lattice. We consider efficient algorithms for thermodynamical characteristics calculation of 2D Ising model. We discuss optimization algorithms for temperature in order to improve effectiveness of replica exchange. We implement and test algorithm on a two-dimensional square Ising Considering the growth of data volumes, there is a need for new research in the field of magnetic data carriers. The researchers use the Monte Carlo method to simulate various spin structures. However, this method has a drawback: in the phase transition region, the simulation process slows down.
Consider the Ising model on a flat square lattice. The probability of any configuration of the investigated models is described by the Gibbs distribution [2]. It is well known that knowing of the statistical sum for a system of interacting spins allows one to strictly calculate all possible mean physical quantities fully describing the state of the system at given parameters. Currently, the studies of the phenomenon of magnetic transformations (transitions) mainly use numerical methods
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(canonical and multicanonical MC) [3]. The Hamiltonian of the Ising spin system in the external
magnetic has the form (
The Monte Carlo method with Markov chains is used as the main component of this research magnetic field. project. 2.1.
The probability of any possible configuration is determined by the Gibbs distribution: =
1 ( )
exp (− ′( ))
In this model, each spin interacts only with its nearest four neighbors through a direct ferromagnetic exchange interaction randomly distributed in the lattice nodes, provided that ∑ =4 =1 = 0
determined by the formula: = min{1, − ∆ } ; =
1 Let us recall that the acceptance probability of the Metropolis-Hastings algorithm is
At low temperatures - a very large positive number. If we propose a spin flip with a positive energy difference, i.e. > 0, we have: ∆ ≫ 0 ⇒ − ∆ ≈ 0 ⇒ the system back to the energy minimum. biased sampling [4].
Most likely, it will go through a sequence of spin flips with a negative energy difference, forcing Therefore, it is not possible to create states according to the Boltzmann distribution, resulting in Replica exchange serves to improve the convergence of the Metropolis-Hastings algorithm in the problem at hand. A number of systems initialized with different temperatures of the MetropolisHastings algorithm exchange configurations during a loop performing value sampling [5].
This is done to allow configurations at high temperatures to move to systems with low temperatures when the simulation process continues, and to save low temperatures from falling into unwanted stable states with a minimum of energy.
The algorithm presented on Figure 1.
(
Hukushima [6] proposed a method, which is simpler than the feedback method, for determining replica temperature values. The scheme starts with an initial arbitrary temperature distribution, captures the extreme temperatures, and iteratively corrects the intermediate temperatures so that the probability of replica exchange for all neighboring temperatures is the same. Since this scheme is based on estimating the energy in each replica as a function of its inverse temperature, we call this method the “energy” method [7,8,9]. This method belongs to the category of approaches that strive for a uniform rate of exchange between replicas.
Let and refer to the inverse temperature and average energy of replica respectively. The goal of the energy method is to adjust so that the replica exchange probabilities of neighboring temperatures are equal.
ℙ( −1, −1 ↔ , ) = ℙ( , ↔ +1, +1). 7)
More precisely, the replicas are divided into two groups: odd and even. Fixing the inverse temperatures of one group, the inverse temperatures of the other group are then corrected one by one [10]. The detailed procedure is outlined in Figure 2.
(
The speeds of the conventional and optimized algorithm were compared on systems with sizes L=100, L=1600, L=4900, L=10000 particles.
The results are shown in Figure 3.
Also, to evaluate the performance of the algorithm, heat capacity plots (Figure 4) were plotted for systems with different sizes.
18000 Optimized
Normal 16000 14000 12000 c 10000 e s t, 8000 6000 4000 2000 0
In this paper, optimization algorithm was considered for replica exchange Monte-Carlo method. We considered “energy” method of optimization. A program was also written to compare optimized and regular algorithms within the framework of the conditions we are interested in. On the basis of the obtained results, it can be concluded that choosing optimized temperature set leads to advantage in execution speed on larger lattices. It will help in further studies of ferromagnetic and antiferromagnetic spin models.
This work was financially supported by the state task of the Ministry of Science and Higher Education of the Russian Federation №075-00400-19-01.
The studies were carried out using the resources of the Center for Shared Use of Scientific Equipment "Center for Processing and Storage of Scientific Data of the Far Eastern Branch of the Russian Academy of Sciences", funded by the Russian Federation represented by the Ministry of Science and Higher Education of the Russian Federation under project No. 075-15-2021-663. Additional computing resources were provided by the FEFU supercomputer cluster (cluster.dvfu.ru).
[5] Katzgraber, Helmut G et al. “Feedback-Optimized Parallel Tempering Monte Carlo.” Journal of
Statistical Mechanics: Theory and Experiment 2006.03 (2006): P03018–P03018. Crossref. Web. [6] K. Hukushima, Domain-wall free energy of spin-glass models: Numerical method and boundary conditions, Phys. Rev. E 60 (1999). [7] Hamze, Firas, Neil Dickson, and Kamran Karimi. “Robust Parameter Selection For Parallel
Tempering.” International Journal of Modern Physics C 21.05 (2010): 603–615. Crossref. Web. [8] David J. Earl, Michael W. Deem «Optimal Allocation of Replicas to Processors in Parallel
Tempering Simulations» The Journal of Chemical Physics, 2005 [9] P. Atkins and J. De Paula, Physical Chemistry for the Life Sciences (Oxford University Press, 2006). [10] H. G. Katzgraber, S. Trebst, D. A. Huse, and M. Troyer, Feedback-optimized parallel tempering Monte Carlo, J. Stat. Mech. P03018 (2006).