Wick polynomials and time-evolution of cumulants

We show how Wick polynomials of random variables can be defined combinatorially as the unique choice which removes all "internal contractions" from the related cumulant expansions, also in a non-Gaussian case. We discuss how an expansion in terms of the Wick polynomials can be used for derivation of a hierarchy of equations for the time-evolution of cumulants. These methods are then applied to simplify the formal derivation of the Boltzmann-Peierls equation in the kinetic scaling limit of the discrete nonlinear Schr\"{o}dinger equation (DNLS) with suitable random initial data. We also present a reformulation of the standard perturbation expansion using cumulants which could simplify the problem of a rigorous derivation of the Boltzmann-Peierls equation by separating the analysis of the solutions to the Boltzmann-Peierls equation from the analysis of the corrections. This latter scheme is general and not tied to the DNLS evolution equations

Derivation of the linear Landau equation and linear Boltzmann equation from the Lorentz model with magnetic field

We consider a test particle moving in a random distribution of obstacles in the plane, under the action of a uniform magnetic field, orthogonal to the plane. We show that, in a weak coupling limit, the particle distribution behaves according to the linear Landau equation with a magnetic transport term. Moreover, we show that, in a low density regime, when each obstacle generates an inverse power law potential, the particle distribution behaves according to the linear Boltzmann equation with a magnetic transport term. We provide an explicit control of the error in the kinetic limit by estimating the contributions of the configurations which prevent the Markovianity. We compare these results with those ones obtained for a system of hard disks in \cite{BMHH}, which show instead that the memory effects are not negligible in the Boltzmann-Grad limit.Comment: 22 pages, 4 figures in Journal of Statistical Physics 201

Renormalization of Generalized KPZ Equation

We use Renormalization Group to prove local well posedness for a generalized KPZ equation introduced by H. Spohn in the context of stochastic hydrodynamics. The equation requires the addition of counter terms diverging with a cutoff as and .Peer reviewe

Kinetic Theory and Renormalization Group Methods for Time Dependent Stochastic Systems

By time dependent stochastic systems we indicate efffective models for physical phenomena where the stochasticity takes into account some features whose analytic control is unattainable and/or unnecessary. In particular, we consider two classes of models which are characterized by the different role of randomness: (1) deterministic evolution with random initial data; (2) truly stochastic evolution, namely driven by some sort of random force, with either deterministic or random initial data. As an example of the setting (1) in this thesis we will deal with the discrete nonlinear Schrödinger equation (DNLS) with random initial data and we will mainly focus on its applications concerning the study of transport coefficients in lattice systems. Since the seminal work by Green and Kubo in the mid 50 s, when they discovered that transport coefficients for simple fluids can be obtained through a time integral over the respective total current correlation function, the mathematical physics community has been trying to rigorously validate these predictions and extend them also to solids. In particular, the main technical difficulty is to obtain at least a reliable asymptotic form of the time behaviour of the Green-Kubo correlation. To do this, one of the possible approaches is kinetic theory, a branch of the modern mathematical physics stemmed from the challenge of deriving the classical laws of thermodynamics from microscopic systems. Nowadays kinetic theory deals with models whose dynamics is transport dominated in the sense that typically the solutions to the kinetic equations, whose prototype is the Boltzmann equation, correspond to ballistic motion intercepted by collisions whose frequency is order one on the kinetic space-time scale. Referring to the articles in the thesis by Roman numerals [I]-[V], in [I] and [II] we build some technical tools, namely Wick polynomials and their connection with cumulants, to pave the way towards the rigorous derivation of a kinetic equation called Boltzmann-Peierls equation from the DNLS model. The paper [III] can be contextualized in the same framework of kinetic predictions for transport coefficients. In particular, we consider the velocity flip model which belongs to the family (2) of our previous classification, since it consists of a particle chain with harmonic interaction and a stochastic term which flips the velocity of the particles. In [III] we perform a detailed study of the position-momentum correlation matrix via two diffeerent methods and we get an explicit formula for the thermal conductivity. Moreover, in [IV] we consider the Lorentz model perturbed by an external magnetic field which can be categorized in the class (1): it is a gas of non interacting particles colliding with obstacles located at random positions in the plane. Here we show that under a suitable scaling limit the system is described by a kinetic equation where the magnetic field affects only the transport term, but not the collisions. Finally, in [IV] we studied a generalization of the famous Kardar-Parisi-Zhang (KPZ) equation which falls into the category (2) being a nonlinear stochastic partial differential equation driven by a space-time white noise. Spohn has recently introduced a generalized vector valued KPZ equation in the framework of nonlinear fluctuating hydrodynamics for anharmonic particle chains, a research field which is again strictly connected to the investigation of transport coefficients. The problem with the KPZ equation is that it is ill-posed. However, in 2013 Hairer succeded to give a rigorous mathematical meaning to the solution of the KPZ via an approximation scheme involving the renormalization of the nonlinear term by a formally infinite constant. In [V] we tackle a vector valued generalization of the KPZ and we prove local in time wellposedness by using a technique inspired by the so-called Wilsonian Renormalization Group

Summability of Connected Correlation Functions of Coupled Lattice Fields

We consider two nonindependent random fields and defined on a countable set Z. For instance, or , where I denotes a finite set of possible "internal degrees of freedom" such as spin. We prove that, if the cumulants of and enjoy a certain decay property, then all joint cumulants between and are -summable in the precise sense described in the text. The decay assumption for the cumulants of and is a restricted summability condition called -clustering property. One immediate application of the results is given by a stochastic process whose state is -clustering at any time t: then the above estimates can be applied with and and we obtain uniform in t estimates for the summability of time-correlations of the field. The above clustering assumption is obviously satisfied by any -clustering stationary state of the process, and our original motivation for the control of the summability of time-correlations comes from a quest for a rigorous control of the Green-Kubo correlation function in such a system. A key role in the proof is played by the properties of non-Gaussian Wick polynomials and their connection to cumulants.Peer reviewe

Dynamics of humoral and cellular response to three doses of anti-SARS-CoV-2 BNT162b2 vaccine in patients with hematological malignancies and older subjects

Background: Few data are available about the durability of the response, the induction of neutralizing antibodies, and the cellular response upon the third dose of the anti-severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) vaccine in hemato-oncological patients. Objective: To investigate the antibody and cellular response to the BNT162b2 vaccine in patients with hematological malignancy. Methods: We measured SARS-CoV-2 anti-spike antibodies, anti-Omicron neutralizing antibodies, and T-cell responses 1 month after the third dose of vaccine in 93 fragile patients with hematological malignancy (FHM), 51 fragile not oncological subjects (FNO) aged 80-92, and 47 employees of the hospital (healthcare workers, (HW), aged 23-66 years. Blood samples were collected at day 0 (T0), 21 (T1), 35 (T2), 84 (T3), 168 (T4), 351 (T pre-3D), and 381 (T post-3D) after the first dose of vaccine. Serum IgG antibodies against S1/S2 antigens of SARS-CoV-2 spike protein were measured at every time point. Neutralizing antibodies were measured at T2, T3 (anti-Alpha), T4 (anti-Delta), and T post-3D (anti-Omicron). T cell response was assessed at T post-3D. Results: An increase in anti-S1/S2 antigen antibodies compared to T0 was observed in the three groups at T post-3D. After the third vaccine dose, the median antibody level of FHM subjects was higher than after the second dose and above the putative protection threshold, although lower than in the other groups. The neutralizing activity of antibodies against the Omicron variant of the virus was tested at T2 and T post-3D. 42.3% of FHM, 80,0% of FNO, and 90,0% of HW had anti-Omicron neutralizing antibodies at T post-3D. To get more insight into the breadth of antibody responses, we analyzed neutralizing capacity against BA.4/BA.5, BF.7, BQ.1, XBB.1.5 since also for the Omicron variants, different mutations have been reported especially for the spike protein. The memory T-cell response was lower in FHM than in FNO and HW cohorts. Data on breakthrough infections and deaths suggested that the positivity threshold of the test is protective after the third dose of the vaccine in all cohorts. Conclusion: FHM have a relevant response to the BNT162b2 vaccine, with increasing antibody levels after the third dose coupled with, although low, a T-cell response. FHM need repeated vaccine doses to attain a protective immunological response