Oscillations, Stability, and Equilibrium in Magnetic Exchange Networks

The generalized N-site magnetic exchange network is described by coupled Bloch equations whose solution is in general stable and characterized by eigenvalues with negative real parts. The analysis of this system is greatly simplified when detailed balance is satisfied, being described by an exchange matrix with eigenvalues that are negative and purely real. Consequently, the multiple-site system in detailed balance approaches its equilibrium condition by purely exponential relaxation. In addition, satisfaction of detailed balance requires that the stationary solution of the Bloch equations describing the longitudinal magnetization of a spin be the equilibrium magnetization of that spin. Violation of detailed balance with cyclic exchange results in the relaxation of the system taking the form of a damped oscillatory function, with the amount of damping generally decreasing as the number of sites increases.