Improved dynamical scaling analysis using the kernel method for nonequilibrium relaxation

The dynamical scaling analysis for the Kosterlitz-Thouless transition in the nonequilibrium relaxation method is improved by the use of Bayesian statistics and the kernel method. This allows data to be fitted to a scaling function without using any parametric model function, which makes the results more reliable and reproducible and enables automatic and faster parameter estimation. Applying this method, the bootstrap method is introduced and a numerical discrimination for the transition type is proposed.

Figure 1

Relaxation of magnetization for the $XY$ model in two dimensions for $T=0.92,0.93,0.94,0.95,0.96,0.97,0.98,0.99,1.00,1.01,1.02,1.05$ plotted on a double-logarithmic scale. For each temperature, 100 data points are chosen so as to give equal intervals on the $X$ axis.

Figure 2

Scaling plot for the data in Fig. 1 with $T_{\text{KT}}=0.8955$ and $\lambda =0.069$.

Figure 3

Relaxation of magnetization for the $XY$ model in two dimensions including $T=0.91,0.90$. The data points for $T=0.90$ have no overlap with those for other temperatures.

Figure 4

Scaling plot for the data in Fig. 3 with $T_{\text{KT}}=0.8977$ and $\lambda =0.070$. The isolated sector corresponds to the data for $T=0.90$.

Figure 5

Example of random sampling for the relaxation of magnetization for the $XY$ model in two dimensions. For each temperature, 100 data points are chosen at random along the $X$ axis.

Figure 6

Scaling plot for the data in Fig. 5.

Figure 7

Relaxation of magnetization for the Ising model in two dimensions for $T=2.268$ and 2.270 with a double-logarithmic scale. Note the upward trend for $T=2.268$ and the downward trend for 2.270.

Figure 8

Relaxation of magnetization for the Ising model in two dimensions for $T=2.270,2.271,2.272,2.273,2.274,2.275,2.276,2.277,2.278,2.279$ with a double-logarithmic scale.

Figure 9

Relaxation of magnetization for the $XY$ model in two dimensions for $T=0.89$ and 0.92 with a double-logarithmic scale. The trend is weak and hard to observe, but the data indicate a downward trend for $T=0.92$ and an almost straight line for 0.89.

Figure 10

Estimated relaxation time $\tau$ for the Ising model in two dimensions. Three different dynamical scaling results are plotted for the function $\tau \left(T\right)$ under the assumption of a second-order transition, KT transition, and without assumption (closed circles).

Figure 11

Estimated relaxation time $\tau$ for the $XY$ model in two dimensions. Three different dynamical scaling results are plotted for the function $\tau \left(T\right)$, under the assumption of a second-order transition, KT transition, and without assumption (closed circles).