Knot state asymptotics I, AJ Conjecture and abelian representations

Consider the Chern-Simons topological quantum field theory with gauge group SU(2) and level k. Given a knot in the 3-sphere, this theory associates to the knot exterior an element in a vector space. We call this vector the knot state and study its asymptotic properties when the level is large. The latter vector space being isomorphic to the geometric quantization of the SU(2)-character variety of the peripheral torus, the knot state may be viewed as a section defined over this character variety. We first conjecture that the knot state concentrates in the large level limit to the character variety of the knot. This statement may be viewed as a real and smooth version of the AJ conjecture. Our second conjecture says that the knot state in the neighborhood of abelian representations is a Lagrangian state. Using microlocal techniques, we prove these conjectures for the figure eight and torus knots. The proof is based on q-difference relations for the colored Jones polynomial. We also provide a new proof for the asymptotics of the Witten-Reshetikhin-Turaev invariant of the lens spaces and a derivation of the Melvin-Morton-Rozansky theorem from the two conjectures.Comment: 47 pages, 2 figure

Direct numerical simulation of unsheared turbulence diffusing toward a free-slip or no-slip surface

The physics involved in the interaction between statistically steady, shearless turbulence and a blocking surface is investigated with the aid of direct numerical simulation. The original conguration introduced by Campagne et al. [ECCOMAS CFD 2006] serves as the basis for comparing cases in which the blocking surface can be either a free-slip surface or a no-slip wall. It is shown that in both cases, the evolutions of the anisotropy state are the same throughout the surface-influenced layer (down to the surface), despite the essentially different natures of the inner layers. The extent of the blocking effect can thereby be measured through a local (surface) quantity identically defined in the two cases. Examination of the evolution and content of the pressure-strain correlation brings information on the mechanisms by which energy is exchanged between the normal and tangential directions: In agreement with an earlier analysis by Perot and Moin [J. Fluid Mech. 295 (1995)], it appears that the level of the pressure strain correlation is governed by a splat/antisplat disequilibrium which is larger in the case of the solid wall due to viscous effects. However, in contradiction with the latter, the pressure-strain correlation remains as a signicant contributor to both Reynolds-stress budgets; it is argued that the net level of the splat/antisplat disequilibrium is set, in the first place, by the normal-velocity skewness of the interacting turbulent field. The influence of viscous friction on the intercomponent energy transfer at the solid wall only comes in the second place and part of it can also be measured by the skewness. The remainder seems to originate from interactions between the strain field and ring-like vortices in the vicinity of the splats

Large scale simulation of turbulence using a hybrid spectral/finite difference solver

Performing Direct Numerical Simulation (DNS) of turbulence on large-scale systems (offering more than 1024 cores) has become a challenge in high performance computing. The computer power increase allows now to solve flow problems on large grids (with close to 10^9 nodes). Moreover these large scale simulations can be performed on non-homogeneous turbulent flows. A reasonable amount of time is needed to converge statistics if the large grid size is combined with a large number of cores. To this end we developed a Navier-Stokes solver, dedicated to situations where only one direction is heterogeneous, and particularly suitable for massive parallel architecture. Based on an hybrid approach spectral/finite-difference, we use a volumetric decomposition of the domain to extend the FFTs computation to a large number of cores. Scalability tests using up to 32K cores as well as preliminary results of a full simulation are presented

Comparison between classical potentials and ab initio for silicon under large shear

The homogeneous shear of the {111} planes along the <110> direction of bulk silicon has been investigated using ab initio techniques, to better understand the strain properties of both shuffle and glide set planes. Similar calculations have been done with three empirical potentials, Stillinger-Weber, Tersoff and EDIP, in order to find the one giving the best results under large shear strains. The generalized stacking fault energies have also been calculated with these potentials to complement this study. It turns out that the Stillinger-Weber potential better reproduces the ab initio results, for the smoothness and the amplitude of the energy variation as well as the localization of shear in the shuffle set