Anomalous energy transport in the FPU‐β chain

Lukkarinen, Jani; Spohn, Herbert

doi:10.1002/cpa.20243

Cited by 68 publications

(102 citation statements)

References 12 publications

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“…Without the stochastic shot noise, this is the well known Fermi-Pasta-Ulam model in contact with heat reservoirs at its ends, which was studied numerically by Lepri et al [2] who found a superdiffusive transport of heat implying an anomalous Fourier's law. This important result has been confirmed by other numerical studies and other approaches [4,8,12,14,15,17,18,23,29] The impulsive stochastic shot noise that we use here can be regarded as a procedure that turns the superdiffusive transport of heat into a diffuse transport leading thus to Fourier's law.…”

Section: Introductionsupporting

confidence: 77%

Fourier's law from a chain of coupled anharmonic oscillators under energy-conserving noise

Landi

Oliveira

2013

Phys. Rev. E

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We analyze the transport of heat along a chain of particles interacting through anharmonic potentials consisting of quartic terms in addition to harmonic quadratic terms and subject to heat reservoirs at its ends. Each particle is also subject to an impulsive shot noise with exponentially distributed waiting times whose effect is to change the sign of its velocity, thus conserving the energy of the chain. We show that the introduction of this energy conserving stochastic noise leads to Fourier's law. The behavior of thels heat conductivity for small intensities of the shot noise and large system sizes are found to obey a finite-size scaling relation. We also show that the heat conductivity is not constant but is an increasing monotonic function of temperature.

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Section: Introductionsupporting

confidence: 77%

Fourier's law from a chain of coupled anharmonic oscillators under energy-conserving noise

Landi

Oliveira

2013

Phys. Rev. E

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“…In fact, the arguments given at the beginning of this paragraph directly give this (putting a = 5/3), and one does not need to find C(t). The kinetic theory approach has been made more rigorous by the work of Lukkarinen and Spohn [96]. They also work with the linearized collision operator and make the relaxation time approximation, and for the quartic FPU chain they confirm the result in [95], namely C(t) ∼ t −3/5 .…”

Section: Kinetic and Peierls-boltzmann Theorymentioning

confidence: 90%

Heat transport in low-dimensional systems

Dhar

2008

Advances in Physics

908

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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“…In [2], [24], [20] [29] a kinetic limit is performed for chains of an-harmonic oscillators, and in [23] a linear Boltzmann equation is rigorously derived for the harmonic chain of oscillators with random masses. In [5] the authors consider a system of harmonic oscillators in ∂ t u α (t, r,k) + v(k) · ∇u α (t, r, k)…”

Section: Introductionmentioning

confidence: 99%