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Abstract: We study the dynamic behavior at high energies of a chain of anharmonicoscillators coupled at its ends to heat baths at possibly differenttemperatures. In our setup, each oscillator is subject to a homogeneousanharmonic pinning potential $V 1q i =|q i|^{2k}-2k$ and harmonic couplingpotentials $V 2q i- q {i-1} = q i- q {i-1}^2-2$ between itself and itsnearest neighbors. We consider the case $k > 1$ when the pinning potential isstronger then the coupling potential. At high energy, when a large fraction ofthe energy is located in the bulk of the chain, breathers appear and block thetransport of energy through the system, thus slowing its convergence toequilibrium.In such a regime, we obtain equations for an effective dynamics by averagingout the fast oscillation of the breather. Using this representation and relatedideas, 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 $k > 2$ wefurther show that the system does not relax exponentially fast to thisequilibrium by demonstrating that zero is in the essential spectrum of thegenerator 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 ofdecrease of the energy when it is concentrated in one of the oscillatorswithout dissipation. Numerical simulations are included which confirm thetheory.

Author: Martin Hairer, Jonathan C. Mattingly


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