Narrow Results

The study of recrystallization and grain growth in annealed metals has been extensively investigated using various computational approaches, with Monte Carlo (MC) simulations proving particularly effective. This research highlights the application of the MC method in simulating microstructures that closely resemble experimentally observed ones. The experimental microstructures were obtained from a 50% cold-worked Al–4% Cu (Duralumin) alloy, annealed at different conditions. The Metropolis algorithm was employed to simulate the microstructures using a two-dimensional Potts model on a square lattice. The simulated results strongly correlate with the actual microstructures, validating the effectiveness of the MC method in exploring grain growth phenomena. This study underscores the potential of MC simulations as a robust tool for investigating recrystallization and grain evolution in annealed metals.

In this paper, we give a systematic review of the theory of Gibbs measures of Potts model on Cayley trees (developed since 2013) and discuss many applications of the Potts model to real world situations: mainly biology, physics, and some examples of alloy behavior, cell sorting, financial engineering, flocking birds, flowing foams, image segmentation, medicine, sociology, etc.

The critical behavior in short time dynamics for the q = 6 and 7 state Potts models in two-dimensions is investigated. It is shown that dynamic finite-size scaling exists for first-order phase transitions.

The q = 8 state Potts model with varying amount of quenched bond randomness is simulated in two dimensions by using a cluster algorithm. It is shown that, by monitoring the autocorrelation times, one can trace down the threshold value of the introduced quenched bond randomness for the rounding of the first-order phase transition.

In order to see the phase conversion taking place in a weak first-order phase transition, we have simulated the q = 3 state Potts model in three dimensions and studied the time evolutions of oriented clusters forming after a rapid temperature quench. Our results indicate that the phase conversion mechanism following a deep temperature quench is spinodal decomposition while a rather shallow quench to temperatures near the phase transition point proceeds through usual nucleation.

The scaling behaviors of the percolation cumulant and the surface renormalization are studied on q = 2 and 7 state Potts models. The results show that the scaling functions can be safely used to determine infinite lattice transition points and the thermal and magnetic exponents indicating that these functions have very small correction to scaling contributions.

In order to test the multicanonical approach in simulations of the spin systems with quenched bond randomness, I simulated the q = 8 state Potts model in two dimensions with various degrees of randomness. It appears quite feasible to simulate spin systems with quenched bond randomness by multicanonical algorithm, but extra care is needed with increasing randomness.

In this work, we have proposed a new geometrical method for calculating the critical temperature and critical exponents by introducing a set of bond breaking probability values. The probability value P_c corresponding to the Coniglio–Klein probability for the transition temperature is obtained among this set of trial probabilities. Critical temperature, thermal and magnetic exponents are presented for d = 2 and d = 3, q = 2 Potts model and for the application of the method to the system with first order phase transition, q = 7 Potts model on different size lattices are employed.

The advantage of this method can be that the bond breaking probability can be applied, where the clusters are defined on a set of dynamic variables, which are different from the dynamic quantities of the actual Hamiltonian or the action of the full system. An immediate application can be to use the method on finite temperature lattice gauge theories.

Nucleation in the two-dimensional q-state Potts model has been studied by means of Monte Carlo simulations using the heat-bath dynamics. The initial metastable state has been prepared by magnetic quench of the ordered low-temperature phase. The magnetic field dependence of the nucleation time has been measured as the function of the magnetic field for different q and lattice sizes at T = 0.5T_c. A size-dependent crossover from the coalescence to nucleation region is observed at all q. The magnetic field dependence of the nucleation time is roughly described by the classical nucleation theory. Our data show increase of the anisotropy in the shape of the critical droplets with increase of q.

A three-state model based on the Potts model is proposed to simulate financial markets. The three states are assigned to "buy", "sell" and "inactive" states. The model shows the main stylized facts observed in the financial market: fat-tailed distributions of returns and long time correlations in the absolute returns. At low inactivity rate, the model effectively reduces to the two-state model of Bornholdt and shows similar results to the Bornholdt model. As the inactivity increases, we observe the exponential distributions of returns.

Monte Carlo simulations using the recently proposed Wang–Landau algorithm are performed to the q = 8 state Potts model in two dimension with various degrees of randomness. We systematically studied the effect of quenched bond randomness to system which has first-order phase transition. All simulations and measurements were done from pure case r = 1 to r = 0.4. Physical quantities such as energy density and ground-state entropy were evaluated at all temperatures. We have also obtained probability distributions of energy to monitor softening of transitions. It appears quite feasible to simulate spin systems with quenched bond randomness by Wang–Landau algorithm.

We have studied the influence of the distribution of bimodal bonds on the phase transition in two-dimensional 8-state Potts model by the recently proposed Wang–Landau (WL) and the Swendsen–Wang (SW) algorithm. All simulations and measurements are done for r = 0.5. Physical quantities such as energy density and specific heat are evaluated at all temperatures. We have also obtained the probability distributions of the energy in order to monitor the transitions. We have observed that some cases of the periodically arranged bond distributions show a single peak, and some cases show double or triple peaks in the specific heat. Besides, it seems that the appearing of these peaks in the specific heat relates to a blocking procedure for periodicity. When the number of interaction pairs between the bimodal bonds is increased on the lattice with the blocking procedure, one can observe a single peak, otherwise, one can observe a double or triple peaks in the specific heat. From the point of view of simulation methods, the WL algorithm also works efficiently in the simulation of the system for a periodically arranged bond distribution as well as the SW algorithm.

Financial market is a complex evolved dynamic system with high volatilities and noises, and the modeling and analyzing of financial time series are regarded as the rather challenging tasks in financial research. In this work, by applying the Potts dynamic system, a random agent-based financial time series model is developed in an attempt to uncover the empirical laws in finance, where the Potts model is introduced to imitate the trading interactions among the investing agents. Based on the computer simulation in conjunction with the statistical analysis and the nonlinear analysis, we present numerical research to investigate the fluctuation behaviors of the proposed time series model. Furthermore, in order to get a robust conclusion, we consider the daily returns of Shanghai Composite Index and Shenzhen Component Index, and the comparison analysis of return behaviors between the simulation data and the actual data is exhibited.

Using multicanonical Monte Carlo simulations, we have calculated the interface free energy between the coexisting ordered and disordered phase at the temperature driven first-order phase transition of the two-dimensional 7-state Potts model. Our result 2f³= 0.0241±0.0010 is about an order of magnitude smaller than other estimates in the recent literature. We discuss the differences in analysis and give a possible explanation for the discrepancy. Finally, we briefly mention similar investigations at the deconfining phase transition of SU(3) lattice gauge theory.

The multi-spin coding of the Monte Carlo simulation of the three-state Potts model on the simple cubic lattice is presented. The ferromagnetic (F) model, the antiferromagnetic (AF) model, and the random mixture of the F and AF couplings are treated. The multispin coding technique is also applied to the block-spin transformation. The block-spin transformation of the F Potts model is simply realized by the majority rule, whereas the AF three-state Potts model is transformed to the block spin having a six-fold symmetry.

The three-dimensional 3-state Potts model has been investigated by examining the average cluster size distributions. Cluster distributions give evidence for the weak first-order nature of the transition for lattice sizes as small as 16³, where the energy histogram method fails to exhibit the true nature of the transition.

The temperature dependence of the dynamics of the fraction F(t) of persistent spins in the triangular Q-state Potts model is investigated by large scale Monte Carlo simulations. After extending Derrida's approach of measuring the fraction of spins that remain in one phase to allQ low-temperature phases, it is shown that the exponent θ of algebraic decay of F(t) is independent of temperature.

We investigated the influence of the distribution of bimodal bonds on phase transition in 8-state Potts model in two dimensions. We show that there is a finite size dependent threshold value of the introduced quenched randomness in the bond distribution for rounding the first-order phase transition.

Studies of dynamical properties of first-order phase transition for a scalar field theory indicate that the phase conversion mechanism itself depends on the strength of the first-order transition: if the transition is strongly (weakly) first-order, bubble nucleation (spinodal decomposition) are favored conversion mechanisms, respectively. These distinct scenarios are of phenomenological impact. In order to see which phase conversion mechanism takes place depending on the strength of transition, we have simulated the q=5 state Potts model in two dimensions with an external magnetic field. The transition gets weakened in its first-order as the external field increases. Our results indicate that the phase conversion mechanism changes from nucleation to spinodal decomposition.

We build the Z₃ invariants fusion rules associated to the (D₄,A₆) conformal algebra. This algebra is known to describe the tri-critical Potts model. The 4-point correlation functions of critical fields are developed in the bootstrap approach, and on the other hand, they are written in terms of integral representation of the conformal blocks. By comparing both expressions, one can determine the structure constants of the operator algebra.