Master Equation for Ising Model

Abstract

A spin system with unperturbed Hamiltonian ${\mathcal{H}}_0=\frac{1}{2}\Sigma \Sigma B_{\mathrm{jk}}{{\sigma }_x}^j{{\sigma }_x}^k-\gamma H\Sigma {{\sigma }_x}^j$ relaxing via the spin-lattice coupling $G=\frac{1}{2}\Sigma \Sigma C_{\mathrm{jk}}{{\sigma }_x}^j({{\sigma }_y}^k+{{\sigma }_z}^k)$ is studied by means of the general density-matrix theory of magnetic relaxation. By making some assumptions about the magnitude and time constants of the lattice correlation functions $〈C_{\mathrm{ij}}\mathrm{}\left(t\right)\mathrm{}{C}_{\mathrm{kl}}\mathrm{}\left(0\right)\mathrm{}〉$, a master equation is obtained. It agrees at high temperatures with a master equation previously suggested by Glauber for the one-dimensional nearest-neighbor case. At high temperatures the magnetic moment relaxes with a single relaxation time, and the spin pair-correlation functions satisfy a closed set of equations. At low temperatures, however, the equations for the magnetization and the correlation functions are coupled to higher-order moments.