Anomalous thermostat and intraband discrete breathers

We investigate the dynamics of a macroscopic system which consists of an anharmonic subsystem embedded in an arbitrary harmonic lattice, including quenched disorder. The coupling between both parts is bilinear. Elimination of the harmonic degrees of freedom leads to a nonlinear Langevin equation with memory kernels Γ(t) and noise term ζ(t) for the anharmonic coordinates q(t)=(qα(t)). For zero temperature, i.e. for ζ(t)≡0, we prove that the support of the Fourier transform of Γ(t) and of the time averaged velocity–velocity correlation functions K(t) of the anharmonic system cannot overlap. As a consequence, the asymptotic solutions can be constant, periodic, quasiperiodic or almost periodic, and possibly weakly chaotic. For a sinusoidal trajectory q(t) with frequency Ω we find that the energy ETET transferred to the harmonic system up to time TT is proportional to TαTα. If Ω equals one of the phonon frequencies ωνων, it is α=2α=2. We prove that there is a zero measure set LL such that for Ω in its full measure complement R∖LR∖L, it is α=0α=0, i.e. there is no energy dissipation. Under certain conditions LL contains a subset L′L′ such that for Ω∈L′ the dissipation rate is nonzero and may be subdissipative (0≤α<1)(0≤α<1) or superdissipative (1<α≤2)(1<α≤2), compared to ordinary dissipation (α=1)(α=1). Consequently, the harmonic bath does act as an anomalous thermostat, in variance with the common belief that elimination of a macroscopically large number of degrees of freedom always generates dissipation, forcing convergence to equilibrium. 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 Ω exist for all Ω in a Cantor set C(k)C(k) of finite Lebesgue measure. This is achieved by estimating the contribution of small denominators appearing for G(t;Ω), related to Γ(t). For Ω∈C(k) the small denominators do not lead to divergencies such that G(t;Ω) is a smooth and bounded function in tt.