We study processes that consist of deterministic evolution punctuated at random times by disturbances with random severity; we call such processes semistochastic. Under appropriate assumptions such a process admits a unique stationary distribution. We develop a technique for establishing bounds on the rate at which the distribution of the random process approaches the stationary distribution. An important example of such a process is the dynamics of the carbon content of a forest whose deterministic growth is interrupted by natural disasters (fires, droughts, insect outbreaks, etc.).

We derive Haff's cooling law for a periodic fluid consisting of two hard disks per unit cell by reducing it to a point particle moving inside a Sinai billiard with finite horizon with an inelastic collision rule. Indeed, our results also apply to general dispersing billiards with piece-wise smooth boundary with finite horizon and no cusps.

We solve the longstanding problem of smoothing a stadium billiard. Besides our results demonstrate why there were no clear conjectures how much the stadium's boundary must be smoothened to destroy chaotic dynamics. To do that we needed to extend standard KAM theory to analyze stability of periodic orbits, because of the low smoothness of the system. In fact, the stadium has a $C^1$ boundary, and we show that $C^2$ smoothing results in appearance of elliptic periodic orbits.

In this paper we propose an algorithm to numerically simulate Markov processes of jump type. While these processes can naturally be generated as solutions to jump-SDEs the algorithm we propose is instead based on the interlacing construction of the process. We show that one can construct in the sense of strong convergence a high order numerical scheme based on high order ODE solvers. This result is in sharp contrast to the well known difficulty of constructing high-order numerical schemes for diffusion processes.

We review the current state of a fundamental problem of rigorous derivation of transport processes in classical statistical mechanics from classical mechanics. Such derivations for diffusion and momentum transport (viscosities) were obtained for minimal models of these processes involving one and two particles respectively. However, a minimal model which demonstrates heat conductivity contains three particles. Its rigorous analysis is currently out of reach for existing mathematical techniques. The gas of localized balls is widely accepted as a basis for a simplest model for derivation of Fourier’s law. We suggest a modification of the localized balls gas and argue that this gas of localized activated balls is a good candidate to rigorously prove Fourier’s law. In particular, hyperbolicity is derived for a reduced version of this model.

A fundamental problem of non-equilibrium statistical mechanics is the derivation of macroscopic transport equations in the hydrodynamic limit. The rigorous study of such limits requires detailed information about rates of convergence to equilibrium for finite sized systems. In this paper, we consider the finite lattice {1, 2, …, N}, with an energy xi ∈ (0, ∞) associated with each site. The energies evolve according to a Markov jump process with nearest neighbour interaction such that the total energy is preserved. We prove that for an entire class of such models the spectral gap of the generator of the Markov process scales as . Furthermore, we provide a complete classification of reversible stationary distributions of product type. We demonstrate that our results apply to models similar to the billiard lattice model considered in Gaspard and Gilbert (2009 J. Stat. Mech.: Theory Exp. 2009 24), and hence provide a first step in the derivation of a macroscopic heat equation for a microscopic stochastic evolution of mechanical origin.

We investigate the macroscopic description of a dilute, gas-like system of particles, which interact through binary collisions that conserve momentum and mass, but which dissipate energy, as in the case of granular media with inelastic collisions. Our starting point is on the mesoscopic level, through the Boltzmann equation. We deduce hydrodynamic equations for the macroscopic description that would reduce to the compressible Navier-Stokes equations if there were no energy dissipation. We do this in a regime where both the Knudsen number (the ratio of mesoscopic to macroscopic length scales) and the restitution deficit (which measures the inelasticity) are small but non-zero. In this regime, we show that for small values of the Knudsen number and small inelasticity it is possible to relate the actual dynamics to a reduced dynamics on a 'slow manifold', which in the limit of zero inelasticity and zero Knudsen number is simply the 'manifold' of local Maxwellians. Instead of expanding the Boltzmann equation itself, we expand this manifold in terms of these two small parameters. In this way, a number of ideas from the theory of dynamical systems, and especially geometric singular perturbation theory, enter our analysis. We discuss the resulting hydrodynamic equations, and compare them with those obtained by other researchers using other methods (suited to other regimes). As we explain here, the particular regime we investigate is especially interesting in the context of pattern formation in driven granular media.

We demonstrate that the defocusing mechanism fails to work if not all focusing components of the boundary are absolutely focusing. More precisely, we construct billiard tables with arbitrary long free path away from a non-absolutely focusing component such that a nonlinearly stable periodic orbit exists. Therefore the only known standard procedure of constructing chaotic ergodic billiards works in general only if all focusing boundary components are absolutely focusing.