Transport Properties of a Chain of Anharmonic Oscillators with Random Flip of Velocities

We discuss, in the context of energy flow in high-dimensional systems and Kolmogorov-Arnol'd-Moser (KAM) theory, the behavior of a chain of rotators (rotors) which is purely Hamiltonian, apart from dissipation at just one end. We derive bounds on the dissipation rate which become arbitrarily small in certain physical regimes, and we present numerical evidence that these bounds are sharp. We relate this to the decoupling of non-resonant terms as is known in KAM problems.

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“…We are now ready for the inductive step. We start with 6) and where f (j−1) depends only on the sites k − j, . .…”

Section: Canonical Transformations For J >mentioning

confidence: 99%

Energy dissipation in Hamiltonian chains of rotators

2017

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“…where Φ i = (1/m)Trp i J i is the energy flux from reservoir i to the chain. Using formula (7) we get the evolution equation for the correlations among the coordinates and momenta, (12) wherez ij = p i q j , and G ij are the elements of a tridiago-…”

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confidence: 99%

Heat transport along a chain of coupled quantum harmonic oscillators

Oliveira

2017

Phys. Rev. E

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We study the heat transport properties of a chain of coupled quantum harmonic oscillators in contact at its ends with two heat reservoirs at distinct temperatures. Our approach is based on the use of an evolution equation for the density operator which is a canonical quantization of the classical Fokker-Planck-Kramers equation. We set up the evolution equation for the covariances and obtain the stationary covariances at the stationary states from which we determine the thermal conductance in closed form when the interparticle interaction is small. The conductance is finite in the thermodynamic limit implying an infinite thermal conductivity.Fifty years ago, Rieder, Lebowitz, and Lieb [1] introduced and exactly solved a microscopic model for thermal conduction, that consisted of a chain of coupled classic harmonic oscillators with its ends in contact with heat reservoirs at distinct temperatures. Using this model they provided a rigorous proof of the well known result that the thermal conductance, the ratio between the heat current and the temperature difference, is finite regardless of the chain length [2]. This result amounts to say that Fourier's law does not hold because the conductivity, which is the product of the conductance and the chain length, becomes infinite when the length increases without bounds. The reason for the occurrence of a finite conductance is that the excitations in ordered systems with harmonic interactions travel balistically. To get the Fourier's law, new ingredients should be added to the harmonic model in order to transform the ballistic into a diffusive motion. Such ingredients include anharmonic potentials [3][4][5][6][7][8], self-consistent reservoirs [9][10][11][12], energy conserving noise [13][14][15], and others [16][17][18][19][20].In the model studied by Rieder, Lebowitz, and Lieb [1], the oscillators were under the action of conservative forces except the first and the last which in addition were subject to dissipating-fluctuating forces representing the contact with heat reservoirs. They are composed by a dissipative force, proportional to the velocity, and a Gaussian white noise with zero mean and variance proportional to the temperature. The equations of motion are understood as Langevin equations, and the equation governing the time evolution of the probability density is a Fokker-Planck-Kramers (FPK) equation [21][22][23][24].Here, we study a quantum version of the model studied by Rieder, Lebowit, and Lieb [1]. We have exactly calculated the thermal conductance in the regime of small interparticle interaction and reached a similar result that the conductance is finite regardless of the length of the chain. However, as should be expected the conductance is not independent of temperature, as is the case of the classical version. It vanishes in the limit of zero temperature and saturates at the classical value at high temperatures. Our approach is based on a quantum version of the FPK equation coming from a canonical quantization of the ordinary FPK equation, recently in...

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“…In [19], Fröhlich and the first author used the techniques of [32] to study diffusion for a lattice particle governed by a Lindblad equation describing jumps in momentum driven by interaction with a heat bath. In some sense, this is the quantum analogue of the classical dynamics of a disordered oscillator systems perturbed by noise in the form of a momentum jump process, considered in [3,4] and reviewed in [5]. In those works, heat transport is considered in the limit of weak noise in a regime for which transport is known to vanish for the disordered oscillator system without noise.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…In those works, heat transport is considered in the limit of weak noise in a regime for which transport is known to vanish for the disordered oscillator system without noise. A key feature of the noise in [3,4] is that energy is conserved in the system with noise; this is necessary so that one can speak about heat flux. By contrast, in the present work, and in [23,27,32,19], energy conservation is broken by the noise.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Diffusion in the Mean for a Periodic Schrödinger Equation Perturbed by a Fluctuating Potential

Schenker

Tilocco

Zhang

2020

Commun. Math. Phys.

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We consider the evolution of a quantum particle hopping on a cubic lattice in any dimension and subject to a potential consisting of a periodic part and a random part that fluctuates stochastically in time. If the random potential evolves according to a stationary Markov process, we obtain diffusive scaling for moments of the position displacement, with a diffusion constant that grows as the inverse square of the disorder strength at weak coupling. More generally, we show that a central limit theorem holds such that the square amplitude of the wave packet converges, after diffusive rescaling, to a solution of a heat equation.

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Transport Properties of a Chain of Anharmonic Oscillators with Random Flip of Velocities

Cited by 42 publications

References 15 publications

Energy dissipation in Hamiltonian chains of rotators

Energy dissipation in Hamiltonian chains of rotators

Heat transport along a chain of coupled quantum harmonic oscillators

Diffusion in the Mean for a Periodic Schrödinger Equation Perturbed by a Fluctuating Potential

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