Fractals and phase transition

We develop the hypothesis that the dynamics of a given system may lead to a fractal dimension d_f different from the original spatial dimension d. This phenomenon is easier to observe near a phase transition. We also speculate how the response function might be sensitive to this change in dimensionality. We discuss how this phenomenon appears in phase transition and growth phenomena. Furthermore, we show that correlations appear as the difference between d − d_f becomes sensitive[1], an effect similar to what occurs in growth phenomena [2-3]. Moreover, we determine [1] the Fisher exponent as η = d − d_f, and the fractal dimension df for the Ising model. We validate it via computer simulations for two dimensions [1]. [1] H. A. Lima, E. E. M. Luis, I. S. S. Carrasco, A. Hansen, F. A. Oliveira. Physical Review E 110, L062107, arXiv preprint arXiv:2402.10167. [2] P. R. H. dos Anjos, W. S. Alves, M. S. Gomes-Filho, D. L. Azevedo and F. A. Oliveira, Frontiers in Physics 9 , 741590 (2021). [3] M. S. Gomes-Filho, Pablo de Castro, D. B. Liarte, and F. A. Oliveira, Entropy, 26, 260 (2024). https://www.mdpi.com/1099-4300/26/3/260