Slow energy dissipation in anharmonic oscillator chains

TL;DR

This paper investigates how high-energy anharmonic oscillator chains exhibit slow energy dissipation due to breather formations, providing mathematical analysis and numerical confirmation of their impact on relaxation to equilibrium.

Contribution

It introduces an effective dynamics approach for high-energy regimes and proves spectral properties indicating slow convergence to equilibrium in anharmonic chains.

Findings

Breathers block energy transport at high energies.

Zero in the spectrum implies non-exponential relaxation.

Numerical simulations confirm theoretical predictions.

Abstract

We study the dynamic behavior at high energies of a chain of anharmonic oscillators coupled at its ends to heat baths at possibly different temperatures. In our setup, each oscillator is subject to a homogeneous anharmonic pinning potential $V_1(q_i) =|q_i|^{2k}/2k$ and harmonic coupling potentials $V_2(q_i- q_{i-1}) = (q_i- q_{i-1})^2/2$ between itself and its nearest neighbors. We consider the case $k > 1$ when the pinning potential is stronger then the coupling potential. At high energy, when a large fraction of the energy is located in the bulk of the chain, breathers appear and block the transport of energy through the system, thus slowing its convergence to equilibrium. In such a regime, we obtain equations for an effective dynamics by averaging out the fast oscillation of the breather. Using this representation and related ideas, we can prove a number of results. When the chain is of length three and $k> 3/2$ we show that there exists a unique invariant measure. If $k > 2$ we further show that the system does not relax exponentially fast to this equilibrium by demonstrating that zero is in the essential spectrum of the generator of the dynamics. When the chain has five or more oscillators and $k> 3/2$ we show that the generator again has zero in its essential spectrum. In addition to these rigorous results, a theory is given for the rate of decrease of the energy when it is concentrated in one of the oscillators without dissipation. Numerical simulations are included which confirm the theory.