Dynamics of ferromagnets: Langevin approach to the mean spherical model

Abstract

Formal similarities between the Hamiltonian of a chain of harmonic oscillators and the effective Hamiltonian of the mean spherical model are exploited by introducing a ’’white noise’’ Langevin equation which reproduces the equilibrium properties of the 3D mean spherical model. Application to the dynamics with no magnetic field is made in three steps: (a) determination of the space–time correlation function at equilibrium; (b) spin relaxation calculations at different constant temperatures; (c) prediction of thermodynamic behavior after sudden coolings from T0?Tc to T?Tc. Dynamic transitions with a final equilibrium state below Tc have a common feature: The approach to global equilibrium requires a time proportional to νγ, where ν is the total number of lattice sites; γ=2/3 for cooling to T?Tc and spin relaxation at T=Tc, whereas γ=1 for spin relaxation at T<Tc. An actually infinite system does not recover symmetry after spin relaxation below Tc, and forms expanding ’’islands’’ of correlated spins after cooling to T?Tc.