Stochastic Ferromagnetism: Analysis and Numerics

Description

This monograph examines magnetization dynamics at elevated temperatures which can be described by the stochastic Landau-Lifshitz-Gilbert equation (SLLG). The first part of the book studies the role of noise in finite ensembles of nanomagnetic particles: we show geometric ergodicity of a unique invariant measure of Gibbs type and study related properties of approximations of the SLLG, including time discretization and Ginzburg-Landau type penalization. In the second part we propose an implementable space-time discretization using random walks to construct a weak martingale solution of the corresponding stochastic partial differential equation which describes the magnetization process of infinite spin ensembles. The last part of the book is concerned with a macroscopic deterministic equation which describes temperature effects on macro-spins, i.e. expectations of the solutions to the SLLG. Furthermore, comparative computational studies with the stochastic model are included. We use constructive tools such as e.g. finite element methods to derive the theoretical results, which are then used for computational studies. The numerical experiments motivate an interesting interplay between inherent geometric and stochastic effects of the SLLG which still lack a rigorous analytical understanding: the role of space-time white noise, possible finite time blow-up behavior of solutions, long-time asymptotics, and effective dynamics. 1 The role of noise in finite ensembles of nanomagnetic particles 1.1 Preliminaries 1.1.1 Geometric ergodicity of Markov chains 1.1.2 Ergodicity with rates for solutions of SDEs 1.1.3 Convergent discretizations of the deterministic LLG equation 1.2 Exponential Ergodicity and Asymptotic Rates 1.2.1 Low-dimensional noise for finitely many interacting spins 1.2.2 High-dimensional noise for finitely many interacting spins 1.2.3 L2-ergodicity with rate 1.2.4 Penalization with multiplicative noise 1.3 Discretizations of the stochastic Landau-Lifshitz-Gilbert equation 1.3.1 A structure-preserving discretization of (1.36): the geometric exponential ergodicity 1.3.2 Strong Convergence of Scheme 1.11 1.3.3 A linear implicit discretization scheme 1.4 Computational studies 1.4.1 Numerical schemes 1.4.2 Long-time dynamics 1.4.3 Interplay of penalization and noise 2 The stochastic Landau-Lifshitz-Gilbert equation 2.1 Preliminaries 2.1.1 Finite elements and temporal discretization 2.1.2 Fractional Sobolev spaces and related compact embeddings 2.1.3 Young integral 2.1.4 Wiener process and the approximating random walk 2.1.5 Convergence of random variables and representation theorems 2.1.6 Stability of solutions of the Landau-Lifshitz-Gilbert equation 2.2 Convergent discretization of SLLG 2.2.1 Unconditional Stability of Scheme 2.9 2.2.2 Convergence of iterates from Scheme 2.9 2.2.3 Existence of a solution to the SLLG equation 2.2.4 A convergent discretization of the SLLG equation which uses random walks 2.3 Computational studies 2.3.1 Numerical implementation 2.3.2 Effects of the space-time white noise in 1D and 2D 2.3.3 Discrete blow-up of the SLLG equation with space-time white noise 3 Effective equations for macrospin magnetization dynamics 3.1 Construction of local strong solutions for the augmented LLG 3.2 Convergence with optimal rates for Scheme A 3.3 Construction of a weak solutions via Scheme 3.5 3.3.1 Solving the nonlinear system in Scheme 3.5 3.4 Computational experiments 3.4.1 µMag standard problem no. 4 with thermal effects 3.4.2 Comparison of the macroscopic model with the SLLG equation Bibliography