Thermal conduction in classical low-dimensional lattices

By S. Lepri, R. Livi, A. Politi. 2001

1,163 citations recorded by the source corpus.

Abstract 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 non-linear systems is briefly discussed. Finally, possible future research themes are outlined.

Subfields: Atomic and Molecular Physics, and Optics, Materials Chemistry, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/febacc1588f1987b100b2249f6980d4486626d0c

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Citations in the corpus

Thermal properties of graphene and nanostructured carbon materials. (2011, methodology): “In the discussion of heat conduction in CNTs, one has to take into account the ambiguity of the intrinsic K definition for 2D and 1D systems [5-8 Although graphene is structurally simpler, I start with experimental data for CNTs, because their thermal properties have been studied for more than a…”
Dimensional crossover of thermal transport in few-layer graphene. (2010, background): “Many recent theoretical studies [9-13] suggest an emerging consensus that the intrinsic thermal conductivity of 2-D and 1-D anharmonic crystals is anomalous and reveals divergence with the size of the system defined either by the number of atoms N or linear dimension L.”
Colloquium: Phononic thermal properties of two-dimensional materials (2017, background): “Through many intensive numerical simulations (Lepri, Livi, and Politi, 2003, Dhar, 2008, Lepri, Livi, and Politi, 2016) and mathematical proofs (Prosen and Campbell, 2000),1 thermal conductivity is not found to be an intensive property.”
Nonlinear Fluctuating Hydrodynamics for Anharmonic Chains (2013, background): “On this topic there is a huge literature, for which we have to refer to the reviews [4,5].”
Thermal conduction phenomena in carbon nanotubes and related nanostructured materials (2013, background): “Divergence of the thermal conductivity with length was also predicted for low-dimensional lattices (Lepri, Livi, and Politi, 2003; Balandin, 2011) and observed in models for CNTs when only first-order umklapp scattering is considered.”
Non-equilibrium steady states: fluctuations and large deviations of the density and of the current (2007, background): “When the two temperatures Ta and Tb are different, the system reaches in the long time limit a non-equilibrium steady state [1, 2, 3, 4], but there does not exist [5, 6] an expression which generalizes (1) for the steady state weights P (C) of the microscopic configurations.”