Abstract
We consider a one dimensional infinite chain of harmonic oscillators whose dynamics is perturbed by a stochastic term conserving energy and momentum. We prove that in the unpinned case the macroscopic evolution of the energy converges to the solution of the fractional diffusion equation $${\partial_t u = -|\Delta|^{3/4}u}$$ . For a pinned system we prove that its energy evolves diffusively, generalizing some results of Basile and Olla (J. Stat. Phys. 155(6):1126–1142, 2014).
Highlights
Superdiffusion of energy and the corresponding anomalous thermal conductivity have been observed numerically in the dynamics of unpinned FPU chains [15,16]
The problem has been studied in models where the Hamiltonian dynamics is perturbed by stochastic terms that conserve energy and momentum, like random exchange of velocity between nearest neighbors particles [1,2]
In the cases when the conductivity is finite it is proven in [4] that energy fluctuations in equilibrium evolve diffusively
Summary
Superdiffusion of energy and the corresponding anomalous thermal conductivity have been observed numerically in the dynamics of unpinned FPU chains [15,16]. While this choice is quite natural in the situation of a pinned chain, it requires some explanation in the unpinned case In the latter situation, if we start with non-centered initial conditions, their respective macroscopic averages will evolve, at the hyperbolic space-time scale, following the linear wave equation. If we start with non-centered initial conditions, their respective macroscopic averages will evolve, at the hyperbolic space-time scale, following the linear wave equation As a result, they will disperse to infinity, since we start with the data whose realization has a finite L2 norm. We mention here the article [6], where a result similar to ours is proven, by very different techniques, for a dynamics with two conserved quantities (energy and volume) in the case when the initial data is given by a Gibbs equilibrium measure.
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