This paper is devoted to the formulation and proof of an abstract alphabet version of the fundamental theorem of Shannon's rate distortion theory. The validity of the theorem is established for both discrete and continuous parameter information sources satisfying a certain regularity condition intermediate in restrictiveness between ergodicity and weak mixing. For ease of presentation, only sin...

We review some of the most recent results on the dynamics of the Hamiltonian Mean Field (HMF) model, a systems of N planar spins with ferromagnetic infiniterange interactions. We show, in particular, how some of the dynamical anomalies of the model can be interpreted and characterized in terms of the weak-ergodicity breaking proposed in frameworks of glassy systems. We also discuss the connecti...

This mostly expository paper shows how weak convergence methods provide simple, elegant proofs of (i) the stabilization of an inverted pendulum under fast vertical oscillations, (ii) the existence of particle traps induced by rapidly varying electric fields and (iii) the adiabatic invariance of ∫ Γ p dx for slowing varying planar Hamiltonian dynamics. Under an appropriate, but very restrictive,...

A dynamical framework is developed with several variations for modeling multiple timescale molecular dynamics at constant temperature. The described approach can be adapted to various applications, including mixtures of heavy and light particles and models with stiff potentials. Canonical sampling properties are proved under the ergodicity assumption. Implications for numerical method developme...

We consider N single server infinite buffer queues with service rate β. Customers arrive at rate N α, choose L queues uniformly, and join the shortest one. The stability condition is α < β. We study in equilibrium the fraction of queues of length at least k ≥ 0. We prove a functional central limit theorem on an infinite-dimensional Hilbert space with its weak topology, with limit a stationary O...

We derive backward and forward fractional Feynman-Kac equations for the distribution of functionals of the path of a particle undergoing anomalous diffusion. Fractional substantial derivatives introduced by Friedrich and co-workers [Phys. Rev. Lett. 96, 230601 (2006)10.1103/PhysRevLett.96.230601] provide the correct fractional framework for the problem. For applications, we calculate the distri...

This survey is based on a series of talks I gave at the conference “Dynamical systems and diophantine approximation” at l’Instut Henri Poincaré in June 2003. I will present asymptotic results (transitivity, ergodicity, weak-mixing) for billiards based on the approximation technique developed by Katok and Zemlyakov. I will also present approximation techniques which allow to prove the abundance ...