Homogenization of the linear Boltzmann equation in a domain with a periodic distribution of holes

Consider a linear Boltzmann equation posed on the Euclidian plane with a periodic system of circular holes and for particles moving at speed 1. Assuming that the holes are absorbing -- i.e. that particles falling in a hole remain trapped there forever, we discuss the homogenization limit of that equation in the case where the reciprocal number of holes per unit surface and the length of the circumference of each hole are asymptotically equivalent small quantities. We show that the mass loss rate due to particles falling into the holes is governed by a renewal equation that involves the distribution of free-path lengths for the periodic Lorentz gas. In particular, it is proved that the total mass of the particle system at time t decays exponentially fast as t tends to infinity. This is at variance with the collisionless case discussed in [Caglioti, E., Golse, F., Commun. Math. Phys. 236 (2003), pp. 199--221], where the total mass decays as Const./t as the time variable t tends to infinity.Comment: 29 pages, 1 figure, submitted; figure 1 corrected in new versio

Long-time properties of MHD turbulence and the role of symmetries

We investigate long-time properties of three-dimensional MHD turbulence in the absence of forcing and examine in particular the role played by the quadratic invariants of the system and by the symmetries of the initial configurations. We observe that, when sufficient accuracy is used, initial conditions with a high degree of symmetries, as in the absence of helicity, do not travel through parameter space over time whereas by perturbing these solutions either explicitly or implicitly using for example single precision for long times, the flows depart from their original behavior and can become either strongly helical, or have a strong alignment between the velocity and the magnetic field. When the symmetries are broken, the flows evolve towards different end states, as predicted by statistical arguments for non-dissipative systems with the addition of an energy minimization principle, as already analyzed in \cite{stribling_90} for random initial conditions using a moderate number of Fourier modes. Furthermore, the alignment properties of these flows, between velocity, vorticity, magnetic potential, induction and current, correspond to the dominance of two main regimes, one helically dominated and one in quasi-equipartition of kinetic and magnetic energy. We also contrast the scaling of the ratio of magnetic energy to kinetic energy as a function of wavenumber to the ratio of eddy turn-over time to Alfv\'en time as a function of wavenumber. We find that the former ratio is constant with an approximate equipartition for scales smaller than the largest scale of the flow whereas the ratio of time scales increases with increasing wavenumber.Comment: 14 pages, 6 figure

Light-driven liquid crystalline nonlinear oscillator under optical periodic forcing

An all-optically driven strategy to govern a liquid crystalline collective molecular nonlinear oscillator is discussed. It does not require external feedbacks of any kind while the oscillator and a time-depending perturbation both are sustained by incident light. Various dynamical regimes such as frequency -locked, quasiperiodic, forced and chaotic are observed in agreement with a theoretical approach developed in the limit of the plane wave approximation.Comment: 5 pages, 6 figures, submitted to Phys. Rev.