We performed two-dimensional simulated tempering (ST) simulations of the two-dimensional Ising model with different lattice sizes in order to investigate the two-dimensional STs applicability to dealing with phase transitions and to study the crossover of critical scaling behavior. The external field, as well as the temperature, was treated as a dynamical variable updated during the simulations. Thus, this simulation can be referred to as Simulated Tempering and Magnetizing (STM). We also performed the Simulated Magnetizing (SM) simulations, in which the external field was considered as a dynamical variable and temperature was not. As has been discussed by previous studies, the ST method is not always compatible with first-order phase transitions. This is also true in the magnetizing process. Flipping of the entire magnetization did not occur in the SM simulations under $T_mathrm{c}$ in large lattice-size simulations. However, the phase changed through the high temperature region in the STM simulations. Thus, the dimensional extension let us eliminate the difficulty of the first-order phase transitions and study wide area of the phase space. We then discuss how frequently parameter-updating attempts should be made for optimal convergence. The results favor frequent attempts. We finally study the crossover behavior of the phase transitions with respect to the temperature and external field. The crossover behavior was clearly observed in the simulations in agreement with the theoretical implications.
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