Stochastic Ferromagnetism

1 - 1 The role of noise in finite ensembles of nanomagnetic particles [Seite 13]
1.1 - 1.1 Preliminaries [Seite 17]
1.1.1 - 1.1.1 Geometric ergodicity of Markov chains [Seite 17]
1.1.2 - 1.1.2 Ergodicity with rates for solutions of SDEs [Seite 27]
1.1.3 - 1.1.3 Convergent discretizations of the deterministic LLG equation [Seite 30]
1.2 - 1.2 Exponential Ergodicity and Asymptotic Rates [Seite 39]
1.2.1 - 1.2.1 Low-dimensional noise for finitely many interacting spins [Seite 39]
1.2.2 - 1.2.2 High-dimensional noise for finitely many interacting spins [Seite 45]
1.2.3 - 1.2.3 L2-ergodicity with rate [Seite 54]
1.2.4 - 1.2.4 Penalization with multiplicative noise [Seite 57]
1.3 - 1.3 Discretizations of the stochastic Landau-Lifshitz-Gilbert equation [Seite 73]
1.3.1 - 1.3.1 A structure-preserving discretization of (1.36): the geometric exponential ergodicity [Seite 73]
1.3.2 - 1.3.2 Strong Convergence of Scheme 1.11 [Seite 80]
1.3.3 - 1.3.3 A linear implicit discretization scheme [Seite 85]
1.4 - 1.4 Computational studies [Seite 91]
1.4.1 - 1.4.1 Numerical schemes [Seite 92]
1.4.2 - 1.4.2 Long-time dynamics [Seite 99]
1.4.3 - 1.4.3 Interplay of penalization and noise [Seite 104]
2 - 2 The stochastic Landau-Lifshitz-Gilbert equation [Seite 109]
2.1 - 2.1 Preliminaries [Seite 112]
2.1.1 - 2.1.1 Finite elements and temporal discretization [Seite 112]
2.1.2 - 2.1.2 Fractional Sobolev spaces and related compact embeddings [Seite 117]
2.1.3 - 2.1.3 Young integral [Seite 120]
2.1.4 - 2.1.4 Wiener process and the approximating random walk [Seite 121]
2.1.5 - 2.1.5 Convergence of random variables and representation theorems [Seite 123]
2.1.6 - 2.1.6 Stability of solutions of the Landau-Lifshitz-Gilbert equation [Seite 129]
2.2 - 2.2 Convergent discretization of SLLG [Seite 135]
2.2.1 - 2.2.1 Unconditional Stability of Scheme 2.9 [Seite 144]
2.2.2 - 2.2.2 Convergence of iterates from Scheme 2.9 [Seite 161]
2.2.3 - 2.2.3 Existence of a solution to the SLLG equation [Seite 169]
2.2.4 - 2.2.4 A convergent discretization of the SLLG equation which uses random walks [Seite 182]
2.3 - 2.3 Computational studies [Seite 192]
2.3.1 - 2.3.1 Numerical implementation [Seite 192]
2.3.2 - 2.3.2 Effects of the space-time white noise in 1D and 2D [Seite 194]
2.3.3 - 2.3.3 Discrete blow-up of the SLLG equation with space-time white noise [Seite 196]
3 - 3 Effective equations for macrospin magnetization dynamics [Seite 202]
3.1 - 3.1 Construction of local strong solutions for the augmented LLG [Seite 206]
3.2 - 3.2 Convergence with optimal rates for Scheme A [Seite 213]
3.3 - 3.3 Construction of a weak solutions via Scheme 3.5 [Seite 215]
3.3.1 - 3.3.1 Solving the nonlinear system in Scheme 3.5 [Seite 222]
3.4 - 3.4 Computational experiments [Seite 226]
3.4.1 - 3.4.1 µMag standard problem no. 4 with thermal effects [Seite 226]
3.4.2 - 3.4.2 Comparison of the macroscopic model with the SLLG equation [Seite 231]
4 - Bibliography [Seite 242]