Anomalous thermostat and intraband discrete breathers

作者：Aubry S^*; Schilling R

来源：Physica D: Nonlinear Phenomena , 2009, 238(20): 2045-2061.

DOI：10.1016/j.physd.2009.07.012

• Disordered nonlinear systems
• Discrete breathers
• Localized modes
• Dissipative behavior
• Anomalous relaxation

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 Gamma(t) and noise term zeta(t) for the anharmonic coordinates q(t) = (q(alpha) (t)). For zero temperature, i.e. for zeta(t) equivalent to 0, we prove that the support of the Fourier transform of Gamma(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 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 zero measure set L such that for Omega in its full measure complement R \ L, it is alpha = 0, i.e. there is no energy dissipation. Under certain conditions L contains a subset L' such that for Omega is an element of L' the dissipation rate is nonzero and may be subdissipative (0 <= alpha < 1) or superdissipative (1 < alpha <= 2), compared to ordinary dissipation (alpha = 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 Omega exist for all Omega in a Cantor set C(k) of finite Lebesgue measure. This is achieved by estimating the contribution of small denominators appearing for G(t; Omega), related to Gamma(t). For Omega is an element of C(k) the small denominators do not lead to divergencies such that G(t; Omega) is a smooth and bounded function in t.

• 出版日期2009-10-15
• 单位中国地震局