Landau Lifshitz 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 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... 

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... 

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... 

Obviously, using the Einstein relation, Eq. (4.25) might have been written down right away as soon as the rotary mobility coefficient had been found. This is equally valid, of course, for both the Landau-Lifshitz and Gilbert representations of the magnetodynamic equation. Using formula (4.16) one finds... 

Our starting point is the Landau Lifshitz or Gilbert (LLG) equation for the dynamics of the magnetization M of a single-domain ferromagnetic particle, namely [48 51],... 

We may summarize the contents of this chapter in more detail as follows. In Section I we demonstrate how the explicit form of Gilbert s equation describing Neel relaxation may be written down from the gyromagnetic equation and how, in the limit of low damping, this becomes the Landau-Lifshitz equation. Next the application of this equation to ferrofluid relaxation is discussed together with the analogy to dielectric relaxation. 

This explicit form of Gilbert s equation for the case of low damping is of the same form as the previous Landau-Lifshitz equation. The neglect of the terms 0(17 ) and higher corresponds to the assumption of Landau and Lifshitz that that is, of small damping. This correspondence... 

This is the form of Gilbert s equation used, though not stated, by Brown in [8]. It is essentially of the same form as the Landau-Lifshitz equation except that both prefixes g and h depend on the damping level. 

As we already know Gilbert s equation, (6.1) can be transformed to the equivalent Landau-Lifshitz form... 

We have based our analysis throughout on Gilbert s equation. Raikher and Shliomis [17] based their analysis of this problem on the Landau-Lifshitz equation which has a different direction of precession convention. This results in a dimensionless damping factor a of opposite sign. If we bear this in mind and make use of the low cr approximations of Eq. (6.78) we see that the above equations coincide precisely with those of Raikher and Shliomis s equation (33) [17] for small cr, when we neglect all terms of order and higher. 

As mentioned in Section II, the magnetodynamic equation underlying the Brown kinetic equation (4.125) can be either that by Landau and Lifshitz or that by Gilbert. To be specific, we adopt the former one, noting their equivalence established by formulas (4.5). Thence, the reference relaxation time in Eq. (4.125) is given by Eq. (4.24). 

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