Exploring stability characteristics of the Landau–Lifshitz–Bloch equation

Abstract

At elevated temperatures, the Landau–Lifshitz–Bloch equation stands as a cornerstone in elucidating micromagnetic phenomena, particularly in the context of materials utilized for the digital storage of information. The Landau–Lifshitz–Bloch model helps researchers in better understanding and manipulating the dynamics of magnetization, and therefore they can successfully design and build advanced magnetic materials and devices with improved performance and energy efficiency. These advancements are connected to Sustainable Development Goals as they can lead to the development of more sustainable and resilient infrastructure, including energy-efficient devices, advanced renewable energy systems, efficient data storage devices, waste reduction and recycling, and high-performance sensors. In light of this, the present study is dedicated to establishing the long-run stability of equilibrium points within the Landau–Lifshitz–Bloch equation. To achieve this, we employ a rigorous analytical approach, leveraging the Lyapunov function and the linearization of associated problems. The Landau–Lifshitz–Bloch equation, renowned for its efficacy in describing the behavior of micromagnetic systems, forms the basis of our investigation. This equation, especially pertinent in high-temperature regimes, serves as a mathematical lens through which we scrutinize the intricate dynamics of magnetic domains. With contemporary applications extending to the storage of digital data, the stability of magnetic domains emerges as a pivotal concern, necessitating a thorough exploration of equilibrium points. Our analytical strategy hinges on the utilization of the Lyapunov function, a mathematical construct employed to assess the stability of equilibrium points in dynamical systems. This function allows us to gauge the evolution of the system over time, providing insights into whether the system converges to a stable state. Furthermore, we employ the linearization of pertinent problems, a technique that simplifies the mathematical representation of the Landau–Lifshitz–Bloch equation near equilibrium points. Through this linearization process, we gain a tractable framework for analyzing the system’s stability properties. The central contribution of this paper lies in the rigorous proof of the asymptotic stability of equilibrium points within the Landau–Lifshitz–Bloch equation.

Keywords

• Asymptotic stability
• Equilibrium points
• Lyapunov function
• Magnetization dynamics