Abstract:
The mathematical physics of mechanical systems in thermal equilibrium is a well studied, and relatively easy, subject, because the Gibbs distribution is in general an adequate guess for the equilibrium state. On the other hand, the mathematical physics of {\em non-equilibrium} systems, such as that of a chain of masses connected with springs to two (infinite) heat reservoirs is more difficult, precisely because no such {\em a priori} guess exists. Recent work has, however, revealed that under quite general conditions, such states can not only be shown to exist, but are {\em unique}, using the H\"ormander conditions and controllability. Furthermore, interesting properties, such as energy flux, exponentially fast convergence to the unique state, and fluctuations of that state have been successfully studied. Finally, the ideas used in these studies can be extended to certain stochastic PDE's using Malliavin calculus to prove regularity of the process.

Abstract:
We propose and study a model with glassy behavior. The state space of the model is given by all triangulations of a sphere with $n$ nodes, half of which are red and half are blue. Red nodes want to have 5 neighbors while blue ones want 7. Energies of nodes with different numbers of neighbors are supposed to be positive. The dynamics is that of flipping the diagonal of two adjacent triangles, with a temperature dependent probability. We show that this system has an approach to a steady state which is exponentially slow, and show that the stationary state is unordered. We also study the local energy landscape and show that it has the hierarchical structure known from spin glasses. Finally, we show that the evolution can be described as that of a rarefied gas with spontaneous generation of particles and annihilating collisions.

Abstract:
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 apparition of resonances when both central rotors are fast. We deal with these resonances using the rapid thermalization of the two external rotors.

Abstract:
We consider networks of massive particles connected by non-linear springs. Some particles interact with heat baths at different temperatures, which are modeled as stochastic driving forces. The structure of the network is arbitrary, but the motion of each particle is 1D. For polynomial interactions, we give sufficient conditions for H\"ormander's "bracket condition" to hold, which implies the uniqueness of the steady state (if it exists), as well as the controllability of the associated system in control theory. These conditions are constructive; they are formulated in terms of inequivalence of the forces (modulo translations) and/or conditions on the topology of the connections. We illustrate our results with examples, including "conducting chains" of variable cross-section. This then extends the results for a simple chain obtained in Eckmann, Pillet, Rey-Bellet (1999).

Abstract:
We define a deterministic ``scattering'' model for heat conduction which is continuous in space, and which has a Boltzmann type flavor, obtained by a closure based on memory loss between collisions. We prove that this model has, for stochastic driving forces at the boundary, close to Maxwellians, a unique non-equilibrium steady state.

Abstract:
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 $C^1$, the essential decorrelation rate is never faster than what is dictated by the {\em smallest} unstable Liapunov multiplier.

Abstract:
We prove that if the initial condition of the Swift-Hohenberg equation $\partial_t u(x,t)=\bigl(\epsilon^2-(1+\partial_ x^2)^2\bigr) u(x,t) -u^3(x,t)$ is bounded in modulus by $Ce^{-\beta x}$ as $x\to+\infty $, the solution cannot propagate to the right with a speed greater than $\sup_{0<\gamma\le\beta}\gamma^{-1}(\epsilon ^2+4\gamma^2+8\gamma^4).$ This settles a long-standing conjecture about the possible asymptotic propagation speed of the Swift-Hohenberg equation. The proof does not use the maximum principle and is simple enough to generalize easily to other equations. We illustrate this with an example of a modified Ginzburg-Landau equation, where the minimal speed is not determined by the linearization alone.

Abstract:
Using, and extending, striking inequalities by V.V. Ivanov on the down-crossings of monotone functions and ergodic sums, we give universal bounds on the probability of finding oscillations of observables in 1-dimensional lattice gases in infinite volume. In particular, we study the finite volume average of the occupation number as one runs through an increasing sequence of boxes of size $2n$ centered at the origin. We show that the probability to see $k$ oscillations of this average between two values $\beta $ and $0<\alpha <\beta $ is bounded by $C R^k$, with $R<1$, where the constants $C$ and $R$ do not depend on any detail of the model, nor on the state one observes, but only on the ratio $\alpha/\beta $.

Abstract:
We study the set of solutions of the complex Ginzburg-Landau equation in $\real^d, d<3$. We consider the global attracting set (i.e., the forward map of the set of bounded initial data), and restrict it to a cube $Q_L$ of side $L$. We cover this set by a (minimal) number $N_{Q_L}(\epsilon)$ of balls of radius $\epsilon$ in $\Linfty(Q_L)$. We show that the Kolmogorov $\epsilon$-entropy per unit length, $H_\epsilon =\lim_{L\to\infty} L^{-d} \log N_{Q_L}(\epsilon)$ exists. In particular, we bound $H_\epsilon$ 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: $H_\epsilon>\OO(\log(1/\epsilon))$

Abstract:
We study damped hyperbolic equations on the infinite line. We show that on the global attracting set $G$ the $\epsilon$-entropy (per unit length) exists in the topology of $W^{1,\infty}$. We also show that the topological entropy per unit length of $G$ exists. These results are shown using two main techniques: Bounds in bounded domains in position space and for large momenta, and a novel submultiplicativity argument in $W^{1,\infty}$.