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The Monte Carlo Computer Experiment to Study the Order and Phase Transitions in the Mixed Phase Region Based on the Example of the 3D Ashkin-Teller Model

Wojtkowiak Z., Musiał G.

Computational Methods in Science and Technology

2021

Vol. 27, No. 3

129--136

In this paper, we demonstrate a new way of performing Monte Carlo (MC) simulations in a mixed phase region that is difficult to study, where with certain probabilities there are different ordering ways. That results in a large oscillation of the values of the computed thermodynamic quantities, which makes their interpretation very problematic. Our results are presented on the example of the 3D Askin-Teller (AT) model, where within a certain range of parameters with equal probabilities there are two different, but equivalent, ways of ordering two of the three order parameters showing independent behavior. The use of our new approach in an MC computer experiment allowed us to use Binder cumulant as well as Challaand the Lee-Kosterlitz-like cumulants. This made it possible to locate phase transitions precisely enough to be able to use the energy distribution histogram method. According to the most effective strategy in the critical region we use our recently proposed cluster MC algorithm and the Metropolis algorithm beyond it, which are suitable for both the first-order and the continuous phase transitions in the 3D AT model. The new approach was demonstrated by determining smooth curves of magnetization and internal energy, and as a consequence by determining the location and character of the phase transition on the line between the mixed phase region and the paramagnetic phase.

Parallel Simulations of the Monte Carlo Type: 3D Ashkin-Teller Model

Musiał G., Dębski L., Kamieniarz G.

Computational Methods in Science and Technology

2007

Vol. 13, nr 1

41-46

The method of parallelization of processing in simulations of the Monte Carlo type is proposed and tested on the 3D Ashkin- -Teller model characterized by a rich phase diagram. The message passing is applied and the MPI library is exploited. It is demonstrated that the method works well in different regions of the phase diagram, where the phase transitions are first-order and continuous. The dependence of speedup and efficiency on the number of parallel processes is studied. The condition for the best speedup and efficiency in these simulations is formulated and discussed using the results obtained on symmetric NUMA multiprocessor and on symmetric multicomputer. The suggestion as to effective use of the method even in highly heterogeneous computer systems is also given.