity κ, diverges with system size L, as κ ∼ L α . For one dimensional interacting systems there is strong numerical evidence for a universal exponent α = 1/3, but there is no exact proof for this so far. A brief discussion of some of the experiments on heat conduction in nanowires and nanotubes is also given. IntroductionIt is now about two hundred years since Fourier first proposed the law of heat conduction that goes by his name. Consider a macroscopic system subjected to different temperatures at its boundaries. One assumes that it is possible to have a * Corresponding author. Email: dabhi@rri.res.in November 19, 2008 19:56 Advances in Physics reva 2 coarse-grained description with a clear separation between microscopic and macroscopic scales. If this is achieved, it is then possible to define, at any spatial point x in the system and at time t, a local temperature field T (x, t) which varies slowly both in space and time (compared to microscopic scales). One then expects heat currents to flow inside the system and Fourier argued that the local heat current density J(x, t) is given bywhere κ is the thermal conductivity of the system. If u(x, t) represents the local energy density then this satisfies the continuity equation ∂u/∂t + ∇.J = 0. Using the relation ∂u/∂T = c, where c is the specific heat per unit volume, leads to the heat diffusion equation:Thus, Fourier's law implies diffusive transfer of energy. In terms of a microscopic picture, this can be understood in terms of the motion of the heat carriers, i.e. , molecules, electrons, lattice vibrations(phonons), etc., which suffer random collisions and hence move diffusively. Fourier's law is a phenomological law and has been enormously succesful in providing an accurate description of heat transport phenomena as observed in experimental systems. However there is no rigorous derivation of this law starting from a microscopic Hamiltonian description and this basic question has motivated a large number of studies on heat conduction in model systems. One important and somewhat surprising conclusion that emerges from these studies is that Fourier's law is probably not valid in one and two dimensional systems, except when the ...

From Thermal Rectifiers to Thermoelectric Devices

“…The advantage of such ap-proach lies in its simplicity and independence from microscopic details of inelastic processes. Probe terminals have been widely used in the literature and proved to be useful to unveil nontrivial aspects of phase-breaking processes [49], heat transport and rectification [22,23,118,52,24,112,13,120], and thermoelectric transport [73,57,56,75,76,124,126,135,136,122,67,15,11,27,25]. The approach can be generalized to any number n p of probe reservoirs.…”

“…As a consequence, ZT exhibits a rapid, liner growth with the system size. While the Fourier-like regime might be an intermediate (in the system size) regime, followed by an asymptotic regime of anomalous thermal conductivity κ ∼ Λ 1/3 [83,52], the range of validity of such regime may expand rapidly as an integrable limit is approached [42]. We point out that it is a priori not excluded that there exist models where the long-time limit t → ∞ and the thermodynamical limit Λ → ∞ do not commute when computing the Drude weights.…”

Nonequilibrium stationary state of a harmonic crystal with alternating masses

“…The study of non-equilibrium steady states (NESS) of macroscopic systems in contact with heat baths at different temperatures has a long history [1][2][3]. There are no known analytic solutions for interacting Hamiltonian systems, except for harmonic crystals.…”

Enhancement and reduction of one-dimensional heat conduction with correlated mass disorder

“…[16,17] In one dimension (1D) in particular, disorder results in the localization of all the modes in in the thermodynamic (L → ∞) limit [18][19][20] although for a finite system, a fraction of the states would always be sufficiently extended to contribute to the heat current. [21] Valuable insights into the interplay between localization and propagation in 1D can be gleaned within the framework of the disordered harmonic chain (DHC), the simplest model of a disordered 1D structure that has been frequently used to understand the effect of disorder on heat conduction. [20,22,23] Broadly speaking, we know that disorder leads to the exponential attenuation of the phonon transmittance Ξ, i.e., Ξ(ω, L) ∼ exp[−L/λ(ω)], where ω and λ are the frequency and attenuation length, respectively, in analogy to the Beer-Lambert law in optics, [23] and is a direct consequence of localization.…”