9,388 research outputs found
Random weighted Sobolev inequalities and application to quantum ergodicity
This paper is a continuation of Poiret-Robert-Thomann (2013) where we studied
a randomisation method based on the Laplacian with harmonic potential. Here we
extend our previous results to the case of any polynomial and confining
potential on . We construct measures, under concentration
type assumptions, on the support of which we prove optimal weighted Sobolev
estimates on . This construction relies on accurate estimates on
the spectral function in a non-compact configuration space. Then we prove
random quantum ergodicity results without specific assumption on the classical
dynamics. Finally, we prove that almost all basis of Hermite functions is
quantum uniquely ergodic.Comment: Clarifications added in the part concerning QU
A Stochastic Representation of the Local Structure of Turbulence
Based on the mechanics of the Euler equation at short time, we show that a
Recent Fluid Deformation (RFD) closure for the vorticity field, neglecting the
early stage of advection of fluid particles, allows to build a 3D
incompressible velocity field that shares many properties with empirical
turbulence, such as the teardrop shape of the R-Q plane. Unfortunately, non
gaussianity is weak (i.e. no intermittency) and vorticity gets preferentially
aligned with the wrong eigenvector of the deformation. We then show that
slightly modifying the former vectorial field in order to impose the long range
correlated nature of turbulence allows to reproduce the main properties of
stationary flows. Doing so, we end up with a realistic incompressible, skewed
and intermittent velocity field that reproduces the main characteristics of 3D
turbulence in the inertial range, including correct vorticity alignment
properties.Comment: 6 pages, 3 figures, final version, published
Born Reciprocity in String Theory and the Nature of Spacetime
After many years, the deep nature of spacetime in string theory remains an
enigma. In this letter we incorporate the concept of Born reciprocity in order
to provide a new point of view on string theory in which spacetime is a derived
dynamical concept. This viewpoint may be thought of as a dynamical chiral phase
space formulation of string theory, in which Born reciprocity is implemented as
a choice of a Lagrangian submanifold of the phase space, and amounts to a
generalization of T-duality. In this approach the fundamental symmetry of
string theory contains phase space diffeomorphism invariance and the underlying
string geometry should be understood in terms of dynamical bi-Lagrangian
manifolds and an apparently new geometric structure, somewhat reminiscent of
para-quaternionic geometry, which we call Born geometry
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