High Frequency Limit for a Chain of Harmonic Oscillators with a Point Langevin Thermostat

@article{Komorowski2018HighFL,
  title={High Frequency Limit for a Chain of Harmonic Oscillators with a Point Langevin Thermostat},
  author={Tomasz Komorowski and Stefano Olla and Lenya Ryzhik and Herbert Spohn},
  journal={Archive for Rational Mechanics and Analysis},
  year={2018},
  volume={237},
  pages={497-543}
}
We consider an infinite chain of coupled harmonic oscillators with a Langevin thermostat at the origin. In the high frequency limit, we establish the reflection-transmission coefficients for the wave energy for the scattering off the thermostat. To our surprise, even though the thermostat fluctuations are time-dependent, the scattering does not couple wave energy at various frequencies. 
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References

SHOWING 1-10 OF 32 REFERENCES
Kinetic limit for a chain of harmonic oscillators with a point Langevin thermostat
Abstract We consider an infinite chain of coupled harmonic oscillators whose Hamiltonian dynamics is perturbed by a random exchange of momentum between particles such that total energy and momentumExpand
Asymptotics of the Solutions of the Stochastic Lattice Wave Equation
We consider the long time limit for the solutions of a discrete wave equation with weak stochastic forcing. The multiplicative noise conserves energy, and in the unpinned case also conservesExpand
Fractional diffusion limit for a kinetic equation with an interface
We consider the limit of a linear kinetic equation, with reflection-transmission-absorption at an interface, with a degenerate scattering kernel. The equation arise from a microscopic chain ofExpand
Transport equations for waves in a half Space
We derive boundary conditions for the phase space energy density of acoustic waves in a half space, in the high frequency limit. These boundary conditions generalize the usual reflection—transmissionExpand
The Phonon Boltzmann Equation, Properties and Link to Weakly Anharmonic Lattice Dynamics
For low density gases the validity of the Boltzmann transport equation is well established. The central object is the one-particle distribution function, f, which in the Boltzmann-Grad limitExpand
High frequency limit of the Helmholtz equations.
We derive the high frequency limit of the Helmholtz equations in terms of quadratic observables. We prove that it can be written as a stationary Liouville equation with source terms. Our method isExpand
WIGNER FUNCTIONS AND STOCHASTICALLY PERTURBED LATTICE DYNAMICS
We consider lattice dynamics with a small stochastic perturbation of order " and prove that for a space-time scale of order " 1 the Wigner function evolves according to a linear transport equationExpand
Energy Transport in Stochastically Perturbed Lattice Dynamics
We consider lattice dynamics with a small stochastic perturbation of order $${\varepsilon}$$ and prove that for a space–time scale of order $${\varepsilon^{-1}}$$ the local spectral density (WignerExpand
Homogenization limits and Wigner transforms
We present a theory for carrying out homogenization limits for quadratic functions (called “energy densities”) of solutions of initial value problems (IVPs) with anti-self-adjoint (spatial)Expand
Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary
Abstract This article describes how high-frequency waves solutions to the scalar wave equation and the Schrodinger equation propagate — in terms of semiclassical measures (also called WignerExpand
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