Generalised Langevin Equation Formulation for Anomalous Diffusion in the Ising Model at the Critical Temperature
| Autor: | Zhong, Wei, Panja, Debabrata, Barkema, Gerard T., Ball, Robin C. |
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| Rok vydání: | 2018 |
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| Zdroj: | Phys. Rev. E 98, 012124 (2018) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1103/PhysRevE.98.012124 |
| Popis: | 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. Comment: 22 pages, 7 figures, 20 figure files |
| Databáze: | arXiv |
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