Generalized Langevin equation formulation for anomalous diffusion in the Ising model at the critical temperature

Area of Science:

• Statistical physics
• Complex systems

Background:

• The Ising model is a fundamental model in statistical physics.
• Anomalous diffusion is observed in various complex systems, including polymers.

Purpose of the Study:

• To investigate anomalous diffusion in 2D and 3D Ising models.
• To establish a theoretical framework explaining anomalous diffusion and response in the Ising model.

Main Methods:

• Monte Carlo spin flip dynamics simulations.
• Development of a memory-kernel based generalized Langevin equation (GLE).
• Derivation of memory kernels from spin-spin correlation functions.

Main Results:

• Demonstrated anomalous diffusion in bulk and tagged magnetizations for 2D and 3D Ising models.
• Identified time-dependent restoring forces decaying as power laws as the cause of anomalous diffusion.
• Showcased the applicability of a GLE formulation, analogous to polymeric systems, for the Ising model.

Conclusions:

• Anomalous diffusion in the Ising model is driven by power-law decaying restoring forces.
• The GLE framework consistently explains both anomalous diffusion and the Ising model's response to external magnetic fields.
• The study highlights striking similarities between critical phenomena in magnetic systems and dynamics in polymeric systems.