Energy Evolution Equation in a Nonlinear Spin System: Derivation and Numerical Modeling

This study investigates the numerical modeling of local energy evolution in a spin system described by a dissipative generalization of the Landau-Lifshitz-type equation. Beginning with a vector equation that governs the dynamics of the spin field S(x,t), we derive a corresponding scalar equation for the local energy density defined as This representation effectively captures the behavior of the energy associated with spin configurations as they evolve over both time and space, taking into account nonlinear effects and dissipative interactions. The resulting equation reflects multiple physical processes, including the influence of an external scalar potential u(x,t), energy transport via convective terms, diffusion due to viscosity, and energy loss associated with the curvature of the spin field, modeled. We analyze how variations in system parameters such as the shape of the external potential and the dissipation coefficient impact the spatial and temporal behavior of the energy. To facilitate this analysis, we introduce a scalar-field-based framework for observing energy evolution without solving the full spin dynamics. Numerical simulations illustrate the method’s effectiveness in detecting localized energy concentrations and monitoring dissipative wave decay.