Metropolis Dynamics Research Articles

Lattice Gas Dynamics: Application to Driven Vortices in Two Dimensional Superconductors

A continuous time Monte Carlo lattice gas dynamics is developed to model driven steady states of vortices in two dimensional superconducting networks. Dramatic differences are found when compared to a simpler Metropolis dynamics. Subtle finite size effects are found at low temperature, with a moving smectic that becomes unstable to an anisotropic liquid on sufficiently large length scales.

Nucleation preexponential in dynamic Ising models at moderately strong fields.

A dynamic Ising model on a two-dimensional square lattice with nearest neighbor interactions is considered in the metastable region at low temperatures. A large number of low-energy cluster configurations is identified, and for those configurations a system of kinetic equations is written. Solution is obtained using symbolic computational approaches. This allows one to identify the full expression for the nucleation rate, including the preexponential. The treatment generalizes the earlier study of a different, lattice-gas spin-flip dynamics [V. A. Shneidman and G. M. Nita, Phys. Rev. Lett. 89, 025701 (2002)], for the cases of Glauber and Metropolis dynamics and for a broader region of fields. In addition, connection with the lowest-energy nucleation paths (which can be studied analytically, without computer assistance) is examined.

Deformation of polyethylene: Monte Carlo simulation

We have simulated semi crystalline polyethylene (PE) using Monte Carlo method with the Metropolis dynamics. The mechanical response is evaluated in the first steps of deformation. The simulation results are compared with the experimental curves at different conditions concerning temperature and deformation rate. The obtained results allow us to suggest that the method of MC together with the model and the dynamics employed give a satisfactory description of the mechanical properties of lineal polymers as the polyethylene. The simulations MC allows to differentiate the effect of the diverse parameters as the temperature, the test time, chain length and density which affect the mechanical behavior. The study of the non-reversible strain up to reading the material fracture is projected for future work.

Rejection-free Monte Carlo algorithms for models with continuous degrees of freedom.

We construct a rejection-free Monte Carlo algorithm for a system with continuous degrees of freedom. We illustrate the algorithm by applying it to the classical three-dimensional Heisenberg model with canonical Metropolis dynamics. We obtain the lifetime of the metastable state following a reversal of the external magnetic field. Our rejection-free algorithm obtains results in agreement with a direct implementation of the Metropolis dynamic and requires orders of magnitude less computational time at low temperatures. The treatment is general and can be extended to other dynamics and other systems with continuous degrees of freedom.

Deformation of polyethylene: Monte Carlo simulation

We have simulated semi crystalline polyethylene (PE) using Monte Carlo method with the Metropolis dynamics. The mechanical response is evaluated in the first steps of deformation. The simulation results are compared with the experimental curves at different conditions concerning temperature and deformation rate. The obtained results allow us to suggest that the method of MC together with the model and the dynamics employed give a satisfactory description of the mechanical properties of lineal polymers as the polyethylene. The simulations MC allows to differentiate the effect of the diverse parameters as the temperature, the test time, chain length and density which affect the mechanical behavior. The study of the non-reversible strain up to reading the material fracture is projected for future work.

Bulk Magnetic Properties of the Fe0.5Mn0.1Al0.4Disordered Alloy: A Monte Carlo Study

In this work we present a simulational study of the bulk magnetic properties of the Fe0.5Mn0.1Al0.4 disordered system. The magnetization per site, specific heat and magnetic susceptibility were obtained using Monte Carlo simulation. Simulation results were performed in the framework of a random site-diluted 3D Ising model with nearest neighbor interactions, and using a single-spin-flip Metropolis dynamics for equilibration and energy minimization, from which canonical ensemble and configurational averages were computed. In the simulation, atoms were randomly distributed on a body-centered cubic lattice in order to simulate the disorder and crystalline structure as obtained in the arc-melted Fe0.5Mn0.1Al0.4 alloy treated at high temperatures for long periods of time and followed by fast quenching. Competitive Fe–Fe ferromagnetic and Fe–Mn and Mn–Mn antiferromagnetic interactions were taken into account, as well as the Al dilutor effect. We conclude that, accordingly with Mossbauer results, the Fe0.5Mn0.1Al0.4 system exhibits a reentrant spin-glass behavior below 30 K characterized by a sharp increase of the mean hyperfine field attributed to the magnetic ordering of the Mn sublattice at these temperatures. Additionally, a ferromagnetic to paramagnetic transition at around 300 K was also evidenced.

Hysteretic behavior of Fe0.9−qMn0.1Alq-disordered alloys: a Monte Carlo study

In this work we have simulated the Fe 0.9− q Mn 0.1 Al q alloy series with Al contents ranging from 10 up to 50 at%, and for several system sizes. In the simulation, the atoms are randomly distributed on a body-centered cubic according to the atomic disorder achieved through quenching techniques for the experimental samples. In computing the thermodynamic quantities such as the magnetization per site as a function of an external applied magnetic field, we have employed a Monte Carlo algorithm based on a Metropolis dynamics implemented with a random site-diluted Ising model. In this model, we have taken into account nearest-neighbor interactions for which both the ferromagnetic (Fe–Fe) and antiferromagnetic (Fe–Mn, Mn–Mn) interactions are present. From the simulation of the hysteresis loops at room temperature, the remanence and the coercive force as a function of the Al concentration have been obtained. Finally, a comparison with the previous experimental data on coercivity obtained by means of vibrating sample magnetometry is also carried out and discussed.

Damage spreading in the mixed spin Ising model.

We apply the damage spreading technique to study a mixed spin Ising model consisting of spin 1/2 and spin 1 with a crystal field interaction on the square lattice within a kind of Metropolis dynamics. The completely different behavior, depending on the value of the crystal field interaction, strongly suggests there may exist a dynamical tricritical point where the phase transition may change from the second order to the first order for this model.

Time decay of the remanent magnetization in the +/-J spin glass model at T=0.

Using the zero-temperature Metropolis dynamics, the time decay of the remanent magnetization in the +/-J Edward-Anderson spin glass model with a uniform random distribution of ferromagnetic and antiferromagnetic interactions has been investigated. Starting from the saturation, the magnetization per spin m reveals a slow decrease with time, which can be approximated by a power law: m(t)=m(infinity)+(t/a(0))(a(1)), a(1)<0. Moreover, its relaxation does not lead it into one of the ground states, and therefore the system is trapped in metastable isoenergetic microstates remaining magnetized. Such behavior is discussed in terms of a random walk that the system performs on its available configuration space.

Damage Spreading in the Ising Model with a Special Metropolis Dynamics Approach**The project supported by the U.S. National Science Foundation under NSF Grant DMR 9970291

The time evolution of the Hamming distance (damage spreading) for the and Ising models on the square lattice is performed with a special metropolis dynamics algorithm. Two distinct regimes are observed according to the temperature range for both models: a low-temperature one where the distance in the long-time limit is finite and seems not to depend on the initial distance and the system size; a high-temperature one where the distance vanishes in the long-time limit. Using the finite size scaling method, the dynamical phase transition (damage spreading transition) temperature is obtained as for the Ising model.

Convergence to Equilibrium for Spin Glasses

We estimate the distance in total variation between the law of a finite state Markov process at time t, starting from a given initial measure, and its unique invariant measure. We derive upper bounds for the time to reach the equilibrium. As an example of application we consider a special case of finite state Markov process in random environment: the Metropolis dynamics of the Random Energy Model. We also study the process of the environment as seen from the process.

Derivation and validation of mesoscopic theories for diffusion of interacting molecules

A mesoscopic theory for diffusion of molecules interacting with a long-range potential is derived for Arrhenius microscopic dynamics. Gradient Monte Carlo simulations are presented on a one-dimensional lattice to assess the validity of the mesoscopic theory. Results are compared for Metropolis and Arrhenius microscopic dynamics. Non-Fickian behavior is demonstrated and it is shown that microscopic dynamics dictate the steady-state concentration profiles.

Image Segmentation Using Markov Random Field Model in Fully Parallel Cellular Network Architectures

Markovian approaches to early vision processes need a huge amount of computing power. These algorithms can usually be implemented on parallel computing structures. Herein, we show that the Markovian labeling approach can be implemented in fully parallel cellular network architectures, using simple functions and data representations. This makes possible to implement our model in parallel imaging VLSI chips.As an example, we have developed a simplified statistical image segmentation algorithm for the Cellular Neural/Nonlinear Networks Universal Machine (CNN-UM), which is a new image processing tool, containing thousands of cells with analog dynamics, local memories and processing units. The Modified Metropolis Dynamics (MMD) optimization method can be implemented into the raw analog architecture of the CNN-UM. We can introduce the whole pseudo-stochastic segmentation process in the CNN architecture using 8 memories/cell. We use simple arithmetic functions (addition, multiplication), equality-test between neighboring pixels and very simple nonlinear output functions (step, jigsaw). With this architecture, the proposed VLSI CNN chip can execute a pseudo-stochastic relaxation algorithm of about 100 iterations in about 100 μs.In the suggested solution the segmentation is unsupervised, where a pixel-level statistical estimation model is used. We have tested different monogrid and multigrid architectures.In our CNN-UM model several complex preprocessing steps can be involved, such as texture-classification or anisotropic diffusion. With these preprocessing steps, our fully parallel cellular system may work as a high-level image segmentation machine, using only simple functions based on the close-neighborhood of a pixel.

Room temperature magnetic properties of quenched-disordered FeAl alloys: A simulational study

We present a Monte Carlo study of the room temperature magnetic properties of quenched-disordered Fe1−qAlq alloys with 0⩽q⩽0.6. Simulations were performed using a random site-diluted classical Heisenberg model with nearest neighbor interactions, and using Metropolis dynamics. Results for the Al concentration of the magnetization and susceptibility predict the occurrence of a critical concentration qc at around 40 at. % Al, at which a transition from a ferromagnetic state (q&amp;lt;qc) to a paramagnetic one (q⩾qc) takes place. The simulated dependence of the average magnetization on the Al content is compared with experimental results of the average hyperfine field obtained by Mössbauer spectroscopy. Finally, an analysis of the finite size scaling properties of this system is presented.

Glassy dynamics of protein folding.

A coarse-grained model of a random polypeptide chain, with only discrete torsional degrees of freedom and Hookean springs connecting pairs of hydrophobic residues is shown to display stretched exponential relaxation under Metropolis dynamics at low temperatures with the exponent beta approximately 1/4, in agreement with the best experimental results. The time dependent correlation functions for fluctuations about the native state, computed in the Gaussian approximation for real proteins, have also been found to have the same functional form. Our results indicate that the energy landscape exhibits universal features over a very large range of energies and is relatively independent of the specific dynamics.

Effective scaling behavior of magnetization and Hamming distance in Ising thin films under a surface field

The recent improvements on the technology for developing high-quality thin magnetic films has renewed the interest in the study of surface effects in both static and dynamic magnetic responses. In this work, we use a Monte-Carlo algorithm with Metropolis dynamics together with a spreading of damage technique to study the interplay between the effects of finite thickness and surface ordering field in thin ferromagnetic Ising (S=1/2) films. We calculate, near the bulk critical temperature and several values of the surface field, the dependence on the film thickness of the average magnetization M and Hamming distance D. We employ a finite size scaling analysis to show that both obey an effective one-parameter scaling but exhibit distinct characteristic surface fields. At their corresponding characteristic surface fields both M and D become roughly thickness independent and we estimate the critical exponent characterizing the behavior of the typical scaling lengths.

Efficient search of critical points in Ising-like systems

We propose a simulation method inspired by self-organization that drives Ising-like magnetic systems rapidly to criticality. We develop a feedback control rule with very few parameters for use with the standard methods of local spin updatings that spontaneously leads the system to the critical temperature. This method for predicting the critical values requires small lattices. It gives good results eg., 1% accuracy for T C in 3-D for 10 3 systems, with reduced computer times, for both Glauber and Metropolis dynamics.

Simulated annealing, acceleration techniques, and image restoration

Typically, the linear image restoration problem is an ill-conditioned, underdetermined inverse problem. Here, stabilization is achieved via the introduction of a first-order smoothness constraint which allows the preservation of edges and leads to the minimization of a nonconvex functional. In order to carry through this optimization task, we use stochastic relaxation with annealing. We prefer the Metropolis dynamics to the popular, but computationally much more expensive, Gibbs sampler. Still, Metropolis-type annealing algorithms are also widely reported to exhibit a low convergence rate. Their finite-time behavior is outlined and we investigate some inexpensive acceleration techniques that do not alter their theoretical convergence properties; namely, restriction of the state space to a locally bounded image space and increasing concave transform of the cost functional. Successful experiments about space-variant restoration of simulated synthetic aperture imaging data illustrate the performance of the resulting class of algorithms and show significant benefits in terms of convergence speed.

The spectral gap of the REM under Metropolis dynamics

We derive upper and lower bounds for the spectral gap of the random energy model under Metropolis dynamics which are sharp in exponential order. They are based on the variational characterization of the gap. For the lower bound, a Poincaré inequality derived by Diaconis and Stroock is used. The scaled asymptotic expression is a linear function of the temperature. The corresponding function for a global version of the dynamics exhibits phase transition instead. We also study the dependence of lower order terms on the volume. In the global dynamics, we observe a phase transition. For the local dynamics, the expressions we have, which are possibly not sharp, do not change their order of dependence on the volume as the temperature changes.

Island growth and phase separation in chemically reactive adsorbed overlayers

Employing the lattice-gas model with attractive interactions between nearest-neighbour particles, we have studied by Monte Carlo simulations the formation of islands in an adsorbed overlayer in two situations. The first situation corresponds to island growth when the adsorbed overlayer is quenched at t = 0 from high temperature in a disordered phase to a temperature below the critical one. Using a realistic algorithm of surface diffusion, and also the Metropolis dynamics for times up to 10 6 Monte Carlo steps, we show that in analogy with the spin-exchange kinetic Ising model the growth law for the average island size can be represented in this case at all the coverages as R( t) ⋍ A + Bt 13. The parameter A is found to be weakly dependent on coverage and on the details of the dynamics employed. On the other hand, the value and the coverage dependence of the parameter B are sensitive to the mechanism of diffusion. The second situation corresponds to phase separation in chemically reactive systems under steady-state conditions. As an example, we analyse the A+B→AB reaction, occurring via the Eley-Rideal mechanism, including A adsorption on vacant sites, A gas →A ads, and A consumption in collisions with gas-phase B particles, A ads+B gas→(AB) gas. For a realistic algorithm of surface diffusion, the average size of A islands is found to be rather low, even if surface diffusion is many orders of magnitude faster than the adsorption/reaction processes. In contrast, the Metropolis dynamics predicts the formation of relatively large islands or even mesoscopically-ordered structures (at θ A ⋍ 0.5). The physics behind these findings is discussed in detail.