From water shifting between solid, liquid, and gas to the competition among different species in the biosphere, and even the competition of financial market forces in human society, phase transition phenomena exist widely in nature. 

How to predict the critical point of a phase transition—the "tipping point" where a system's properties change abruptly—is essential for avoiding sudden system collapses. To develop a unified method to predict these points has been a challenge as systems behave differently under equilibrium and non-equilibrium conditions.

In a study published in Physical Review Letters, researchers from the Institute of Physics (IOP) of the Chinese Academy of Sciences developed a unified method to predict critical "tipping points" in magnetic systems.

Researchers analyzed the frequency-dependent response of a magnetic system to small external perturbations, and found that equilibrium phase transitions such as field-driven magnetization switching exhibited a divergence in the static response as the system nears its critical point. 

However, researchers found that this static indicator failed for nonequilibrium phase transitions such as magnetization switching driven by spin-transfer torque. For these systems, they demonstrated that the dynamic response at systems' ferromagnetic resonance frequency diverged instead.

Furthermore, researchers demonstrated that these static and dynamic indicators can be unified within the general framework of first-order linear differential systems, showing the broad applicability of their theory. Both indicators were also found to be robust against thermal noise, which is important for practical applications.

"This work provides a unified, low-cost, and robust strategy for determining critical points in both types of phase transitions," said Prof. HAN Xiufeng from IOP. "We believe this framework can be generalized and has potential for applications in magnetoresistive random-access memory devices, critical-enhanced sensors, and other complex systems.