A geometrical interpretation of critical exponents

Abstract

We develop the hypothesis that the dynamics of a given system may lead to the activity being constricted to a subset of space, characterized by a fractal dimension smaller than the space dimension. We also address how the response function might be sensitive to this change in dimensionality. We discuss how this phenomenon is observable in growth processes and near critical points for systems in equilibrium. In particular, we determine the fractal dimension $d_f$ for the Ising model and validate it via computer simulations for two dimensions.

Figures (6)

• Figure 1: The dynamic fractal properties of the $2d$ Ising model. We exhibit the fractal dimension $d_f(T)$ as a function of temperature $T$ for a squared lattice with $L=2048$, and $L=5120$. The horizontal dashed blue line is the theoretical value $d_f=1.75$ at $T=T_c$ . The continuous black line is the the function $f(T)=7/4+a(T-T_c)+b(T-T_c)^2$, after adjusting to the data
• Figure 2: The magnetization $M(T)$ as a function of $T_c-T$, in log scale. $M(T)$ is the time average of $m(T,t)$. The blue dashed straight line is the result of a fit of $M(T) \propto (T_c-T)^\beta$ to the data. From the fit we get $T_c=1.00056(7)$ and $\beta=0.125(1)$, which are close to the exact values $T_c=1$ and $\beta=1/8$.
• Figure 3: (a) The correlation function $G (r)$ as a function of distance $r$ for the specified temperatures. The dashed lines represent the results of the fit using Eq. (\ref{['G']}). (b) Zoom in the result for $T=T_c$, highlighting the agreement between the data and the fit, where the parameters $\rho = 421(3)$ and $\eta = 0.258(1)$ are obtained.
• Figure 4: The correlation length $\rho (T)$ as a function of temperature $T$. Here $L=1024$. In the inset we observe the singularity at $T_c$ which is the best indication of a phase transition. The points are from the simulations and the traced line is the adjustable function Eq. (\ref{['rho']}), we exhibit the points for $T <T_c$ and $T>T_c$. From that we obtain $T_c=1.00105(4)$ and $\nu=1.004(4)$.
• Figure 5: Spin Matrix (a) $T=0.8T_c$, in this homogeneous medium $|\nabla^2G(r)| << | \kappa^2 G(r)|$ ; (b) $T=1.005T_c$, heterogeneous medium $|\nabla^2G(r)| >> | \kappa^2 G(r)|$, clusters require fractal analysis .