Slow energy dissipation in anharmonic oscillator chains

• Published in: Communications on pure and applied mathematics 2009-08, Vol.62 (8), p.999-1032
• Main Authors: Hairer, Martin, Mattingly, Jonathan C.
• Format: Article
• Language: English
• Subjects: Energy dissipation; Exact sciences and technology; Harmonic analysis; Mathematical analysis; Mathematics; Measure and integration; Operator theory; Ordinary differential equations; Oscillators; Sciences and techniques of general use; Statistical mechanics; Theory
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.20280

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 V1(qi) = |qi|2k/2k and harmonic coupling potentials V2(qi − qi − 1) = (qi − qi − 1)2/2 between itself and its nearest neighbors. We consider the case k > 1 when the pinning potential is stronger than 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 3 and $k > {3 \over 2}$, 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 0 is in the essential spectrum of the generator of the dynamics. When the chain has five or more oscillators and $k > {3 \over 2}$, we show that the generator again has 0 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 that confirm the theory. © 2009 Wiley Periodicals, Inc.