Monte Carlo method kinetics

Agranovich V M, Efremov N A and Kirsanov V V 1980 Computer simulation of kinetics of excitation bimolecular quenching by Monte-Carlo method Fiz. Tverd. Tela 22 2118-27... 

A. Maltz, E. V. Albano. Kinetic phase transitions in dimer-dimer surface reaction models studied by means of mean-field and Monte Carlo methods. Surf Sci 277-A A-42S, 1992. 

C. C. Battaille, D. J. Srolvitz, J. E. Butler. A kinetic Monte Carlo method for the atomic-scale simulation of chemical vapor deposition application to diamond. J App Phys 52 6293, 1997. 

The Monte Carlo method as described so far is useful to evaluate equilibrium properties but says nothing about the time evolution of the system. However, it is in some cases possible to construct a Monte Carlo algorithm that allows the simulated system to evolve like a physical system. This is the case when the dynamics can be described as thermally activated processes, such as adsorption, desorption, and diffusion. Since these processes are particularly well defined in the case of lattice models, these are particularly well suited for this approach. The foundations of dynamical Monte Carlo (DMC) or kinetic Monte Carlo (KMC) simulations have been discussed by Eichthom and Weinberg (1991) in terms of the theory of Poisson processes. The main idea is that the rate of each process that may eventually occur on the surface can be described by an equation of the Arrhenius type ... 

Then the modelization of the hydrolysis kinetics requires at each time the knowledge of a and N. a can be calculated by writing the different relations of dissociation equilibria of water,polyacid and NH3 (produced by the hydrolysis reaction). We have proposed to determine at each reaction step and simulate the whole kinetics by using a Monte-Carlo method. (see ref.8 ). 

We review Monte Carlo calculations of phase transitions and ordering behavior in lattice gas models of adsorbed layers on surfaces. The technical aspects of Monte Carlo methods are briefly summarized and results for a wide variety of models are described. Included are calculations of internal energies and order parameters for these models as a function of temperature and coverage along with adsorption isotherms and dynamic quantities such as self-diffusion constants. We also show results which are applicable to the interpretation of experimental data on physical systems such as H on Pd(lOO) and H on Fe(110). Other studies which are presented address fundamental theoretical questions about the nature of phase transitions in a two-dimensional geometry such as the existence of Kosterlitz-Thouless transitions or the nature of dynamic critical exponents. Lastly, we briefly mention multilayer adsorption and wetting phenomena and touch on the kinetics of domain growth at surfaces. 

It Is to be remarked that the process described by the Infinite set of kinetic (coagulation) equations can be simulated by Monte-Carlo methods ( ). The Information on the number of molecules of the respective size Is stored In the computer memory and weighting for selection of molecules Is applied given by the number and reactivity of groups In the respective molecule. 

Voter, A.F. Introduction to the kinetic Monte Carlo method, In Radiation Effects in Solids (eds K.E. Sickafus, E.A. Kotomin and B.P. Uberuaga), Springer, NATO Publishing Unit, Dordrecht, The Netherlands, 2006, pp. 1-24. 

These two methods are different and are usually employed to calculate different properties. Molecular dynamics has a time-dependent component, and is better at calculating transport properties, such as viscosity, heat conductivity, and difftisivity. Monte Carlo methods do not contain information on kinetic energy. It is used more in the lattice model of polymers, protein stmcture conformation, and in the Gibbs ensemble for phase equilibrium. 

Interestingly, in the experiments devoted solely to computational chemistry, molecular dynamics calculations had the highest representation (96-98). The method was used in simulations of simple liquids, (96), in simulations of chemical reactions (97), and in studies of molecular clusters (98). One experiment was devoted to the use of Monte Carlo methods to distinguish between first and second-order kinetic rate laws (99). One experiment used DFT theory to study two isomerization reactions (100). 

Calculations [36] with the help of the Monte Carlo method, have shown (see, for example, Fig. 2) that, in the case most unfavourable for the theory, n(0) = A1(0), eqns. (27) describe the kinetics of electron tunneling at 50 and 80% of reagents decay with an accuracy of 2 and 4%, respectively [that is, 2% of the initial concentration, n(0) = 7V(0)]. Thus, eqns. (27) give a quantitative description of the kinetics of electron tunneling reactions up to a decay of 80%. 

Kinetic Monte Carlo Method with Dynamic Relaxation... 

The search for local minima in the neighborhood of a given local minimum is usually performed by the excitation of the system from this state followed by the relaxation of the system. If the relaxation of the excited system results in a state different from the initial state (and explored earlier), then a new local minimum is found, otherwise the evolution of the excited system is continued. The ways of moving out of the initial state can be different in temperature accelerated dynamics (TAD) by Sorensen and Voter [78], MD is used at high temperatures in the activation-relaxation technique (ART) by Mousseau and Barkema [79] and the local activated Monte Carlo method (LAMC) [80], the system evolves along the direction opposite to the direction of the force in the long-scale kinetic Monte Carlo... 

Contents 1. Introduction 176 2. Static NMR Spectra and the Description of Dynamic Exchange Processes 178 2.1. Simulation of static NMR spectra 178 2.2. Simulation of DNMR spectra with average density matrix method 180 3. Calculation of DNMR Spectra with the Kinetic Monte Carlo Method 182 3.1. Kinetic description of the exchange processes 183 3.2. Kinetic Monte Carlo simulation of DNMR spectra for uncoupled spin systems 188 3.3. Kinetic Monte Carlo simulation of coupled spin systems 196 3.4. The individual density matrix 198 3.5. Calculating the FID of a coupled spin system 200 3.6. Vector model and density matrix in case of dynamic processes 205 4. Summary 211 Acknowledgements 212 References 212... 

CALCULATION OF DNMR SPECTRA WITH THE KINETIC MONTE CARLO METHOD... 

Microscopic Kinetic Description of the Adsorption Process in Gas Chromatography - Monte Carlo Methods... 

All the statistical characteristics of copolymer chain structure and composition inhomogeneity, (including the ones reported in the above papers) can be easily calculated by means of the Markov chain formalism for any of kinetic models presented in Sect. 2. Then it does not seem advisable for the solution of such problems to apply the Monte-Carlo method with which the simulation of the copolymer chain growth was carried out [83-93]. 

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