On the Lattice Thermal Conduction

Nakazawa, Hiroshi

doi:10.1143/ptps.45.231

Cited by 80 publications

(105 citation statements)

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“…They found that the system had an infinite conductivity and a constant temperature profile away from the ends, results later generalized to the higher dimensional case by Nakazawa. (6) In this paper, we provide a rigorous proof that the steady state of the self-consistent system has indeed the properties found by BRV for the d=1 case in refs. 3 and 4, and we extend the results to cover all d \ 1.…”

Section: Introductionsupporting

confidence: 64%

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Fourier's Law for a Harmonic Crystal with Self-Consistent Stochastic Reservoirs

Bonetto

Lebowitz

Lukkarinen

2004

Journal of Statistical Physics

169

237

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We consider a d-dimensional harmonic crystal in contact with a stochastic Langevin type heat bath at each site. The temperatures of the ''exterior'' left and right heat baths are at specified values T L and T R , respectively, while the temperatures of the ''interior'' baths are chosen self-consistently so that there is no average flux of energy between them and the system in the steady state. We prove that this requirement uniquely fixes the temperatures and the self consistent system has a unique steady state. For the infinite system this state is one of local thermal equilibrium. The corresponding heat current satisfies Fourier's law with a finite positive thermal conductivity which can also be computed using the Green-Kubo formula. For the harmonic chain (d=1) the conductivity agrees with the expression obtained by Bolsterli, Rich, and Visscher in 1970 who first studied this model. In the other limit, d ± 1, the stationary infinite volume heat conductivity behaves as (a d d) −1 where a d is the coupling to the intermediate reservoirs. We also analyze the effect of having a non-uniform distribution of the heat bath couplings. These results are proven rigorously by controlling the behavior of the correlations in the thermodynamic limit.

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Section: Introductionsupporting

confidence: 64%

“…6, we can Fourier transform this system in the periodic direction and obtain a system of decoupled chains. More precisely, let for k=1,..., NOE…”

Section: Higher Dimensionsmentioning

confidence: 99%

Fourier's Law for a Harmonic Crystal with Self-Consistent Stochastic Reservoirs

Bonetto

Lebowitz

Lukkarinen

2004

Journal of Statistical Physics

169

237

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“…(2) However, the space decay of the limit position-momentum covariance in ref. 20 is exponential which differs from the power decay in our problem (see Remark 4.2(iii)). Therefore, the equilibrium measures are distinct.…”

Section: Introductionmentioning

confidence: 80%

On Two-Temperature Problem for Harmonic Crystals

Dudnikova

Komech

Mauser³

2004

Journal of Statistical Physics

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We consider the dynamics of a harmonic crystal in d dimensions with n components, d, n \ 1. The initial date is a random function with finite mean density of the energy which also satisfies a Rosenblatt-or Ibragimov-Linnik-type mixing condition. The random function is translation-invariant in x 1 ,..., x d − 1 and converges to different translation-invariant processes as x d Q ± ., with the distributions m ± . We study the distribution m t of the solution at time t ¥ R. The main result is the convergence of m t to a Gaussian translation-invariant measure as t Q .. The proof is based on the long time asymptotics of the Green function and on Bernstein's ''room-corridor'' argument. The application to the case of the Gibbs measures m ± =g ± with two different temperatures T ± is given. Limiting mean energy current density is − (0,..., 0, C(T + − T − )) with some positive constant C > 0 what corresponds to Second Law.

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“…There are no known analytic solutions for interacting Hamiltonian systems, except for harmonic crystals. When the atoms in the crystal all have the same mass, this NESS can be obtained explicitly [4,5]. It gives a uniform "temperature", i.e p 2 j /m, in the bulk of the system.…”

Section: Introductionmentioning

confidence: 99%

Nonequilibrium stationary state of a harmonic crystal with alternating masses

2012

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We analyze the non-equilibrium steady states (NESS) of a one dimensional harmonic chain of N atoms with alternating masses connected to heat reservoirs at unequal temperatures. We find that the temperature profile defined through the local kinetic energy T (j) ≡ < p 2 j >/m j , oscillates with period two in the bulk of the system. Depending on boundary conditions, either the heavier or the lighter particles in the bulk are hotter. We obtain exact expressions for the bulk temperature profile and steady state current in the limit N → ∞. These depend on whether N is odd or even.We also study similar temperature oscillations in the NESS of systems with noise in the dynamics.These die out as N → ∞.

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On the Lattice Thermal Conduction

Cited by 80 publications

References 0 publications

Fourier's Law for a Harmonic Crystal with Self-Consistent Stochastic Reservoirs

Fourier's Law for a Harmonic Crystal with Self-Consistent Stochastic Reservoirs

On Two-Temperature Problem for Harmonic Crystals

Nonequilibrium stationary state of a harmonic crystal with alternating masses

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