Langevin and Fokker-Planck equation with nonconventional boundary conditions for the description of domain-wall dynamics in ferromagnetic systems

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Abstract
It is shown that domain-wall dynamics in ferromagnetic systems can be described in terms of a Langevin equation for the domain-wall velocity v. A detailed analysis is given of the complex structure of this equation around v=0, consequent to the peculiar role played by the domain-wall coercive field, which represents, on the one hand, a threshold for domain-wall movement and exhibits, on the other hand, stochastic fluctuations when the domain wall is in motion. By this analysis, a Fokker-Planck equation with specific boundary conditions is derived and solved in terms of a complete, orthogonal set of eigenfunctions. On this basis, the amplitude probability distribution and autocorrelation function of the v process are calculated and discussed, and their relationship with the Barkhausen effect observed in ferromagnetic materials is considered.