Anomalous Thermostat and Intraband Discrete Breathers

Abstract

We investigate the dynamics of a macroscopic system which consists of an anharmonic subsystem embedded in an arbitrary harmonic lattice, including quenched disorder. Elimination of the harmonic degrees of freedom leads to a nonlinear Langevin equation for the anharmonic coordinates. For zero temperature, we prove that the support of the Fourier transform of the memory kernel and of the time averaged velocity-velocity correlations functions of the anharmonic system can not overlap. As a consequence, the asymptotic solutions can be constant, periodic,quasiperiodic or almost periodic, and possibly weakly chaotic. For a sinusoidal trajectory with frequency $\Omega$ we find that the energy $E_T$ transferred to the harmonic system up to time $T$ is proportional to $T^{\alpha}$. If $\Omega$ equals one of the phonon frequencies $\omega_\nu$, it is $\alpha=2$. We prove that there is a full measure set such that for $\Omega$ in this set it is $\alpha=0$, i.e. there is no energy dissipation. Under certain conditions there exists a zero measure set such that for $\Omega \in this set the dissipation rate is nonzero and may be subdissipative $(0 \leq \alpha < 1)$ or superdissipative $(1 <\alpha \leq 2)$. Consequently, the harmonic bath does act as an anomalous thermostat. Intraband discrete breathers are such solutions which do not relax. We prove for arbitrary anharmonicity and small but finite coupling that intraband discrete breathers with frequency $\Omega$ exist for all $\Omega$ in a Cantor set $\mathcal{C}(k)$ of finite Lebesgue measure. This is achieved by estimating the contribution of small denominators appearing in the memory kernel. For $\Omega\in\mathcal{C}(k)$ the small denominators do not lead to divergencies such that this kernel is a smooth and bounded function in $t$.