In solid-state physics, the Landau–Lifshitz equation (LLE), named for Lev Landau and Evgeny Lifshitz, is a partial differential equation describing time evolution of magnetism in solids, depending on 1 time variable and 1, 2, or 3 space variables.
Landau–Lifshitz equation The LLE describes an anisotropic magnet. The equation is described in (Faddeev & Takhtajan 2007, p. 457) as follows: it is an equation for a vector field S, in other words a function on R1+n taking values in R3. The equation depends on a fixed symmetric 3-by-3 matrix J, usually assumed to be diagonal; that is, J = diag ( J 1 , J 2 , J 3 ) {\displaystyle J=\operatorname {diag} (J_{1},J_{2},J_{3})} . The LLE is then given by Hamilton's equation of motion for the Hamiltonian
H = 1 2 ∫ [ ∑ i ( ∂ S ∂ x i ) 2 − J ( S ) ] d x ( 1 ) {\displaystyle H={\frac {1}{2}}\int \left[\sum _{i}\left({\frac {\partial \mathbf {S} }{\partial x_{i}}}\right)^{2}-J(\mathbf {S} )\right]\,dx\qquad (1)}
(where J(S) is the quadratic form of J applied to the vector S) which is
∂ S ∂ t = S ∧ ∑ i ∂ 2 S ∂ x i 2 + S ∧ J S . ( 2 ) {\displaystyle {\frac {\partial \mathbf {S} }{\partial t}}=\mathbf {S} \wedge \sum _{i}{\frac {\partial ^{2}\mathbf {S} }{\partial x_{i}^{2}}}+\mathbf {S} \wedge J\mathbf {S} .\qquad (2)}
In 1+1 dimensions, this equation is
∂ S ∂ t = S ∧ ∂ 2 S ∂ x 2 + S ∧ J S . ( 3 ) {\displaystyle {\frac {\partial \mathbf {S} }{\partial t}}=\mathbf {S} \wedge {\frac {\partial ^{2}\mathbf {S} }{\partial x^{2}}}+\mathbf {S} \wedge J\mathbf {S} .\qquad (3)}
In 2+1 dimensions, this equation takes the form
∂ S ∂ t = S ∧ ( ∂ 2 S ∂ x 2 + ∂ 2 S ∂ y 2 ) + S ∧ J S ( 4 ) {\displaystyle {\frac {\partial \mathbf {S} }{\partial t}}=\mathbf {S} \wedge \left({\frac {\partial ^{2}\mathbf {S} }{\partial x^{2}}}+{\frac {\partial ^{2}\mathbf {S} }{\partial y^{2}}}\right)+\mathbf {S} \wedge J\mathbf {S} \qquad (4)}
which is the (2+1)-dimensional LLE. For the (3+1)-dimensional case, the LLE looks like
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