Magnetization of the 2D Ising model. (a) Arrhenius plot of relaxation time of magnetization in units of MC passes vs . Symbols: MC data; line: Arrhenius fit. Inset: MC trajectory segment of the reduced magnetization at . (b) Free energy profile of the reduced magnetization at temperatures (top to bottom at ).
Variance in the estimator of the mean magnetization as a function of the simulation length measured in units of MC passes. The symbols show the results of a regular MC simulation (top, inverted triangles) and ST simulations with , 5, 7, and 9 temperatures (top to bottom) and corresponding upper temperatures , 2.2, 2.3, and 2.4. The solid lines are the theoretical predictions, , with and defined in Eq. (8) for the MC data and in Eq. (17) for the ST data . The dashed lines are fits of . Note that for , the actual error is slightly larger than predicted.
Efficiency gain in ST (squares) and REMC simulations (triangles) over regular MC sampling of the magnetization of the finite 2D Ising model at . The theoretical prediction, Eq. (1), is shown as a line. The bottom axis indicates the upper temperature in the ST and REMC simulations. The top axis indicates the corresponding number of evenly spaced temperatures in the ST simulations, and replicas in the REMC simulations.
Normalized autocorrelation function of the magnetization irrespective of temperature in ST simulations (solid lines) with , 5, 7, and 9 evenly spaced temperatures and upper temperatures of , 2.2, 2.3, and 2.4. Dashed lines show the theoretical predictions . For reference, the top curve shows for a regular MC run at temperature .
Normalized autocorrelation function of the temperature in ST simulations with nearest-neighbor temperature exchanges (solid lines), , 5, 7, and 9 evenly spaced temperatures and upper temperatures of , 2.2, 2.3, and 2.4. Dashed lines show the theoretical predictions with from Eq. (23). Also shown is the curve for with global temperatures changes permitting distant jumps in the temperature.
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