Exponential convergence to equilibrium for the homogeneous Landau equation with hard potentials

This paper deals with the long time behaviour of solutions to the spatially homogeneous Landau equation with hard potentials . We prove an exponential in time convergence towards the equilibrium with the optimal rate given by the spectral gap of the associated linearized operator. This result improves the polynomial in time convergence obtained by Desvillettes and Villani \cite{DesVi2}. Our approach is based on new decay estimates for the semigroup generated by the linearized Landau operator in weighted (polynomial or stretched exponential) $L^p$-spaces, using a method develloped by Gualdani, Mischler and Mouhot \cite{GMM}.Comment: 20 pages. Minor corrections, improvement on the presentatio

Quantitative and qualitative Kac's chaos on the Boltzmann's sphere

We investigate the construction of chaotic probability measures on the Boltzmann's sphere, which is the state space of the stochastic process of a many-particle system undergoing a dynamics preserving energy and momentum. Firstly, based on a version of the local Central Limit Theorem (or Berry-Esseen theorem), we construct a sequence of probabilities that is Kac chaotic and we prove a quantitative rate of convergence. Then, we investigate a stronger notion of chaos, namely entropic chaos introduced in \cite{CCLLV}, and we prove, with quantitative rate, that this same sequence is also entropically chaotic. Furthermore, we investigate more general class of probability measures on the Boltzmann's sphere. Using the HWI inequality we prove that a Kac chaotic probability with bounded Fisher's information is entropically chaotic and we give a quantitative rate. We also link different notions of chaos, proving that Fisher's information chaos, introduced in \cite{HaurayMischler}, is stronger than entropic chaos, which is stronger than Kac's chaos. We give a possible answer to \cite[Open Problem 11]{CCLLV} in the Boltzmann's sphere's framework. Finally, applying our previous results to the recent results on propagation of chaos for the Boltzmann equation \cite{MMchaos}, we prove a quantitative rate for the propagation of entropic chaos for the Boltzmann equation with Maxwellian molecules.Comment: 51 pages, to appear in Ann. Inst. H. Poincar\'e Probab. Sta

Sometimes the impact factor outshines the H index

Journal impact factor (which reflects a particular journal's quality) and H index (which reflects the number and quality of an author's publications) are two measures of research quality. It has been argued that the H index outperforms the impact factor for evaluation purposes. Using articles first-authored or last-authored by board members of Retrovirology, we show here that the reverse is true when the future success of an article is to be predicted. The H index proved unsuitable for this specific task because, surprisingly, an article's odds of becoming a 'hit' appear independent of the pre-eminence of its author. We discuss implications for the peer-review process