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CNA Seminar
Misha Neklyudov
University of Tubingen
Title: The role of noise in ensembles of nanomagnetic particles

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 distribution. Then, we discuss two implicit discretizations to approximate transition functions both, at finite and infinite times: the first 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 finite times. The second scheme is computationally more efficient since it is linear; it is shown to converge weakly at optimal rate for all finite times. We use a general result 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 finite 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.

Recording (avi): http://mm.math.cmu.edu/recordings/cna/CNA-MishaNeklyudov-Oct-02-2012.avi
Recording (mp4): http://mm.math.cmu.edu/recordings/cna/CNA-MishaNeklyudov-Oct-02-2012.mp4
Date: Tuesday, October 2, 2012
Time: 1:30 pm
Location: Wean Hall 7218
Submitted by:  David Kinderlehrer