Math @ Duke

Publications [#163256] of Jonathan C. Mattingly
Papers Published
 Hairer, Martin and Mattingly, Jonathan C., Slow energy dissipation in anharmonic oscillator chains,
Communications on Pure and Applied Mathematics, vol. 62 no. 8
(2008),
pp. 9991032, ISSN 00103640 [MR2531551], [arXiv:0712.3884]
(last updated on 2009/10/14)
Abstract: We study the dynamic behaviour 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 $V_2(q_i q_{i1}) = (q_i q_{i1})^2/2$ between itself and its
nearest neighbours.
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.


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