Misha Neklyudov, University of Tubingen

Abstract:

The dynamics of nanomagnetic particles is described by the stochastic Landau-Lifshitz-Gilbert equation. We show that, in the case of finite number of spins, the system relaxes exponentially fast to the unique invariant measure which is described by a Boltzmann distribution. Furthermore, we provide Arrhenius type law for the rate of the convergence to the dis- tribution. Then, we discuss two implicit discretizations to approximate transition functions both, at nite and innite times: the rst scheme is shown to inherit the geometric `unit-length' property of single spins, as well as the Lyapunov structure, and is shown to be geometrically ergodic; moreover, iterates converge strongly with rate for nite times. The second scheme is computationally more efficient since it is linear; it is shown to converge weakly at optimal rate for all nite times. We use a general re-sult of Shardlow and Stuart to then conclude convergence to the invariant measure of the limiting problem for both discretizations. At last, we discuss the corresponding SPDE and present construction of the solution through nite elements method. The noise is assumed to be of the trace class. Computational examples will be reported to illustrate the theory. This is a joint work with A. Prohl.