Hairer, Martin and Mattingly, Jonathan C. (2009) Slow energy dissipation in anharmonic oscillator chains. Communications on Pure and Applied Mathematics, Vol.62 (No.8 Sp. Iss. SI). pp. 999-1032. doi:10.1002/cpa.20280

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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 I homogeneous anharmonic pinning potential V-1(q(i)) = vertical bar q(i)vertical bar(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 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 it 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 > 3/2, we show that there exists it 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 > 3/2, 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. (C) 2009 Wiley Periodicals, Inc.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Journal or Publication Title:	Communications on Pure and Applied Mathematics
Publisher:	John Wiley & Sons
ISSN:	0010-3640
Official Date:	August 2009
Dates:	Date Event August 2009 Published
Volume:	Vol.62
Number:	No.8 Sp. Iss. SI
Number of Pages:	34
Page Range:	pp. 999-1032
DOI:	10.1002/cpa.20280
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	Engineering and Physical Sciences Research Council (EPSRC), Sloan Foundation, National Science Foundation (U.S.) (NSF)
Grant number:	EP/D071593/1 (EPSRC), DMS 04-49910, DMS 06-16710

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