Kinetic Theory and Renormalization Group Methods for Time Dependent Stochastic Systems

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Title: Kinetic Theory and Renormalization Group Methods for Time Dependent Stochastic Systems
Author: Marcozzi, Matteo
Contributor organization: University of Helsinki, Faculty of Science, Department of Mathematics and Statistics
Helsingin yliopisto, matemaattis-luonnontieteellinen tiedekunta, matematiikan ja tilastotieteen laitos
Helsingfors universitet, matematisk-naturvetenskapliga fakulteten, institutionen för matematik och statistik
Publisher: Helsingin yliopisto
Date: 2016-09-30
Language: eng
Thesis level: Doctoral dissertation (article-based)
Abstract: 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.
Subject: mathematics
Rights: Julkaisu on tekijänoikeussäännösten alainen. Teosta voi lukea ja tulostaa henkilökohtaista käyttöä varten. Käyttö kaupallisiin tarkoituksiin on kielletty.

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