The Role of Noise in Finite Ensembles of Nanomagnetic Particles

Abstract

The dynamics of finitely many nanomagnetic particles are described by the stochastic Landau–Lifshitz–Gilbert equation. We show that the system relaxes exponentially quickly to the unique invariant measure which is described by a Boltzmann distribution. We present two approaches to verify this result. The first uses the general theory of (Meyn and Tweedy, Adv Appl Prob 24:542–574, 1992; Meyn and Tweedy, Adv Appl Prob 25:487–517 1993; Meyn and Tweedy, Adv Appl Prob 25:518–548, 1993) for Markov chains, which involves the concepts of a Lyapunov structure, and irreducibility of transition probabilities; we show exponential convergence in a supremum topology, but lack explicit rates. The second approach shows exponential ergodicity in a weaker L2 topology, with an explicit rate of convergence of the Arrhenius type law. Then, we discuss two implicit discretizations to approximate transition functions at both 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 a rate for finite times. The second scheme is computationally more efficient since it is linear; it is shown to converge weakly at an optimal rate for all finite times. We use a general result of Shardlow and Stuart (Siam J Numer Anal 37(4):1120–1137 2000) to conclude convergence to the invariant measure of the limiting problem for both discretizations.