The role of noise ensembles of
Misha Neklyudov, University of Tubingen
The dynamics of nanomagnetic particles is described by the
Landau-Lifshitz-Gilbert equation. We show that, in the case of
number of spins, the system relaxes exponentially fast to the unique
invariant measure which is described by a Boltzmann distribution.
provide Arrhenius type law for the rate of the convergence to the
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;
shown to converge weakly at optimal rate for all nite times. We use
general re-sult of Shardlow and Stuart to then conclude convergence
invariant measure of the limiting problem for both discretizations.
last, we discuss the corresponding SPDE and present construction of
solution through nite elements method. The noise is assumed to be
the trace class. Computational examples will be reported to
the theory. This is a joint work with A. Prohl.