Rényi information flow in the Ising model with single-spin dynamics

The $n$-index Rényi mutual information and transfer entropies for the two-dimensional kinetic Ising model with arbitrary single-spin dynamics in the thermodynamic limit are derived as functions of ensemble averages of observables and spin-flip probabilities. Cluster Monte Carlo algorithms with different dynamics from the single-spin dynamics are thus applicable to estimate the transfer entropies. By means of Monte Carlo simulations with the Wolff algorithm, we calculate the information flows in the Ising model with the Metropolis dynamics and the Glauber dynamics, respectively. We find that not only the global Rényi transfer entropy, but also the pairwise Rényi transfer entropy, peaks in the disorder phase.

Figure 1

Plot of $N{T}_{\mathrm{pw}}$ for the Metropolis dynamics against temperature for several system sizes. The statistical errors are much smaller than the symbol sizes. The inset shows the maximum of $T_{\mathrm{pw}}$ as a function of $1/L$, in which the horizontal dashed line indicates $T_c=2.2692$.

Figure 2

Plot of $N{T}_{\mathrm{gl}}$ for the Metropolis dynamics against temperature for several system sizes. The dashed vertical line indicates $T_c=2.2692$. The statistical errors are much smaller than the symbol sizes.

Figure 3

$\langle P_i\rangle$ (upper panel) and $\langle S_j{P}_i\rangle$ (lower panel) plotted against temperature for the 2D Ising model with the Metropolis dynamics. The statistical errors are much smaller than the symbol sizes.

Figure 4

$\langle P_i\rangle$ (upper panel) and $\langle S_j{P}_i\rangle$ (lower panel) plotted against temperature for the 2D Ising model with the Glauber dynamics. The statistical errors are much smaller than the symbol sizes.

Figure 5

Plot of $I_{\mathrm{pw}}^{\mathrm{R}}$ (upper panel) and $I_{\mathrm{gl}}^{\mathrm{R}}/N$ (lower panel) against temperature.

Figure 6

$N^3{T}_{\mathrm{pw}}^{\mathrm{R}}$ (upper panel) and $N^3{T}_{\mathrm{gl}}^{\mathrm{R}}$ (lower panel) plotted against temperature for the 2D Ising model with Metropolis dynamics for several system sizes $L$. The statistical errors are much smaller than the symbol sizes.

Figure 7

$N^3{T}_{\mathrm{pw}}^{\mathrm{R}}$ (upper panel) and $N^3{T}_{\mathrm{gl}}^{\mathrm{R}}$ (lower panel) as a function of temperature for the 2D Ising model with the Glauber dynamics for several system sizes. The statistical errors are much smaller than the symbol sizes.