Energy transport along Fermi-Pasta-Ulam chains containing binary isotopic disorder: Zero-temperature systems

Abstract

Dissipation from harmonic energy eigenstates is used to characterize energy transport in binary isotopically disordered (BID) Fermi-Pasta-Ulam (FPU-beta) chains. Using a continuum analog for the corresponding harmonic portion of the Hamiltonian, the time-independent wave amplitude is calculated for a plane wave having wavelength \lambda that is incident upon the disordered section, and the solution is mapped onto the discrete chain. Due to Anderson localization, energy is initially localized near the incident end of the chain, and in the absence of anharmonicity the wave amplitude is stationary in time. For sufficient anharmonicity, however, mode transitions lead to dissipation. Energy transport along the chain is quantified using both the second moment M of the site energy, and the number of masses contributing to transport, which was estimated from the localization parameter. Over the time scales studied, M increased linearly in time, yielding an effective transport coefficient G. At low and intermediate impurity concentration, G(c) can be characterized by a competition between the rate of mode transitions and the time for energy to propagate a distance equal to the localization length \xi. At the highest concentrations (1.6 \le c\lambda \le 16.0), there is significant mode transition suppression in BID systems, and the transport coefficient G(c) becomes proportional to \xi(c).