Nonlinear Sciences – Chaotic Dynamics
Scientific paper
2001-05-10
Nonlinear Sciences
Chaotic Dynamics
16 pages, 6 figures
Scientific paper
10.1088/0305-4470/34/46/304
We investigate the energy relaxation process produced by thermal baths at zero temperature acting on the boundary atoms of chains of classical anharmonic oscillators. Time-dependent perturbation theory allows us to obtain an explicit solution of the harmonic problem: even in such a simple system nontrivial features emerge from the interplay of the different decay rates of Fourier modes. In particular, a crossover from an exponential to an inverse-square-root law occurs on a time scale proportional to the system size $N$. A further crossover back to an exponential law is observed only at much longer times (of the order $N^3$). In the nonlinear chain, the relaxation process is initially equivalent to the harmonic case over a wide time span, as illustrated by simulations of the $\beta$ Fermi-Pasta-Ulam model. The distinctive feature is that the second crossover is not observed due to the spontaneous appearance of breathers, i.e. space-localized time-periodic solutions, that keep a finite residual energy in the lattice. We discuss the mechanism yielding such solutions and also explain why it crucially depends on the boundary conditions.
Lepri Stefano
Livi Roberto
Piazza Francesco
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