A coherence theorem for pseudonatural transformations

We prove coherence theorems for bicategories, pseudofunctors and pseudonatural transformations. These theorems boil down to proving the coherence of some free $(4,2)$-categories. In the case of bicategories and pseudofunctors, existing rewriting techniques based on Squier's Theorem allow us to conclude. In the case of pseudonatural transformations this approach only proves the coherence of part of the structure, and we use a new rewriting result to conclude. To this end, we introduce the notions of white-categories and partial coherence

Mean field limit for the one dimensional Vlasov-Poisson equation

We consider systems of $N$ particles in dimension one, driven by pair Coulombian or gravitational interactions. When the number of particles goes to infinity in the so called mean field scaling, we formally expect convergence towards the Vlasov-Poisson equation. Actually a rigorous proof of that convergence was given by Trocheris in \cite{Tro86}. Here we shall give a simpler proof of this result, and explain why it implies the so-called "Propagation of molecular chaos". More precisely, both results will be a direct consequence of a weak-strong stability result on the one dimensional Vlasov-Poisson equation that is interesting by it own. We also prove the existence of global solutions to the $N$ particles dynamic starting from any initial positions and velocities, and the existence of global solutions to the Vlasov-Poisson equation starting from any measures with bounded first moment in velocity.Comment: 16 pages, to appear in S\'eminaire Laurent Schwartz (2012-2013

ALL(pT)A_{LL}(p_T) for single hadron photoproduction at high pTp_T

In order to understand the gluon contribution to the nucleon spin, some experiments can study the production of hadrons at high transverse momemtum from lepton-nucleon or nucleon-nucleon scattering. RHIC has recently measured such double spin asymmetries $A_{LL}(p_T)$ for pion production at high center of mass energies, and inclusion of its data to global fits based on NLO collinear pQCD calculations gives some constraints on the gluon polarization in the range $0.05<x_g<0.2$. To complement these results at COMPASS range, we will present preliminary COMPASS results on double longitudinal spin asymmetries $A_{LL}(p_T)$ for single hadron production measured on deuteron and proton target at $Q^2<1$ GeV$^2$, $p_T>1$ GeV/c and center of mass energy $\sqrt{s}\approx 18$ GeV. All COMPASS data taken from 2002 to 2011 by scattering 160 GeV polarized muons on longitudinally polarized $^6$LiD and NH$_3$ targets have been used, and the number of hadrons collected with $p_T>1$ GeV/c for this analysis amounts to about 10 millions. The obtained asymmetries will be compared to theoretical predictions of at NLO without gluon resummation calculation.Comment: Talk presented CIPANP2015. 6 pages, LaTeX, 7 pdf figures and 1 png figur

Uniform contractivity in Wasserstein metric for the original 1D Kac's model

We study here a very popular 1D jump model introduced by Kac: it consists of $N$ velocities encountering random binary collisions at which they randomly exchange energy. We show the uniform (in $N$) exponential contractivity of the dynamics in a non-standard Monge-Kantorovich-Wasserstein: precisely the MKW metric of order 2 on the energy. The result is optimal in the sense that for each $N$, the contractivity constant is equal to the $L^2$ spectral gap of the generator associated to Kac's dynamic. As a corollary, we get an uniform but non optimal contractivity in the MKW metric of order $4$. We use a simple coupling that works better that the parallel one. The estimates are simple and new (to the best of our knowledge).Comment: 5 page