Stochastic Ferromagnetism (eBook)

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L.Banas, Univ. Bielefeld, Germany; Z.Brzezniak, Univ. York, UK; M.Neklyudov, Univ. Sydney, Australia;A.Prohl, Univ. Tübingen, Germany.

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