Non-equilibrium two-phase coexistence in a confined granular layer

We report the observation of the homogenous nucleation of crystals in a dense layer of steel spheres confined between two horizontal plates vibrated vertically. Above a critical vibration amplitude, two-layer crystals with square symmetry were found to coexist in steady state with a surrounding granular liquid. By analogy to equilibrium hard sphere systems, the phase behavior can be explained through entropy maximization. However, dramatic non-equilibrium effects are present, including a significant difference in the granular temperatures of the two phases.Comment: 4 pages, 3 figures, RevTex4 forma

Smectic blue phases: layered systems with high intrinsic curvature

We report on a construction for smectic blue phases, which have quasi-long range smectic translational order as well as three dimensional crystalline order. Our proposed structures fill space by adding layers on top of a minimal surface, introducing either curvature or edge defects as necessary. We find that for the right range of material parameters, the favorable saddle-splay energy of these structures can stabilize them against uniform layered structures. We also consider the nature of curvature frustration between mean curvature and saddle-splay.Comment: 15 pages, 11 figure

L1-Poincaré inequalities for differential forms on Euclidean spaces and Heisenberg groups

In this paper, we prove interior Poincar\ue9 and Sobolev inequalities in Euclidean spaces and in Heisenberg groups, in the limiting case where the exterior (resp. Rumin) differential of a differential form is measured in L1 norm. Unlike for Lp, p>1, the estimates are doomed to fail in top degree. The singular integral estimates are replaced with inequalities which go back to Bourgain-Brezis in Euclidean spaces, and to Chanillo-Van Schaftingen in Heisenberg groups

Smectic Phases with Cubic Symmetry: The Splay Analog of the Blue Phase

We report on a construction for smectic blue phases, which have quasi-long range smectic translational order as well as long range cubic or hexagonal order. Our proposed structures fill space with a combination of minimal surface patches and cylindrical tubes. We find that for the right range of material parameters, the favorable saddle-splay energy of these structures can stabilize them against uniform layered structures.Comment: 4 pages, 4 eps figures, RevTe

ORLICZ SPACES AND ENDPOINT SOBOLEV-POINCAR \u301EINEQUALITIES FOR DIFFERENTIAL FORMS INHEISENBERG GROUPS

In this paper we prove Poincar \u301e and Sobolev inequalities for differ-ential forms in the Rumin\u2019s contact complex on Heisenberg groups. Inparticular, we deal with endpoint values of the exponents, obtaining fi-nally estimates akin to exponential Trudinger inequalities for scalar func-tion. These results complete previous results obtained by the authors awayfrom the exponential case. From the geometric point of view, Poincar \u301eand Sobolev inequalities for differential forms provide a quantitative for-mulation of the vanishing of the cohomology. They have also applicationsto regularity issues for partial differential equations

The dynamics of thin vibrated granular layers

We describe a series of experiments and computer simulations on vibrated granular media in a geometry chosen to eliminate gravitationally induced settling. The system consists of a collection of identical spherical particles on a horizontal plate vibrating vertically, with or without a confining lid. Previously reported results are reviewed, including the observation of homogeneous, disordered liquid-like states, an instability to a `collapse' of motionless spheres on a perfect hexagonal lattice, and a fluctuating, hexagonally ordered state. In the presence of a confining lid we see a variety of solid phases at high densities and relatively high vibration amplitudes, several of which are reported for the first time in this article. The phase behavior of the system is closely related to that observed in confined hard-sphere colloidal suspensions in equilibrium, but with modifications due to the effects of the forcing and dissipation. We also review measurements of velocity distributions, which range from Maxwellian to strongly non-Maxwellian depending on the experimental parameter values. We describe measurements of spatial velocity correlations that show a clear dependence on the mechanism of energy injection. We also report new measurements of the velocity autocorrelation function in the granular layer and show that increased inelasticity leads to enhanced particle self-diffusion.Comment: 11 pages, 7 figure

Monge's transport problem in the Heisenberg group

We prove the existence of solutions to Monge transport problem between two compactly supported Borel probability measures in the Heisenberg group equipped with its Carnot-Caratheodory distance assuming that the initial measure is absolutely continuous with respect to the Haar measure of the group

Buckling Instabilities of a Confined Colloid Crystal Layer

A model predicting the structure of repulsive, spherically symmetric, monodisperse particles confined between two walls is presented. We study the buckling transition of a single flat layer as the double layer state develops. Experimental realizations of this model are suspensions of stabilized colloidal particles squeezed between glass plates. By expanding the thermodynamic potential about a flat state of $N$ confined colloidal particles, we derive a free energy as a functional of in-plane and out-of-plane displacements. The wavevectors of these first buckling instabilities correspond to three different ordered structures. Landau theory predicts that the symmetry of these phases allows for second order phase transitions. This possibility exists even in the presence of gravity or plate asymmetry. These transitions lead to critical behavior and phases with the symmetry of the three-state and four-state Potts models, the X-Y model with 6-fold anisotropy, and the Heisenberg model with cubic interactions. Experimental detection of these structures is discussed.Comment: 24 pages, 8 figures on request. EF508

An insight into polarization states of solid-state organic lasers

The polarization states of lasers are crucial issues both for practical applications and fundamental research. In general, they depend in a combined manner on the properties of the gain material and on the structure of the electromagnetic modes. In this paper, we address this issue in the case of solid-state organic lasers, a technology which enables to vary independently gain and mode properties. Different kinds of resonators are investigated: in-plane micro-resonators with Fabry-Perot, square, pentagon, stadium, disk, and kite shapes, and external vertical resonators. The degree of polarization P is measured in each case. It is shown that although TE modes prevail generally (P>0), kite-shaped micro-laser generates negative values for P, i.e. a flip of the dominant polarization which becomes mostly TM polarized. We at last investigated two degrees of freedom that are available to tailor the polarization of organic lasers, in addition to the pump polarization and the resonator geometry: upon using resonant energy transfer (RET) or upon pumping the laser dye to an higher excited state. We then demonstrate that significantly lower P factors can be obtained.Comment: 12 pages, 12 figure