LandauLifshitzGilbert

The Landau-Lifshitz-Gilbert equation is a fundamental equation in magnetism that describes the time evolution of the magnetization M(r, t) in ferromagnetic materials within the continuum approximation. It combines the precessional motion of the magnetization around an effective magnetic field with a damping mechanism that aligns M toward the field direction over time.

In its common form for the magnetization vector M of saturation M_s, the equation is written as

dM/dt = -γ M × H_eff + (α / M_s) M × dM/dt.

Here γ is the gyromagnetic ratio, α is a dimensionless damping parameter, and H_eff is the effective magnetic field that includes external fields as well as contributions from magnetic anisotropy, exchange interactions, and demagnetizing (dipolar) effects. An equivalent form can be written for the normalized magnetization m = M / M_s, often used in simulations.

The effective field H_eff is defined by the functional derivative of the magnetic energy F[M], typically H_eff = -δF/δM, and thus H_eff may incorporate multiple energy terms such as exchange stiffness, magnetocrystalline anisotropy, Zeeman energy, and long-range dipolar interactions.

The LLG equation describes two intertwined processes: a precession of M around H_eff with frequency set by γ, and a damping toward alignment with H_eff driven by α. It is widely used in micromagnetics and spintronics to model phenomena such as domain-wall motion, magnetization switching, and ultrafast magnetization dynamics. In many applications, the equation is solved numerically with appropriate discretization and time-integration schemes, sometimes augmented with additional torques (e.g., spin-transfer torque) in more complete models.