Analysis of dynamic Ising model by a variational approximate method: Estimation of transfer entropy

Transfer entropy is an important index for investigating the causal relationship between random variables. It is known that a phase transition can be detected by transfer entropy. In this work, I propose a method for obtaining the transfer entropy by applying the variational approximation method to a dynamic Ising model. Our method is also effective for dynamics that do not satisfy the detailed balance condition. Our method finds the transfer entropy by solving a transcendental equation. When compared with the transfer entropy obtained by Monte Carlo sampling, it was found that they qualitatively matched near the transition point; moreover, the match was obtained with good accuracy in the region deviated from the transition point.

Figure 1

Comparison of the results obtained using the proposed method (open circles) and Monte Carlo simulation results (filled circles). Figures in the first row represent the results obtained by expressing $m$ as a function of $J$. Figures in the second row represent the $N$ times transfer entropy expressed as a function of $J$ in the case of asynchronous update. Figures in the third row indicate the transfer entropy by synchronous update, expressed as a function of $J$. Figures in the left column represent results of Ising model on a square lattice. Figures on the right column represent the results of Ising model on a cubic lattice. Refer to text for explanation of vertical lines.

Figure 2

Schematic diagram of a square lattice. The symbol $•$ on the center represents $i\mathrm{th}$ vertex. The symbols $\blacksquare$ represent vertices adjacent to the $i\mathrm{th}$ vertex and correspond to $\partial i$. The symbols $◯$ represent vertices adjacent to $\blacksquare$ and correspond to $\partial \left(\partial i\right)$.