Gilbert equation

We will present the equation of motion for a classical spin (the magnetic moment of a ferromagnetic single-domain particle) in the context of the theory of stochastic processes. The basic Langevin equation is the stochastic Landau-Lifshitz(-Gilbert) equation [5,45]. More details on this subject and various techniques to solve this equation can be found in the reviews by Coffey et al. [46] and Garcia-Palacios [8]. 

The Gilbert equation can be cast into the Landau-Lifshitz form [48]... 

In the work of Brown [5] and Kubo and Hashitsume [45] the starting equation is the Gilbert equation (3.43), in which the effective field is increased by a fluctuating field yielding the stochastic Gilbert equation. This equation can, as in the deterministic case, be cast into the Landau-Lifshitz form as... 

The phenomenological equation that determines the motion of the magnetic moment of a ferromagnetic sample is known generically as the Landau-Lifshitz-Gilbert equation and has two basic modifications. The first form... 

The dynamic response of a magnetic particle subject to an applied field follows from the Gilbert equation of motion [7]... 

This equation is mathematically equivalent to the Landau-Lifshitz-Gilbert equation (LLG)... 

In order to treat thermal effects on small time scales, a random thermal field, Hth, is added to the effective field in the Landau-Lifshitz Gilbert equation. The thermal field is a Gaussian random process with the following statistical properties... 

Neel s calculation of t was criticized by Brownbecause the system is not explicitly treated as a gyromagnetic one and it supposes a discrete orientation approximation (m is essentially along the easy directions). He supposed that the magnetic moment orientations may be described by a Gilbert equation augmented by a random field term h(t), which is assumed to be white noise ... 

The use of Gilbert s equation and the meaning of the parameters used in it need some discussions. In fact, the Gilbert equation is concerned with spins and their influence on the determination of the relaxation time of the magnetic moment m set. Brown has overcome the difficulty of different directions of spins by supposing that the particle spins are parallel, with the same moment and that they are rigidly coupled. However, at the particle surface, disorders exist for the spin directions and variations of moments can be present (see Section B). 

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