Mean-field limit and Semiclassical Expansion of a Quantum Particle System

We consider a quantum system constituted by $N$ identical particles interacting by means of a mean-field Hamiltonian. It is well known that, in the limit $N\to\infty$, the one-particle state obeys to the Hartree equation. Moreover, propagation of chaos holds. In this paper, we take care of the $\hbar$ dependence by considering the semiclassical expansion of the $N$-particle system. We prove that each term of the expansion agrees, in the limit $N\to\infty$, with the corresponding one associated with the Hartree equation. We work in the classical phase space by using the Wigner formalism, which seems to be the most appropriate for the present problem.Comment: 44 pages, no figure

A Kac model for kinetic annihilation

In this paper we consider the stochastic dynamics of a finite system of particles in a finite volume (Kac-like particle system) which annihilate with probability $\alpha \in (0,1)$ or collide elastically with probability $1-\alpha$. We first establish the well-posedness of the particle system which exhibits no conserved quantities. We rigorously prove that, in some thermodynamic limit, a suitable hierarchy of kinetic equations is recovered for which tensorized solution to the homogenous Boltzmann with annihilation is a solution. For bounded collision kernels, this shows in particular that propagation of chaos holds true. Furthermore, we make conjectures about the limit behaviour of the particle system when hard-sphere interactions are taken into account.Comment: 40 page

Derivation of the Fick's Law for the Lorentz Model in a low density regime

We consider the Lorentz model in a slab with two mass reservoirs at the boundaries. We show that, in a low density regime, there exists a unique stationary solution for the microscopic dynamics which converges to the stationary solution of the heat equation, namely to the linear profile of the density. In the same regime the macroscopic current in the stationary state is given by the Fick's law, with the diffusion coefficient determined by the Green-Kubo formula.Comment: 33 pages, 7 figure

Analytical approach to relaxation dynamics of condensed Bose gases

The temporal evolution of a perturbation of the equilibrium distribution of a condensed Bose gas is investigated using the kinetic equation which describes collision between condensate and noncondensate atoms. The dynamics is studied in the low momentum limit where an analytical treatment is feasible. Explicit results are given for the behavior at large times in different temperature regimes.Comment: 25 pages, 3 figures. Typos corrected. Final version to appear in Annals of Physic

Semiclassical Propagation of Coherent States for the Hartree equation

In this paper we consider the nonlinear Hartree equation in presence of a given external potential, for an initial coherent state. Under suitable smoothness assumptions, we approximate the solution in terms of a time dependent coherent state, whose phase and amplitude can be determined by a classical flow. The error can be estimated in $L^2$ by C \sqrt {\var}, \var being the Planck constant. Finally we present a full formal asymptotic expansion

Measurement of the inclusive isolated-photon cross section in pp collisions at √s = 13 TeV using 36 fb−1 of ATLAS data

The differential cross section for isolated-photon production in pp collisions is measured at a centre-of-mass energy of 13 TeV with the ATLAS detector at the LHC using an integrated luminosity of 36.1 fb. The differential cross section is presented as a function of the photon transverse energy in different regions of photon pseudorapidity. The differential cross section as a function of the absolute value of the photon pseudorapidity is also presented in different regions of photon transverse energy. Next-to-leading-order QCD calculations from Jetphox and Sherpa as well as next-to-next-to-leading-order QCD calculations from Nnlojet are compared with the measurement, using several parameterisations of the proton parton distribution functions. The predictions provide a good description of the data within the experimental and theoretical uncertainties. [Figure not available: see fulltext.

A Kac model for fermions

We introduce a stochastic $N$-particle system and show that, as $N\to \infty$, an effective description ruled by the homogeneous fermionic Uehling-Uhlenbeck equation is recovered. The particle model we consider is the same as the Kac model for the homogeneous Boltzmann equation with an additional exclusion constraint taking into account the Pauli Exclusion Principle.Comment: 46 page

Strong semiclassical approximation of Wigner functions for the Hartree dynamics

We consider the Wigner equation corresponding to a nonlinear Schroedinger evolution of the Hartree type in the semiclassical limit $\hbar\to 0$. Under appropriate assumptions on the initial data and the interaction potential, we show that the Wigner function is close in $L^2$ to its weak limit, the solution of the corresponding Vlasov equation. The strong approximation allows the construction of semiclassical operator-valued observables, approximating their quantum counterparts in Hilbert-Schmidt topology. The proof makes use of a pointwise-positivity manipulation, which seems necessary in working with the $L^2$ norm and the precise form of the nonlinearity. We employ the Husimi function as a pivot between the classical probability density and the Wigner function, which -- as it is well known -- is not pointwise positive in general.Comment: 24 page

Periodontal Disease and Vitamin D Deficiency in Pregnant Women: Which Correlation with Preterm and Low-Weight Birth?

Periodontal disease seems to be correlated with low vitamin D serum levels, preterm birth (PTB) and low birth weight (LBW), although the literature still lacks a consensus. This study aimed to investigate this correlation in a cohort of pregnant women over 20 weeks of gestation from the University Hospital "Maggiore della Carità", Novara, Italy. We assessed serum levels of vitamin D and oral health status through the following indexes: Oral Hygiene Index (OHI), Plaque Control Record (PCR), Gingival Bleeding Index (GBI), and Community Periodontal Index of Treatment Needs (CPTIN). Moreover, we assessed the number of PTB and LBW among the newborns. Out of 121 pregnant women recruited, 72 (mean age 29.91 ± 3.64 years) were included. There was a statistically significant correlation between preterm and OHI > 3 (p = 0.033), and between LBW and OHI > 3 (p = 0.005) and CPITN = 3 (p = 0.027). Both pregnant women with vitamin D deficiency ((25-hydroxy-vitamin D) < 30 ng/mL) and PTB plus LBW newborns were significantly correlated (p < 0.05) with poor levels of all oral health status indexes during pregnancy. Furthermore, these conditions (women with hypovitaminosis D and combination of PTB and LBW) were shown to be significantly correlated (p < 0.001). Taken together, our findings reported a high prevalence of PTB and LBW with poor oral health and vitamin D deficiency in pregnant women