Energy dissipation in Hamiltonian chains of rotators

Abstract

We discuss, in the context of energy flow in high-dimensional systems and Kolmogorov-Arnol'd-Moser (KAM) theory, the behavior of a chain of rotators (rotors) which is purely Hamiltonian, apart from dissipation at just one end. We derive bounds on the dissipation rate which become arbitrarily small in certain physical regimes, and we present numerical evidence that these bounds are sharp. We relate this to the decoupling of non-resonant terms as is known in KAM problems.

Breathers as metastable states for the discrete NLS equation

“…We now recall in more detail the situation considered in [4]. The authors start from a Hamiltonian system of N rotators coupled to their nearest neighbors.…”

“…We will focus primarily on the case N = 3 for simplicity, but in principle, our methods apply to systems with arbitrarily many degrees of freedom, and we plan to return to the consideration of the general case in a future work. We will choose initial conditions for this system in which essentially all of the energy is in mode u 1 , and will add a weak dissipative term to the last mode as in [4] by adding to Eq. (1) a term of the form…”

“…For example, this type of transport is important for understanding heat flow in lattice systems [9]. In [4], the authors find that in a system of N rotators, coupled to their nearest neighbors, there are regions of phase space in which the transport is very slow and they give theoretical and numerical estimates indicating that the slow transport is due to the metastability of periodic breathers in the system. We note, that breathers and the transport of energy in Hamiltonian lattices, have also been among the topics of R. de la Llave's research ( [7], [6]).…”

On the relaxation rate of short chains of rotors interacting with Langevin thermostats

“…1 For chains of length n, it is conjectured in [3, Remark 5.3] that the exponent is 1/(2 n/2 − 2), which is indeed 1/2 when n = 3, 4. This conjecture is supported by [5], where the rate of energy dissipation in deterministic chains of rotors of arbitrary lengths is studied.…”

Decay of Hamiltonian Breathers Under Dissipation

“…We will choose initial conditions for this system in which essentially all of the energy is in mode u 1 , and will add a weak dissipative term to the last mode as in [1,2] by adding to Eq. (1.1) a term of the form iγ δ n, j u j ,…”