1 The role of noise in finite ensembles of nanomagnetic
particles .................................................. 7
1.1 Preliminaries ............................................. 11
1.1.1 Geometric ergodicity of Markov chains .............. 11
1.1.2 Ergodicity with rates for solutions of SDEs ........ 21
1.1.3 Convergent discretizations of the deterministic
LLG equation ....................................... 24
1.2 Exponential Ergodicity and Asymptotic Rates ............... 33
1.2.1 Low-dimensional noise for finitely many
interacting spins .................................. 33
1.2.2 High-dimensional noise for finitely many
interacting spins .................................. 39
1.2.3 L2-ergodicity with rate ............................ 48
1.2.4 Penalization with multiplicative noise ............. 51
1.3 Discretizations of the stochastic Landau-Lifshitz-
Gilbert equation .......................................... 67
1.3.1 A structure-preserving discretization of (1.36):
the geometric exponential ergodicity ............... 67
1.3.2 Strong Convergence of Scheme 1.11 .................. 74
1.3.3 A linear implicit discretization scheme ............ 79
1.4 Computational studies ..................................... 85
1.4.1 Numerical schemes .................................. 86
1.4.2 Long-time dynamics ................................. 93
1.4.3 Interplay of penalization and noise ................ 98
2 The stochastic Landau-Lifshitz-Gilbert equation .......... 103
2.1 Preliminaries ............................................ 106
2.1.1 Finite elements and temporal discretization ....... 106
2.1.2 Fractional Sobolev spaces and related compact
embeddings ........................................ 111
2.1.3 Young integral .................................... 114
2.1.4 Wiener process and the approximating random walk .. 115
2.1.5 Convergence of random variables and
representation theorems ........................... 117
2.1.6 Stability of solutions of the Landau-Lifshitz-
Gilbert equation .................................. 123
2.2 Convergent discretization of SLLG ........................ 129
2.2.1 Unconditional Stability of Scheme 2.9 ............. 138
2.2.2 Convergence of iterates from Scheme 2.9 ........... 155
2.2.3 Existence of a solution to the SLLG equation ...... 163
2.2.4 A convergent discretization of the SLLG equation
which uses random walks ........................... 176
2.3 Computational studies .................................... 186
2.3.1 Numerical implementation .......................... 186
2.3.2 Effects of the space-time white noise in 1D and
2D ................................................ 188
2.3.3 Discrete blow-up of the SLLG equation with
space-time white noise ............................ 190
3 Effective equations for macrospin magnetization
dynamics ................................................. 196
3.1 Construction of local strong solutions for the
augmented LLG ............................................ 200
3.2 Convergence with optimal rates for Scheme A .............. 207
3.3 Construction of a weak solutions via Scheme 3.5 .......... 209
3.3.1 Solving the nonlinear system in Scheme 3.5 ........ 216
3.4 Computational experiments ................................ 220
3.4.1 μMag standard problem no. 4 with thermal
effects ........................................... 220
3.4.2 Comparison of the macroscopic model with the
SLLG equation ..................................... 225
Bibliography .................................................. 236
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