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 $\Omega \in thissetthedissipationrateisnonzeroandmaybesubdissipative$(0 \leq α< 1)$orsuperdissipative$(1 <α\leq 2)$.Consequently,theharmonicbathdoesactasananomalousthermostat.Intrabanddiscretebreathersaresuchsolutionswhichdonotrelax.Weproveforarbitraryanharmonicityandsmallbutfinitecouplingthatintrabanddiscretebreatherswithfrequency$Ω$existforall$Ω$inaCantorset$\mathcal{C}(k)$offiniteLebesguemeasure.Thisisachievedbyestimatingthecontributionofsmalldenominatorsappearinginthememorykernel.For$Ω\in\mathcal{C}(k)$thesmalldenominatorsdonotleadtodivergenciessuchthatthiskernelisasmoothandboundedfunctionin$t$.

Physica D in print