On the continuous limit for a system of classical spins
type: Item
instance of: scholarly article
title: On the continuous limit for a system of classical spins
zbMATH Open document ID: 0614.35087
DOI: 10.1007/BF01220998
author: Catherine Sulem
author: P. L. Sulem
author: C. Bardos
published in: Communications in Mathematical Physics
publication date: 1986-01-01T00:00:00Z
review text: A classical model for an isotropic ferromagnet is provided by a collection of three-dimensional spin vectors with unit length and arbitrary directions, located at the nodes of a d-dimensional cubic lattice. A simple hypothesis consists in assuming that each spin processes in the local magnetic field created by the closest neighbours. When interested in the phenomena at scales large compared to the lattices mesh size, we are led to consider the equation \[ \partial S/\partial t=S\wedge \Delta S \] where \(S=S(x,t)\) is a three-dimensional continuous vector field. The main results established in the paper concern the existence in the large of weak solutions, obtained as (weak) limits of solutions of the discrete problem when the mesh size goes to zero. For smooth initial conditions, there exists, locally in time, a unique smooth solution. This solution is global in time if initially the spins do display small deviations. In this case, the long time dynamics is essentially linear.
Mathematics Subject Classification ID: 35Q99
zbMATH DE Number: 3994159
zbMATH Keywords: spin dynamics
zbMATH Keywords: uniqueness
zbMATH Keywords: long time dynamics
zbMATH Keywords: isotropic ferromagnet
zbMATH Keywords: spin vectors
zbMATH Keywords: existence in the large
zbMATH Keywords: weak solutions
zbMATH Keywords: limits
zbMATH Keywords: discrete problem
MaRDI profile type: Publication
Wikidata QID: Q56623042
cites work: Q5630031
cites work: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids
cites work: Global, small amplitude solutions to nonlinear evolution equations
cites work: Q5890172
cites work: Global existence of small solutions to nonlinear evolution equations
full work available at URL: https://doi.org/10.1007/bf01220998
OpenAlex ID: W2094983924
description: scientific article; zbMATH DE number 3994159