Standard Metropolis Algorithm Research Articles

Monte Carlo simulations of glass-forming liquids beyond Metropolis.

Monte Carlo simulations are widely employed to measure the physical properties of glass-forming liquids in thermal equilibrium. Combined with local Monte Carlo moves, the Metropolis algorithm can also be used to simulate the relaxation dynamics, thus offering an efficient alternative to molecular dynamics. Monte Carlo simulations are, however, more versatile because carefully designed Monte Carlo algorithms can more efficiently sample the rugged free energy landscape characteristic of glassy systems. After a brief overview of Monte Carlo studies of glass-formers, we define and implement a series of Monte Carlo algorithms in a three-dimensional model of polydisperse hard spheres. We show that the standard local Metropolis algorithm is the slowest and that implementing collective moves or breaking detailed balance enhances the efficiency of the Monte Carlo sampling. We use time correlation functions to provide a microscopic interpretation of these observations. Seventy years after its invention, the Monte Carlo method remains the most efficient and versatile tool to compute low-temperature properties in supercooled liquids.

Exploring the equilibrium and dynamic phase transition properties of the Ising ferromagnet on a decorated triangular lattice.

We study the equilibrium and dynamic phase transition properties of a two-dimensional Ising model on a decorated triangular lattice under the influence of a time-dependent magnetic field composed of a periodic square wave part plus a time-independent bias term. Using Monte Carlo simulations with a standard Metropolis algorithm, we determine the equilibrium critical behavior in zero field. At a fixed temperature corresponding to the multidroplet regime, we locate the relaxation time and the dynamic critical half period at which a dynamic phase transition takes place between ferromagnetic and paramagnetic states. Benefiting from finite-size scaling theory, we estimate the dynamic critical exponent ratios for the dynamic order parameter and its scaled variance, respectively. The response function of the average energy is found to follow a logarithmic scaling as a function of lattice size. At the critical half period and in the vicinity of a small bias field regime, the average of the dynamic order parameter obeys a scaling relation with a dynamic scaling exponent which is very close to the equilibrium critical isotherm value. Finally, in the slow critical dynamics regime, investigation of metamagnetic fluctuations in the presence of bias field reveals a symmetric double-peak behavior for the scaled variance contours of the dynamic order parameter and average energy. Our results strongly resemble those previously reported for kinetic Ising models.

Ground state and thermodynamic properties of the coupled double-Ising model: application to rare-earth tetraborides

A combination of the standard Metropolis algorithm and the parallel tempering method is used to study ground-state and thermodynamic properties of the coupled double Ising (CDI) model on the Shastry–Sutherland lattice (SSL). This model is derived from the complex spin-electron model, in which electrons described by the Hubbard model and spins described by the Ising model interact via the anisotropic spin-dependent interaction of the Ising type, under the following three conditions: (a) the Hubbard interaction between electrons in the conduction band is sufficiently large; (b) the number of conduction electrons is equal to the number of the lattice points; (c) there is the easy axis spin anisotropy in the system. The model is solved numerically for the selected combinations of spins from the electron () and spin () Ising branch which represent very realistically the situation in TmB4, ErB4 and HoB4 compounds, where in addition all three conditions are simultaneously satisfied. It is shown that the interlpay between the electron and spin branch of the CDI model leads to the suppression of the 1/3 magnetization plateau (the main magnetization plateau found in the ordinary Ising model on the SSL) and the stabilization of magnetization plateaus with , which is in perfect agreement with the experimental measurements in TmB4 and ErB4 compounds. A nice correspondence between theoretical and experimental results is also obtained for the temperature dependence of the specific heat capacity, where the low-temperature peak corresponding to the electron branch of the CDI model is followed by high-temperature peak of the spin branch. This opens a new route for exploring the physics of other complex spin-electron systems in which the above-mentioned conditions (a)–(c) are fulfilled.

Quantum metrics with very low action inR+R2gravity

We have run numerical simulations of Euclidean lattice quantum gravity for metrics which are time-independent and spherically symmetric. The radial variable is discretized as $r=h{L}_{\text{Planck}}$, with $h=0,1,\dots{},N$ and $N$ up to ${10}^{5}$. The Lagrangian is of the form $\sqrt{g}(R+\ensuremath{\alpha}{R}^{2})$ (in units $c=\ensuremath{\hbar}=G=1$) and the action is positive-definite, allowing the use of a standard Metropolis algorithm with update probability $\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}\mathrm{\ensuremath{\Delta}}S)$. By minimizing the $R+{R}^{2}$ action with respect to conformal modes, Bonanno and Reuter have recently found analytical evidence of a nontrivial ``rippled'' ground state resembling a kinetic condensate of QCD. Our simulations at low but finite temperature ($T={\ensuremath{\beta}}^{\ensuremath{-}1}$) also display strong localized oscillations of the metric, whose total action $S$ remains $\ensuremath{\ll}\ensuremath{\hbar}$ thanks to the indefinite sign of $R$. The average metric $⟨{g}_{rr}⟩$ is significantly different from flat space. The scaling properties of $S$ and $⟨{g}_{rr}⟩$ are investigated in dependence on $N$ and $\ensuremath{\beta}$.

Hierarchical Transformed Scale Mixtures for Flexible Modeling of Spatial Extremes on Datasets With Many Locations

–Flexible spatial models that allow transitions between tail dependence classes have recently appeared in the literature. However, inference for these models is computationally prohibitive, even in moderate dimensions, due to the necessity of repeatedly evaluating the multivariate Gaussian distribution function. In this work, we attempt to achieve truly high-dimensional inference for extremes of spatial processes, while retaining the desirable flexibility in the tail dependence structure, by modifying an established class of models based on scale mixtures Gaussian processes. We show that the desired extremal dependence properties from the original models are preserved under the modification, and demonstrate that the corresponding Bayesian hierarchical model does not involve the expensive computation of the multivariate Gaussian distribution function. We fit our model to exceedances of a high threshold, and perform coverage analyses and cross-model checks to validate its ability to capture different types of tail characteristics. We use a standard adaptive Metropolis algorithm for model fitting, and further accelerate the computation via parallelization and Rcpp. Lastly, we apply the model to a dataset of a fire threat index on the Great Plains region of the United States, which is vulnerable to massively destructive wildfires. We find that the joint tail of the fire threat index exhibits a decaying dependence structure that cannot be captured by limiting extreme value models. Supplementary materials for this article are available online.

Investigation of domain structure in ferroelectric thin films by means of the Ising model

The domain structure in thin ferroelectric films was investigated in the framework of the modified Ising model with short-range exchange and long-range dipole-dipole interactions. The depolarizing field was also taken into account for the study of size effects in thin films. This field is the dominant factor influencing on the existence of phase transitions in the films. We used Monte Carlo method with the standard Metropolis algorithm to generate configurations on three-dimensional cubic lattices. The domains sizes were calculated in the longitudinal and transverse directions relatively to the spontaneous polarization direction. Temperature dependences of the domain sizes are obtained for different values of interaction constants. It was proved that the introduction of the depolarizing field leads to the appearance of a dead layer on the film boundaries. The dead layer thickness increases with increasing temperature. At a certain temperature (not equal to the Curie temperature), the dead layer is destroyed. The temperature dependences of the dielectric susceptibility at different film thicknesses have been calculated. It was shown that these dependencies have two maxima. The first maximum corresponds to a phase transition, the second one exists only for only thin films and corresponds to the destruction of the dead layer.

Formation of magnetization plateaus in the 3D Ising model with the long-range RKKY interaction: application to rare-earth tetraborides

The standard Metropolis algorithm and the parallel tempering method are used to study the influence of the long-range RKKY interaction on the formation of magnetization plateaus in the 3D Ising model with magnetically coupled Shastry-Sutherland layers. It is shown that depending on the Fermi momentum k F the model exhibits the rich spectrum of magnetic solutions which is demonstrated by the appearance of the large number of magnetization plateaus on the magnetization curves. In particular, the following set of individual fractional magnetization plateaus with macroscopic stability regions is observed: m ∕m s = 1/8, 1/7, 1/6, 1/5, 2/9, 1/4, 1/3, 3/8, 2/5, 5/12, 3/7, 1/2, 5/9 and 2/3. In the regime of k F ~ 2π ∕1.24, corresponding to the real situation in TmB 4 , we have confirmed the existence of the magnetization plateau with m ∕m s = 1∕8, the nature of which was intensively discussed in recent experimental works. Since the change of k F can be induced by doping, our results provide predictions of the complete sequences of magnetization plateaus that could appear in tetraboride solid solutions. And finally, the current results are confronted with our previous ones obtained for the two-dimensional Ising model with the RKKY interaction in order to reveal the role of system dimension on the formation of magnetization plateaus.

Influence of Interplane Interactions on Formation of Magnetization Plateaus in Generalized 3D Ising Models with Magnetically Coupled Shastry-Sutherland Layers

The combination of the standard Metropolis algorithm and the parallel tempering method is used to examine the influence of interplane interactions on the formation of magnetization plateaus in two types of generalized Ising models with magnetically coupled Shastry-Sutherland layers. In the first model, we consider the first (J1) and second (J2) intraplane and first (I1) and second (I2) interplane nearest neighbour (NN) interactions, while in the second one, we consider all intraplane and interplane interactions between spins inside an sphere of radius r < 6, and the interaction between two spins at Ri and Rj lattice sites is modeled by a simple one-parametric formula with exponentially decaying amplitudes $J_{\text {ij}} \sim e^{-\alpha |\textbf {R}_{\textbf {i}}-\textbf {R}_{\textbf {j}}|}$ . It is shown that the NN interplane interaction I1 does not change qualitatively the magnetic phase diagram of the model found for J1 and J2 (I1 = 0), while the next NN interaction I2 influences strongly both the stability regions of magnetic phases found for J1 and J2 (I1 = 0, I2 = 0) as well as the formation of new phases. In particular, we have found that the ferromagnetic interaction I2 suppresses the stability regions of magnetic plateau phases found for nonzero J1 and J2, while the antiferromagnetic I2 interaction forms, in addition to the standard 1/3 and 1/2 plateaus, the following set of new macroscopic magnetization plateaus: m/ms = 1/6,1/4,2/3, and 3/4. Practically the same plateaus are formed by the second model in the limit of strongly decaying spin interactions ( $\alpha \sim 4$ ), while in the opposite limit (α 1/2 and m/ms < 1/3 disappear and the magnetization curves become continuous in these regions.

Efficient and Scalable Approach to Equilibrium Conditional Simulation of Gibbs Markov Random Fields

We study the performance of an automated hybrid Monte Carlo (HMC) approach for conditional simulation of a recently proposed, single-parameter Gibbs Markov random field. This is based on a modified version of the planar rotator (MPR) model and is used for efficient gap filling in gridded data. HMC combines the deterministic over-relaxation method and the stochastic Metropolis update with dynamically adjusted restriction and performs automatic detection of the crossover to the targeted equilibrium state. We focus on the ability of the algorithm to efficiently drive the system to equilibrium at very low temperatures even with sparse conditioning data. These conditions are the most challenging computationally, requiring extremely long relaxation times if simulated by means of the standard Metropolis algorithm. We demonstrate that HMC has considerable benefits in terms of both computational efficiency and prediction performance of the MPR method.

Magnetic phase diagram of the Ising model with the long-range RKKY interaction

The standard Metropolis algorithm and the parallel tempering method are used to examine magnetization processes in the Ising model with the long-range RKKY interaction on the Shastry-Sutherland lattice. It is shown that the Ising model with RKKY interaction exhibits, depending on the value of the Fermi wave vector $k_F$, the reach spectrum of magnetic solutions, which is manifested in the appearance of new magnetization plateaus on the magnetization curve. In particular, we have found the following set of individual magnetization plateaus with fractional magnetization $m/m_s$=1/18, 1/9, 1/8, 1/5, 1/4, 1/3, 3/8, 5/12, 1/2, 3/5, 2/3, which for different values of $k_F$ form various sequences of plateaus, changing from very complex, appearing near the point $k_F=2\pi/1.2$, to very simple appearing away this point. Since the change of $k_F$ can be induced by doping (the substitution of rare-earth ion by other magnetic ion that introduces the additional electrons to the conduction band) the model is able to predict the complete sequences of magnetization plateaus, which could appear in the tetraboride solid solutions.

Critical Relaxation of a Three-Dimensional Fully Frustrated Ising Model

Critical relaxation from the low-temperature ordered state of the three-dimensional fully frustrated Ising model on a simple cubic lattice is studied by the short-time dynamics method. Cubic systems with periodic boundary conditions and linear sizes of L = 32, 64, 96, and 128 in each crystallographic direction are studied. Calculations were carried out by the Monte Carlo method using the standard Metropolis algorithm. The static critical exponents for the magnetization and correlation radius and the dynamic critical exponents are calculated.

Критическая релаксация трехмерной полностью фрустрированной модели Изинга

AbstractCritical relaxation from the low-temperature ordered state of the three-dimensional fully frustrated Ising model on a simple cubic lattice is studied by the short-time dynamics method. Cubic systems with periodic boundary conditions and linear sizes of L = 32, 64, 96, and 128 in each crystallographic direction are studied. Calculations were carried out by the Monte Carlo method using the standard Metropolis algorithm. The static critical exponents for the magnetization and correlation radius and the dynamic critical exponents are calculated.

Critical properties of 2d disordered 3-state antiferromagnetic potts model ON TRIANGULAR LATTICE

By introducing a small amount of non-magnetic impurities into an antiferromagnetic (AF) two-dimensional (2D) Potts model on a triangular lattice it is that the impurities in spin systems described by this model result in the change of a first order to a second-order phase transition. The systems with linear sizes L × L = N, L = 9-144 are considered. Investigations are performed using the standard Metropolis algorithm along with Monte-Carlo single-cluster Wolff algorithm. On the basis of the theory of finite-size scaling, critical exponents (CE) are calculated: the heat capacity α, the susceptibility γ, the order parameter β, and the CE of the correlation radius ν.

The Monte Carlo simulation of 2D ANNNI-model

In this, study we present the data for 2D Axial Next Nearest Neighbor Ising model (ANNNI-model) obtained from Monte Carlo (MC) simulations using the standard Metropolis algorithm. The temperature dependences of thermodynamic parameters for a cubic lattice with linear sizes L=32 at different values of the competing interaction parameter |J1/J|=0.1÷1.0. Transition temperatures of ferromagnetic ordering to the paramagnetic state at |J1/J|&lt;0.3 and to the modulated state at 0.3&lt;|J1/J|&lt;0.5 are shown to shift towards low temperatures with an increase in a competing interaction parameter absolute value. Conversely, transition temperatures of the modulate state to the paramagnetic ordering grow. The modulated ordering in the 2D ANNNImodel appears in the temperature range 0.1&lt;T&lt;2.0 at 0.2&lt;|J1/J|≤1.0. Modulated structure parameters are computed using a mathematic apparatus of Fourier transform spectral analysis. According to the Fourier analysis results, the wave number grows with an increase in the competing interaction parameter absolute value. Summarizing obtained results, we plot a phase diagram of 2D anisotropic Ising model with competing interactions.

Entropic formulation for the protein folding process: Hydrophobic stability correlates with folding rates

Here we focus on the conformational search for the native structure when it is ruled by the hydrophobic effect and steric specificities coming from amino acids. Our main tool of investigation is a 3D lattice model provided by a ten-letter alphabet, the stereochemical model. This minimalist model was conceived for Monte Carlo (MC) simulations when one keeps in mind the kinetic behavior of protein-like chains in solution. We have three central goals here. The first one is to characterize the folding time (τ) by two distinct sampling methods, so we present two sets of 103 MC simulations for a fast protein-like sequence. The resulting sets of characteristic folding times, τ and τq were obtained by the application of the standard Metropolis algorithm (MA), as well as by an enhanced algorithm (MqA). The finding for τq shows two things: (i) the chain-solvent hydrophobic interactions {hk} plus a set of inter-residues steric constraints {ci,j} are able to emulate the conformational search for the native structure. For each one of the 103MC performed simulations, the target is always found within a finite time window; (ii) the ratio τq∕τ≅1∕10 suggests that the effect of local thermal fluctuations, encompassed by the Tsallis weight, provides to the chain an innate efficiency to escape from energetic and steric traps. We performed additional MC simulations with variations of our design rule to attest this first result, both algorithms the MA and the MqA were applied to a restricted set of targets, a physical insight is provided. Our second finding was obtained by a set of 600 independent MC simulations, only performed with the MqA applied to an extended set of 200 representative targets, our native structures. The results show how structural patterns should modulate τq, which cover four orders of magnitude; this finding is our second goal. The third, and last result, was obtained with a special kind of simulation performed with the purpose to explore a possible connection between the hydrophobic component of protein stability and the native structural topology. We simulated those same 200 targets again with the MqA, only. However, this time we evaluated the relative frequency {ϕq} in which each target visits its corresponding native structure along an appropriate simulation time. Due to the presence of the hydrophobic effect in our approach we obtained a strong correlation between the stability and the folding rate (R=0.85). So, as faster a sequence found its target, as larger is the hydrophobic component of its stability. The strong correlation fulfills our last goal. This final finding suggests that the hydrophobic effect could not be a general stabilizing factor for proteins.

Phase transitions and thermodynamic properties of triangular strongly-diluted antiferromagnetic Potts model

By introducing a substantial amount of non-magnetic impurities into an antiferromagnetic 2d Potts model on a triangular lattice it is that the impurities in spin systems described by this model result in the change of a first-order to a second-order phase transition. The systems with linear sizes L×L=N, L=18−48 are considered. Investigations are performed using the standard Metropolis algorithm along with Monte-Carlo single-cluster Wolff algorithm.

Magnetic properties of core–shell (1/2–3/2) nanoparticle: Monte Carlo simulation

In this work, a Monte Carlo simulation, based on standard Metropolis algorithm, has been applied to investigate magnetic properties of a ferrimagnetic nanoparticle, with a cubic shape and core–shell structure. The nanoparticle is constituted of spin-1/2 in the core surrounded by spin-3/2 in the shell. It has been shown that the exchange coupling plays an important role on the phase diagrams. Some interesting features have been observed for the temperature dependence of total magnetization curves of particle. In particular, the effects of the interface coupling and the shell coupling on both the compensation temperature and the magnetization profiles are investigated. In addition, the effect of the single ion anisotropy as well as the hysteresis loops behavior has also been discussed in detail.

Monte Carlo methods beyond detailed balance

Monte Carlo algorithms are nearly always based on the concept of detailed balance and ergodicity. In this paper we focus on algorithms that do not satisfy detailed balance. We introduce a general method for designing non-detailed balance algorithms, starting from a conventional algorithm satisfying detailed balance. This approach is first applied to a very simple model, which shows the basic viability of the method. Then we apply it to the Ising model, where we find that the method is an improvement compared to the standard Metropolis algorithm, be it with a modest gain of a factor 2.3.

Non-equilibrium dynamics of a ferrimagnetic core–shell nanocubic particle

The non-equilibrium dynamics of a single cubic core–shell ferrimagnetic nanoparticle system under a time dependent oscillating magnetic field is elucidated by making use of a classical Monte Carlo simulation technique with a standard Metropolis algorithm. Many interesting and unusual thermal and magnetic behaviors are observed, for instance, the locations of dynamic phase transition points change significantly depending upon amplitude and period of the external magnetic field as well as other Hamiltonian parameters in related planes. Much effort has also been devoted to the influences of the varying shell thickness on the thermal and magnetic properties of the particle, and outstanding physical findings are reported in order to better understand the dynamic process of the studied nanoparticle system.

QUALITATIVE ASPECTS OF THE PHASE DIAGRAM OF J1–J2 MODEL ON THE CUBIC LATTICE

The qualitative aspects of the phase diagram of the Ising model on the cubic lattice, with ferromagnetic (F) nearest-neighbor interactions (J1) and antiferromagnetic (AF) next-nearest-neighbor couplings (J2) are analyzed in the plane temperature versus α, where α = J2/|J1| is the frustration parameter. We used the original Wang–Landau sampling (WLS) and the standard Metropolis algorithm to confront past results of this model obtained by the effective-field theory (EFT) for the cubic lattice. Our numerical results suggest that the predictions of the EFT are in general qualitatively correct, but the low-temperature re-entrant behavior, observed in the frontier separating the F and the collinear order, is an artifact of the EFT approach and should disappear when we consider Monte Carlo (MC) simulations of the model. In addition, our results indicate that the continuous phase transition between the F and the paramagnetic (P) phases, that occurs for 0.0 ≤α&lt; 0.25, belongs to the universality class of the three-dimensional pure Ising Model.