Generalised Langevin Equation Formulation for Anomalous Diffusion in the Ising Model at the Critical Temperature

Abstract

We consider the two- (2D) and three-dimensional (3D) Ising model on a square lattice at the critical temperature $T_c$, under Monte-Carlo spin flip dynamics. The bulk magnetisation and the magnetisation of a tagged line in the 2D Ising model, and the bulk magnetisation and the magnetisation of a tagged plane in the 3D Ising model exhibit anomalous diffusion. Specifically, their mean-square displacement increases as power-laws in time, collectively denoted as $\sim t^c$, where $c$ is the anomalous exponent. We argue that the anomalous diffusion in all these quantities for the Ising model stems from time-dependent restoring forces, decaying as power-laws in time --- also with exponent $c$ --- in striking similarity to anomalous diffusion in polymeric systems. Prompted by our previous work that has established a memory-kernel based Generalised Langevin Equation (GLE) formulation for polymeric systems, we show that a closely analogous GLE formulation holds for the Ising model as well. We obtain the memory kernels from spin-spin correlation functions, and the formulation allows us to consistently explain anomalous diffusion as well as anomalous response of the Ising model to an externally applied magnetic field in a consistent manner.

22 pages, 7 figures, 20 figure files

References in corpus (15)

• Anomalous transport in the crowded world of biological cells
• Anomalous dynamics of cell migration
• Anomalous Dynamics of Unbiased Polymer Translocation through a Narrow Pore
• An introduction to Monte Carlo methods
• Pore-blockade Times for Field-Driven Polymer Translocation
• Anomalous Polymer Dynamics Is Non-Markovian: Memory Effects and The Generalized Langevin Equation Formulation
• Fractional Brownian motion approach to polymer translocation: the governing equation of motion
• Passage Times for Polymer Translocation Pulled through a Narrow Pore
• Polymer Translocation out of Planar Confinements
• Rouse Modes of Self-avoiding Flexible Polymers
• Non-Equilibrium Dynamics of Single polymer Adsorption to Solid Surfaces
• Anomalous diffusion of a tethered membrane: A Monte Carlo investigation
• Probabilistic Phase Space Trajectory Description for Anomalous Polymer Dynamics
• Dynamical Scaling of Polymerized Membranes
• Dynamical Eigenmodes of a Polymerized Membrane

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• Dynamical critical behavior of the two-dimensional three-state Potts model