Thermal fluctuations cause a random walk of the system in a multidimensional energy space, which ultimately produces a decay of the initial magnetization state. If the value of the thermal energy is comparable to the relevant energy barriers the system will be unstable and will randomly oscillate between the energy minima, presenting superparamagnetism. If this energy is small compared to the energy barrier value, the magnetization will stay in a local minimum during many precession periods and eventually will surmount the energy crest separating two energy minima and appear in the valley of another minimum. This phenomenon is known as thermal activation and is always present in any finite temperature experiment [Basso 00]. Due to the fact that the thermal switching over large energy barriers occurs rarely, the switching events are also known as infrequent events. Thermal activation can be also responsible for domain wall motion on pinning potentials due to defects [Chen 99]. The magnetization thermal stability poses an important problem and is a limitation from the point of view of magnetic-based technologies such as MRAM and magnetic recording. Due to this, the study of thermal activation is a subject of current interest. Moreover, the thermal activation is important in order to understand the magnetic relaxation and the dynamical coercivity phenomena from a fundamental point of view.
In the most general case, when temperature fluctuations are small as compared to energy barriers, the probability of magnetic switching is governed by the Arrhenius-Néel law:
To calculate the prefactor we need some simplifications. The Transition State Theory (TST) simplifies the thermodynamics of the system assuming that the trajectories pass through the saddle point. This theory has its origin in chemical reaction kinetics and was originally developed by H. Kramers [Kramers 40], who demonstrated that the prefactor can be calculated from the probability diffusion equation or Fokker-Planck equation (FPE). Due to this, the original energy minimum is known as reactive state and the final minimum as product state. The first evaluation in a magnetic system was for a single domain particles with uniaxial anisotropy and applied field parallel to the easy axis [Brown 63b]. In this case an asymptotic analytical expression was obtained from the smallest eigenvalue of the FPE equation:
For a general system and Intermediate to High Damping (IHD) case Langer developed an expression for the escape rate [Langer 68]: