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\(5 / 6\)-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field. (English) Zbl 1435.82022

The paper is devoted to the study of a one-dimensional infinite chain of coupled charged classical harmonic oscillators in a magnetic field with a small stochastic perturbation. Chains of coupled oscillators are typical models showing superdiffusive transport of energy, and it has been argued by H. Spohn [J. Stat. Phys. 154, No. 5, 1191–1227 (2014; Zbl 1291.82119)] that for general anharmonic chains the macroscopic diffusion of energy is governed by a fractional diffusion equation with possible exponents 3/4 and 5/6. In recent work it has been shown that the exponent 3/4 is indeed obtained in a harmonic chains of oscillators with a stochastic exchange of momentum between neighbouring sites [M. Jara et al., Ann. Appl. Probab. 19, No. 6, 2270–2300 (2009; Zbl 1232.60018); Commun. Math. Phys. 339, No. 2, 407–453 (2015; Zbl 1329.82116)].
In the present contribution, the authors prove that the exponent 5/6 is recovered for a coupled charged harmonic chain of oscillators in a magnetic field with noise. The local perturbation of the dynamics is described considering stochastic differential equations for position and velocity in which a noise term expressed by means of standard Wiener processes and conserving total energy is introduced. The proof relies on a two-step scaling limit, as already exploited in previous work.
In the first instance, the authors prove that in the weak-noise limit the local density of energy is governed by the phonon linear Boltzmann equation. They then consider a properly rescaled solution of the Boltzmann equation and show that it converges to the solution of the fractional diffusion equation with exponent 5/6, thus showing superdiffusion of energy in a chain of oscillators. The scaling limit is essentially determined by the asymptotic behavior of the derivative of the dispersion relation and of the mean value of the scattering kernel for small wave numbers. A key technical ingredient useful for the proof is to consider the microscopic local density of energy, called Wigner distribution as in the physical literature, associated to the eigenvectors of the deterministic dynamics including the effect of the magnetic field, rather than the eigenvectors of the harmonic Hamiltonian dynamics.

MSC:

82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
60K50 Anomalous diffusion models (subdiffusion, superdiffusion, continuous-time random walks, etc.)
60G51 Processes with independent increments; Lévy processes
35Q20 Boltzmann equations
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
35R60 PDEs with randomness, stochastic partial differential equations
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References:

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