Slow energy dissipation in anharmonic oscillator chains

dc.contributor.author

Hairer, M

dc.contributor.author

Mattingly, JC

dc.date.accessioned

2016-10-12T11:26:45Z

dc.date.issued

2009-08-01

dc.description.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(qi) = |qi| 2k/2k and harmonic coupling potentials V 2(qi-qi-1) = (qi-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 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 > 3/2, 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 > 3/2, 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.

dc.identifier.eissn

0010-3640

dc.identifier.issn

0010-3640

dc.identifier.uri

https://hdl.handle.net/10161/12940

dc.publisher

Wiley

dc.relation.ispartof

Communications on Pure and Applied Mathematics

dc.relation.isversionof

10.1002/cpa.20280

dc.title

Slow energy dissipation in anharmonic oscillator chains

dc.type

Journal article

duke.contributor.orcid

Mattingly, JC|0000-0002-1819-729X

pubs.begin-page

999

pubs.end-page

1032

pubs.issue

8

pubs.organisational-group

Duke

pubs.organisational-group

Mathematics

pubs.organisational-group

Statistical Science

pubs.organisational-group

Trinity College of Arts & Sciences

pubs.publication-status

Published

pubs.volume

62

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