A Tutorial on Advanced Dynamic Monte Carlo Methods for Systems With Discrete State Spaces

Abstract

Advanced algorithms are necessary to obtain faster-than-real-time dynamic simulations in a number of different physical problems that are characterized by widely disparate time scales. Recent advanced dynamic Monte Carlo algorithms that preserve the dynamics of the model are described. These include the n-fold way algorithm, the Monte Carlo with Absorbing Markov Chains (MCAMC) algorithm, and the Projective Dynamics (PD) algorithm. To demonstrate the use of these algorithms, they are applied to some simplified models of dynamic physical systems. The models studied include a model for ion motion through a pore such as a biological ion channel and the metastable decay of the ferromagnetic Ising model. Non-trivial parallelization issues for these dynamic algorithms, which are in the class of parallel discrete event simulations, are discussed. Efforts are made to keep the article at an elementary level by concentrating on a simple model in each case that illustrates the use of the advanced dynamic Monte Carlo algorithm.

Filling of three-dimensional space by two-dimensional sheet growth

“…The KMC scheme uses an algorithm which was first proposed for simulating Ising spin systems [32], coupled systems of chemical reaction [33] and crystal growth [34]. These applications and further details are described in reviews by Chatterjee and Vlachos, Voter, and Novotny [35][36][37] A. Verification: single sheet growth…”

Effect of the size distribution of magnetic nanoparticles on metastability in magnetization relaxation

“…[16,17]. Three decay regimes were proposed [16][17][18]. (i) In a strong-field regime, the magnetic field is so strong that the metastable state is short-lived (R c < a), where a is the lattice spacing and R c is the radius of the critical droplet.…”

“…As shown in Refs. [16][17][18], for a given lattice size L, at substantially low temperatures, the Ising model is in the SD regime for |H| < 4.…”

“…It is scientifically interesting and technologically important to understand magnetization relaxation as a function of magnetic field, magnetic anisotropy, and the distributions of magnetic anisotropy barriers. Despite its importance, theoretical understanding of metastability in magnetization relaxation has been, so far, limited to the Ising model (where the magnetic anisotropy energy does not exist) [16][17][18][19].…”

“…We consider the following two cases: (i) when a uniaxial magnetic anisotropy parameter is constant and (ii) when the parameter has square and Gaussian distributions. We examine the decay or the average lifetime of the metastable state for the Blume-Capel model without and with the distributions of magnetic anisotropy parameter, using kinetic Monte Carlo (KMC) simulations and the absorbing Markov chains (AMC) method [18,20]. It is found that the decay of the metastable state is determined by the smallest value of the magnetic anisotropy parameter, and that the behavior of the magnetization relaxation deviates from the well-known Arrhenius law due to the distributions.…”