Dynamics of nanomagnetic particle systems

The dynamics of nanomagnetic particles is described by the stochastic Landau-Lifshitz-Gilbert (SLLG) equation.

In the first part of the talk we will discuss the long time behaviour of the finite-dimensional SLLG equation. Firstly, we explain how statistical mechanics argument defines the form of the noise of the equation. Then we will consider different approximations of the equation such as structure preserving discretisation and penalisation approximation. We discuss the convergence of approximations and their consistency with the long time behaviour of the system.

In the second part of the talk we will look at the infinite dimensional case. Firstly, we present a numerical scheme convergent to the solution of SLLG equation. Then we show some numerical results and discuss open problems, such as existence of invariant measure, existence of solution in the case of space-time white noise, etc. In particular we will explain why Krylov-Bogoliubov Theorem is not directly applicable to the proof of existence of invariant measure even in the case of coloured noise. In the end we will present certain transformation of SLLG equation which allows to represent the noise as the sum of additive noise and energy conservative noise. Computational examples will be reported to illustrate the theory.

The talk is based on the recently published book (jointly with L. Banas, Z. Brzezniak, A. Prohl) and on the work in progress of the author.