A Guide to Monte Carlo Simulations in Statistical Physics: Reweighting methods

Phenomenological kinetics (PK) is widely used in the study of the reaction rates in heterogeneous catalysis, and it is an important aid in reactor design. PK makes simplifying assumptions: It neglects the role of fluctuations, assumes that there is no correlation between the locations of the reactants on the surface, and considers the reacting mixture to be an ideal solution. In this article we test to what extent these assumptions damage the theory. In practice the PK rate equations are used by adjusting the rate constants to fit the results of the experiments. However, there are numerous examples where a mechanism fitted the data and was shown later to be erroneous or where two mutually exclusive mechanisms fitted well the same set of data. Because of this, we compare the PK equations to “computer experiments” that use kinetic Monte Carlo (kMC) simulations. Unlike in real experiments, in kMC the structure of the surface, the reaction mechanism, and the rate constants are known. Therefore, any discrepancy between PK and kMC must be attributed to an intrinsic failure of PK. We find that the results obtained by solving the PK equations and those obtained from kMC, while using the same rate constants and the same reactions, do not agree. Moreover, when we vary the rate constants in the PK model to fit the turnover frequencies produced by kMC, we find that the fit is not adequate and that the rate constants that give the best fit are very different from the rate constants used in kMC. The discrepancy between PK and kMC for the model of CO oxidation used here is surprising since the kMC model contains no lateral interactions that would make the coverage of the reactants spatially inhomogeneous. Nevertheless, such inhomogeneities are created by the interplay between the rate of adsorption, of desorption, and of vacancy creation by the chemical reactions.

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Does phenomenological kinetics provide an adequate description of heterogeneous catalytic reactions

It has been well established that spatially extended, bistable systems that are driven by an oscillating field exhibit a nonequilibrium dynamic phase transition (DPT). The DPT occurs when the field frequency is of the order of the inverse of an intrinsic lifetime associated with the transitions between the two stable states in a static field of the same magnitude as the amplitude of the oscillating field. The DPT is continuous and belongs to the same universality class as the equilibrium phase transition of the Ising model in zero field [G. Korniss et al., Phys. Rev. E 63, 016120 (2001); H. Fujisaka et al., Phys. Rev. E 63, 036109 (2001)]. However, it has previously been claimed that the DPT becomes discontinuous at temperatures below a tricritical point [M. Acharyya, Phys. Rev. E 59, 218 (1999)]. This claim was based on observations in dynamic Monte Carlo simulations of a multipeaked probability density for the dynamic order parameter and negative values of the fourth-order cumulant ratio. Both phenomena can be characteristic of discontinuous phase transitions. Here we use classical nucleation theory for the decay of metastable phases, together with data from large-scale dynamic Monte Carlo simulations of a two-dimensional kinetic Ising ferromagnet, to show that these observations in this case are merely finite-size effects. For sufficiently small systems and low temperatures, the continuous DPT is replaced, not by a discontinuous phase transition, but by a crossover to stochastic resonance. In the infinite-system limit, the stochastic-resonance regime vanishes, and the continuous DPT should persist for all nonzero temperatures.

https://arxiv.org/pdf/cond-mat/0207275.pdf

Absence of first-order transition and tricritical point in the dynamic phase diagram of a spatially extended bistable system in an oscillating field.

We have presented dynamic phase transition features and stationary-state behavior of a ferrimagnetic small nanoparticle system with a core–shell structure. By means of detailed Monte Carlo simulations, a complete picture of the phase diagrams and magnetization profiles has been presented and the conditions for the occurrence of a compensation point Tcomp in the system have been investigated. According to Neel nomenclature, the magnetization curves of the particle have been found to obey P-type, N-type and Q-type classification schemes under certain conditions. Much effort has been devoted to investigating the hysteretic response of the particle, and we observed the existence of triple hysteresis loop behavior, which originates from the existence of a weak ferromagnetic core coupling Jc/Jsh, as well as a strong antiferromagnetic interface exchange interaction Jint/Jsh. Most of the calculations have been performed for a particle in the presence of oscillating fields of very high frequencies and high amplitudes in comparison with exchange interactions, which resembles a magnetic system under the influence of ultrafast switching fields. Particular attention has also been paid to the influence of the particle size on the thermal and magnetic properties, as well as magnetic features such as coercivity, remanence and the compensation temperature of the particle. We have found that, in the presence of ultrafast switching fields, the particle may exhibit a dynamic phase transition from paramagnetic to a dynamically ordered phase with increasing ferromagnetic shell thickness.

http://iopscience.iop.org/article/10.1088/0953-8984/24/43/436004/pdf

Dynamic phase transition properties and hysteretic behavior of a ferrimagnetic core-shell nanoparticle in the presence of a time dependent magnetic field

Realistic multisite lattice-gas modeling and KMC simulation of catalytic surface reactions: Kinetics and multiscale spatial behavior for CO-oxidation on metal (100) surfaces

We present a study, within a mean-field approach, of the kinetics of a classical mixed Ising ferrimagnetic model on a square lattice, in which the two interpenetrating square sublattices have spins $\ensuremath{\sigma}=\ifmmode\pm\else\textpm\fi{}1/2$ and $S=\ifmmode\pm\else\textpm\fi{}1,0.$ The kinetics is described by a Glauber-type stochastic dynamics in the presence of a time-dependent oscillating external field and a crystal field interaction. We can identify two types of solutions: a symmetric one, where the total magnetization $M$ oscillates around zero, and an antisymmetric one where $M$ oscillates around a finite value different from zero. There are regions of the phase space where both solutions coexist. The dynamical transition from one regime to the other can be of first or second order depending on the region in the phase diagram. Depending on the value of the crystal field we found up to two dynamical tricritical points where the transition changes from continuous to discontinuous. Also, we perform a similar study on the Blume-Capel $(S=\ifmmode\pm\else\textpm\fi{}1,0)$ model and find strong differences between its behavior and the one of the mixed model.

https://arxiv.org/pdf/cond-mat/9806036.pdf

Kinetics of a mixed Ising ferrimagnetic system

We present a kinetic Monte Carlo study of the dynamical response of a Ziff-Gulari-Barshad model for CO oxidation with CO desorption to periodic variation of the CO pressure. We use a square-wave periodic pressure variation with parameters that can be tuned to enhance the catalytic activity. We produce evidence that, below a critical value of the desorption rate, the driven system undergoes a dynamic phase transition between a CO2 productive phase and a nonproductive one at a critical value of the period and waveform of the pressure oscillation. At the dynamic phase transition the period-averaged CO2 production rate is significantly increased and can be used as a dynamic order parameter. We perform a finite-size scaling analysis that indicates the existence of power-law singularities for the order parameter and its fluctuations, yielding estimated critical exponent ratios beta/nu approximately 0.12 and gamma/nu approximately 1.77. These exponent ratios, together with theoretical symmetry arguments and numerical data for the fourth-order cumulant associated with the transition, give reasonable support for the hypothesis that the observed nonequilibrium dynamic phase transition is in the same universality class as the two-dimensional equilibrium Ising model.

https://arxiv.org/pdf/cond-mat/0408070.pdf

Response of a catalytic reaction to periodic variation of the CO pressure: increased CO2 production and dynamic phase transition.

We study by kinetic Monte Carlo simulations the dynamic behavior of a Ziff-Gulari-Barshad model with CO desorption for the reaction $\mathrm{C}\mathrm{O}+\mathrm{O}\ensuremath{\rightarrow}\mathrm{C}{\mathrm{O}}_{2}$ on a catalytic surface. Finite-size scaling analysis of the fluctuations and the fourth-order order-parameter cumulant show that below a critical CO desorption rate, the model exhibits a nonequilibrium first-order phase transition between low and high CO coverage phases. We calculate several points on the coexistence curve. We also measure the metastable lifetimes associated with the transition from the low CO coverage phase to the high CO coverage phase, and vice versa. Our results indicate that the transition process follows a mechanism very similar to the decay of metastable phases associated with equilibrium first-order phase transitions and can be described by the classic Kolmogorov-Johnson-Mehl-Avrami theory of phase transformation by nucleation and growth. In the present case, the desorption parameter plays the role of temperature, and the distance to the coexistence curve plays the role of an external field or supersaturation. We identify two distinct regimes, depending on whether the system is far from or close to the coexistence curve, in which the statistical properties and the system-size dependence of the lifetimes are different, corresponding to multidroplet or single-droplet decay, respectively. The crossover between the two regimes approaches the coexistence curve logarithmically with system size, analogous to the behavior of the crossover between multidroplet and single-droplet metastable decay near an equilibrium first-order phase transition.

https://arxiv.org/pdf/cond-mat/0411335.pdf

Decay of metastable phases in a model for the catalytic oxidation of CO.

The magnetic behavior of a mixed Ising ferrimagnetic system on a square lattice, in which the two interpenetrating square sublattices have spins $\ensuremath{\sigma}$ $(\ifmmode\pm\else\textpm\fi{}1/2)$ and spins S $(\ifmmode\pm\else\textpm\fi{}1,0),$ in the presence of an oscillating magnetic field, has been studied with Monte Carlo techniques. The model includes nearest- and next-nearest-neighbor interactions, a crystal field, and the oscillating external field. By studying the hysteretic response of this model to an oscillating field, we found that it qualitatively reproduces the increasing of the coercive field at the compensation temperature observed in real ferrimagnets, a crucial feature for magneto-optical applications. This behavior is basically independent of the frequency of the field and the size of the system. The magnetic response of the system is related to a dynamical transition from a paramagnetic to a ferromagnetic phase and to the different temperature dependence of the relaxation times of both sublattices.

http://rikvold-group.fsu.edu/research/matsci/MCatSCRI/proceedings/Buendia/buendia.pdf

Magnetic behavior of a mixed Ising ferrimagnetic model in an oscillating magnetic field

In this paper we calculate the free energy of a mixed Ising ferrimagnetic system by a mean-field approach. The system consists of two interpenetrating square sublattices, one with spins $\ifmmode\pm\else\textpm\fi{}1,0$ and the other with spins $\ifmmode\pm\else\textpm\fi{}1/2.$ We obtain the phase diagram of the system, identify the stable and metastable phases, calculate the temperature dependence of the magnetization, and give an estimate of the free-energy barrier between stable and metastable phases. By comparing our results with Monte Carlo simulations of the same model, we show that this approach gives an excellent estimate of the compensation temperatures and the magnetizations in a quite wide range of temperatures, up to quite close to the transition temperature, with a negligible computational effort. We found that when an external magnetic field is present, compensation temperatures only appear in the metastable phase, and that a system can have different compensation temperatures depending on how it is prepared. Some of these features have already been observed in experimental studies of molecular-based ferrimagnets. Our results suggest that the free-energy barrier becomes independent of the external field just at the compensation temperature.

E. Machado

Papers

Kinetics of a mixed Ising ferrimagnetic system

Response of a catalytic reaction to periodic variation of the CO pressure: increased CO2 production and dynamic phase transition.

Decay of metastable phases in a model for the catalytic oxidation of CO.

Magnetic behavior of a mixed Ising ferrimagnetic model in an oscillating magnetic field

Metastability and compensation temperatures for a mixed Ising ferrimagnetic system