137 research outputs found
Liapunov Multipliers and Decay of Correlations in Dynamical Systems
The essential decorrelation rate of a hyperbolic dynamical system is the
decay rate of time-correlations one expects to see stably for typical
observables once resonances are projected out. We define and illustrate these
notions and study the conjecture that for observables in , the essential
decorrelation rate is never faster than what is dictated by the {\em smallest}
unstable Liapunov multiplier
Extensive Properties of the Complex Ginzburg-Landau Equation
We study the set of solutions of the complex Ginzburg-Landau equation in
. We consider the global attracting set (i.e., the forward map of
the set of bounded initial data), and restrict it to a cube of side .
We cover this set by a (minimal) number of balls of radius
in \Linfty(Q_L). We show that the Kolmogorov -entropy
per unit length,
exists. In particular, we bound by \OO(\log(1/\epsilon), which
shows that the attracting set is smaller than the set of bounded analytic
functions in a strip. We finally give a positive lower bound:
H_\epsilon>\OO(\log(1/\epsilon))Comment: 24 page
Non-equilibrium steady states for chains of four rotors
We study a chain of four interacting rotors (rotators) connected at both ends
to stochastic heat baths at different temperatures. We show that for
non-degenerate interaction potentials the system relaxes, at a stretched
exponential rate, to a non-equilibrium steady state (NESS). Rotors with high
energy tend to decouple from their neighbors due to fast oscillation of the
forces. Because of this, the energy of the central two rotors, which interact
with the heat baths only through the external rotors, can take a very long time
to dissipate. By appropriately averaging the oscillatory forces, we estimate
the dissipation rate and construct a Lyapunov function. Compared to the chain
of length three (considered previously by C. Poquet and the current authors),
the new difficulty with four rotors is the appearance of resonances when both
central rotors are fast. We deal with these resonances using the rapid
thermalization of the two external rotors.Comment: Minor changes to reflect the published versio
Non-Equilibrium Statistical Mechanics of Strongly Anharmonic Chains of Oscillators
We study the model of a strongly non-linear chain of particles coupled to two
heat baths at different temperatures. Our main result is the existence and
uniqueness of a stationary state at all temperatures. This result extends those
of Eckmann, Pillet, Rey-Bellet to potentials with essentially arbitrary growth
at infinity. This extension is possible by introducing a stronger version of
H\"ormander's theorem for Kolmogorov equations to vector fields with
polynomially bounded coefficients on unbounded domains.Comment: ~60 pages, 3 figure
Strange Heat Flux in (An)Harmonic Networks
We study the heat transport in systems of coupled oscillators driven out of
equilibrium by Gaussian heat baths. We illustrate with a few examples that such
systems can exhibit ``strange'' transport phenomena. In particular, {\em
circulation} of heat flux may appear in the steady state of a system of three
oscillators only. This indicates that the direction of the heat fluxes can in
general not be "guessed" from the temperatures of the heat baths. Although we
primarily consider harmonic couplings between the oscillators, we explain why
this strange behavior persists under weak anharmonic perturbations
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