Section: New Results

Diffusive and superdiffusive behavior in one-dimensional chains of oscillators

In order to understand abnormally diffusive phenomena which are physically observed in nanotube technologies, one mathematical approach consists in starting from deterministic system of Newtonian particles, and then perturb this system with a stochastic component which provides enough ergodicity to the dynamics. It is already well known that these stochastic chains model correctly the behavior of the conductivity [24]. In [1], [2] (published in Communications in Mathematical Physics) M. Simon with her coauthors C. Bernardin, P. Gonçalves, M. Jara, T. Komorowski, S. Olla and M. Sasada have observed both behaviors, normal and anomalous diffusion, in the context of low dimensional asymmetric systems. They manage to describe the microscopic phenomena at play which are responsible for each one of these phenomena, and they go beyond the predictions that have recently been done in [31], [32]. Moreover, in [2], the authors manage to treat rigorously, for the first time, the case of an anharmonic potential: more precisely, they consider a small quartic anharmonicity and show that the result obtained in the harmonic (linear) case persists up to some small critical value of the nonlinear perturbation.