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Superdiffusion of energy in a chain of harmonic oscillators with noise. (English) Zbl 1329.82116

Superdiffusion of energy and anomalous thermal conduction have been observed numerically in the dynamics of unpinned low-dimensional lattices and nonlinear oscillators. This is generally attributed to small scattering rates. In case of additional pinning potentials, it is expected that the system shows normal diffusion. Recently, systems with Hamiltonians are studied, the dynamics of which is disturbed by stochastic processes.
The article considers a one-dimensional infinite chain of harmonic oscillators whose dynamics is perturbed by a stochastic process preserving energy and momentum. The authors prove that in the unpinned case the macroscopic evolution of the energy converges to the solution of the fractional diffusion equation \(\partial_tu=-|\Delta|^{3/4}u\). For a pinned system they prove that its energy evolves diffusively, generalizing some results of G. Basile and S. Olla [J. Stat. Phys. 155, No. 6, 1126–1142 (2014; Zbl 1297.82004)]. (abstract by the authors)
The limit evolution of the Wigner distribution of the energy \(W_E(t)\) is considered in detail, when the initial data have square summable realisations. \(W_E(t)\) represents the energy density in both the spatial variable and the frequency mode. It is essential that the evolution of \(W_E(t)\) is not autonomous but involves a second distribution \(Y_E(t)\), whose real and imaginary parts represent the difference between kinetic and potential energy and the energy current, respectively. The principal advantage of working with \(W_E(t)\) and \(Y_E(t)\) is that their evolution may be described by a system of ordinary differential equations.

MSC:

82C70 Transport processes in time-dependent statistical mechanics
82D99 Applications of statistical mechanics to specific types of physical systems
81Q80 Special quantum systems, such as solvable systems

Citations:

Zbl 1297.82004
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References:

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