Landau–Lifshitz model
Encyclopedia
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.
. It is given by Hamilton's equation of motion for the Hamiltonian
(where J(S) is the quadratic form of J applied to the vector S)
which is
In 1+1 dimensions this equation is
In 2+1 dimensions this equation takes the form
which is the (2+1)-dimensional LLE. For the (3+1)-dimensional case LLE looks like
Solid-state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from...
, the Landau–Lifshitz equation (LLE), named for Lev Landau
Lev Landau
Lev Davidovich Landau was a prominent Soviet physicist who made fundamental contributions to many areas of theoretical physics...
and Evgeny Lifshitz
Evgeny Lifshitz
Evgeny Mikhailovich Lifshitz was a leading Soviet physicist of Jewish origin and the brother of physicist Ilya Mikhailovich Lifshitz. Lifshitz is well known in general relativity for coauthoring the BKL conjecture concerning the nature of a generic curvature...
, is a partial differential equation describing time evolution of magnetism
Magnetism
Magnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well...
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 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,. It is given by Hamilton's equation of motion for the Hamiltonian(where J(S) is the quadratic form of J applied to the vector S)
which is
In 1+1 dimensions this equation is
In 2+1 dimensions this equation takes the form
which is the (2+1)-dimensional LLE. For the (3+1)-dimensional case LLE looks like
Integrable reductions
In general case LLE (2) is nonintegrable. But it admits the two integrable reductions:- a) in the 1+1 dimensions, that is Eq. (3), it is integrable
- b) when
. In this case the (1+1)-dimensional LLE (3) turns into the continuous classical Heisenberg ferromagnet equation (see e.g. Heisenberg model (classical)Heisenberg model (classical)The Classical Heisenberg model is the n = 3 case of the n-vector model, one of the models used in statistical physics to model ferromagnetism, and other phenomena.-Definition:...
) which is already integrable.
See also
- Nonlinear Schrödinger equationNonlinear Schrödinger equationIn theoretical physics, the nonlinear Schrödinger equation is a nonlinear version of Schrödinger's equation. It is a classical field equation with applications to optics and water waves. Unlike the Schrödinger equation, it never describes the time evolution of a quantum state...
- Heisenberg model (classical)Heisenberg model (classical)The Classical Heisenberg model is the n = 3 case of the n-vector model, one of the models used in statistical physics to model ferromagnetism, and other phenomena.-Definition:...
- Spin waveSpin waveSpin waves are propagating disturbances in the ordering of magnetic materials. These low-lying collective excitations occur in magnetic lattices with continuous symmetry. From the equivalent quasiparticle point of view, spin waves are known as magnons, which are boson modes of the spin lattice...
- MicromagnetismMicromagnetismMicromagnetics deals with the interactions between magnetic moments on sub-micrometre length scales. These are governed by several competing energy terms. Dipolar energy is the energy which causes magnets to align north to south pole...
- Ishimori equationIshimori equationThe Ishimori equation is a partial differential equation proposed by the Japanese mathematician . Its interest is as the first example of a nonlinear spin-one field model in the plane that is integrable .-Equation:The IE has the form...
- MagnetMagnetA magnet is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, and attracts or repels other magnets.A permanent magnet is an object...
- FerromagnetismFerromagnetismFerromagnetism is the basic mechanism by which certain materials form permanent magnets, or are attracted to magnets. In physics, several different types of magnetism are distinguished...