Anomalous Heat Conduction in Three-Dimensional Nonlinear Lattices

Shiba, Hayato; Ito, Nobuyasu

doi:10.1143/jpsj.77.054006

Cited by 30 publications

(28 citation statements)

References 45 publications

(111 reference statements)

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“…I am also grateful to J. Deutsche and O. Narayan for permission in using data from their paper [111] and H. Shiba and N. Ito for using data from their paper [170]. Finally, I thank a large number of colleagues who read the first draft of this review and made valuable suggestions towards improving it.…”

Section: Acknowledgements Referencesmentioning

confidence: 95%

“…However one of the authors of the paper has expressed doubts about whether convergence has been attained at these sizes [168] and this seems very likely to be the case. The most recent simulations by Shiba and Ito [170] considered the same Hamiltonian as in Eq. (120) and used the same parameter set as [165], namely k 4 = 0.1, T L = 20, T R = 10.…”

Section: Interacting Systems In Two Dimensionsmentioning

confidence: 99%

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Heat transport in low-dimensional systems

Dhar

2008

Advances in Physics

926

1,114

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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 ...

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Section: Acknowledgements Referencesmentioning

confidence: 95%

Section: Interacting Systems In Two Dimensionsmentioning

confidence: 99%

Heat transport in low-dimensional systems

Dhar

2008

Advances in Physics

926

1,114

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“…For this sake, we simulate the relaxation for different wavenumbers k for φ 4 + on-site potential (16). The results are presented in Fig.…”

Section: Standard Evaluation Of Integralmentioning

confidence: 99%

“…The classic solution for the problem suggested by Peierls [3] was questioned after seminal numeric experiment of Fermi, Pasta and Ulam [4]; the latter has disproved common belief concerning fast thermalization and mixing in nonintegrable systems with weak nonlinearity. Despite large amount of work done, necessary and sufficient conditions for microscopic model of solid to obey macroscopically the Fourier law with finite and size -independent heat conduction coefficient [4][5][6][7][8][9][10][11][12][13][14][15][16] are not known yet. Numerous anomalies in the heat transfer in microscopic models of dielectrics were revealed by means of direct numeric simulation over recent years, including qualitatively different behavior of models of different types (with and without on-site potential) and dimensionality [1].…”

Section: Introductionmentioning

confidence: 99%

Nonstationary heat conduction in one-dimensional models with substrate potential

et al. 2012

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The paper investigates non-stationary heat conduction in one-dimensional models with substrate potential. In order to establish universal characteristic properties of the process, we explore three different models -Frenkel-Kontorova (FK), phi4+ (φ 4 +) and phi4-(φ 4 −). Direct numeric simulations reveal in all these models a crossover from oscillatory decay of short-wave perturbations of the temperature field to smooth diffusive decay of the long-wave perturbations. Such behavior is inconsistent with parabolic Fourier equation of the heat conduction and clearly demonstrates the necessity of hyperbolic models. The crossover wavelength decreases with increase of average temperature. The decay patterns of the temperature field almost do not depend on the amplitude of the perturbations, so the use of linear evolution equations for temperature field is justified. In all model investigated, the relaxation of thermal perturbations is exponential -contrary to linear chain, where it follows a power law. However, the most popular lowest-order hyperbolic generalization of the Fourier law, known as Cattaneo-Vernotte (CV) or telegraph equation (TE) is not valid for description of the observed behavior of the models with on-site potential. In part of the models a characteristic relaxation times exhibit peculiar scaling with respect to the temperature perturbation wavelength. Quite surprisingly, such behavior is similar to that of well-known model with divergent heat conduction (Fermi-Pasta-Ulam chain) and rather different from the model with normal heat conduction (chain of rotators). Thus, the data on the non-stationary heat conduction in the systems with on-site potential do not fit commonly accepted concept of universality classes for heat conduction in one-dimensional models.

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“…1). In our recent paper, 5) we have shown with nonequilibirium simulations that the FPU-β model (2 . 1) with the fcc structure has diverging thermal conductivity with the system size dependence of κ(N z ) ∼ N 0.51(1) z , which means α ∼ 0.49(1).…”

Section: Relation To the Nonequilibrium Simulation Resultsmentioning

confidence: 94%

Long-Time Tail Problem and Anomalous Transport in Three-Dimensional Nonlinear Lattices

Shiba

Ito

2009

Prog. Theor. Phys. Suppl.

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Equilibrium autocorrelation functions of heat flux in a model of three-dimensional insulating solids are investigated through nonequilibrium particle dynamics simulations. We employ the Fermi-Pasta-Ulam(FPU)-β nonlinear lattice in which there exist only nearestneighbor interactions with harmonic and bi-quadratic nonlinear terms. In an fcc lattice of the size of 192 3 , power-law decay of the equilibrium autocorrelation function, which is often referred to as a long-time tail, is confirmed, but its decay exponent has turned out to be 0.74(1) which is different from 1.5 expected from hydrodynamic phenomenology for threedimensional systems. This value, 0.74(1), of a decay exponent implies that thermal conductivity of this system diverges with size. Finite size effects are studied using one-dimensional FPU-β lattices. §1. IntroductionAround 1955, Fermi, Pasta, and Ulam opened a new era of nonequilibrium statistical mechanics 1) using the MANIAC 1 computer. The model they have used is called Fermi-Pasta-Ulam (FPU) nonlinear chains which are coupled oscillators whose interaction potential has a form of cubic or bi-quadratic terms in addition to harmonic one. Although the model is nonlinear, they have found recurrence of the normal modes, which suggests that the irreversibility might not exist in this model system.Since then, about 60 years have passed, and the above result has generated new concepts and fields like solitons and integrable systems. However, macroscopic irreversible behaviors such as heat conduction in nonlinear lattice systems have not been fully understood yet.Recently, heat conduction in nonlinear oscillator chains has been investigated numerically. 2) It has been found clearly that heat conductivity shows diverging tendency in the low-dimensional cases where the total momentum is conserved. Such divergence may originate in the long-time tail of the autocorrelation functions, J(t)J(0) ∼ t −d/2 , analogous to the case in fluids which was first found by Alder et al. 3) in hard-core systems.It has been implicitly supposed that thermal conductivity would converge in the three-dimensional cases. However, no work has confirmed this conjecture. Recently, the authors have performed large-scale nonequilibrium molecular dynamics simulations, and observed divergence of thermal conductivity in three-dimensional nonlinear lattices. 4), 5) The divergence appears up to the system size of 128×128×256 lattice sites in the simple cubic lattice. We postulated that this behavior originates in long-time tails in the unconventional autocorrelation function of heat flux. In this paper, taking the data of the tail directly from numerical simulations, we argue the by guest on June 24, 2016 http://ptps.oxfordjournals.org/ Downloaded from

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Anomalous Heat Conduction in Three-Dimensional Nonlinear Lattices

Cited by 30 publications

References 45 publications

Heat transport in low-dimensional systems

Heat transport in low-dimensional systems

Nonstationary heat conduction in one-dimensional models with substrate potential

Long-Time Tail Problem and Anomalous Transport in Three-Dimensional Nonlinear Lattices

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