Stochastic theory simulation

T. Matsuda, G. D. Smith, R. G. Winkler, D. Y. Yoon. Stochastic dynamics simulations of n-alkane melts confined between solid surfaces Influence of surface properties and comparison with Schetjens-Fleer theory. Macromolecules 28 65- 13, 1995. 

A final comment on the interpretation of stochastic simulations We are so accustomed to writing continuous functions—differential and integrated rate equations, commonly called deterministic rate equations—that our first impulse on viewing these stochastic calculations is to interpret them as approximations to the familiar continuous functions. However, we have got this the wrong way around. On a molecular level, events are discrete, not continuous. The continuous functions work so well for us only because we do experiments on veiy large numbers of molecules (typically 10 -10 ). If we could experiment with very much smaller numbers of molecules, we would find that it is the continuous functions that are approximations to the stochastic results. Gillespie has developed the stochastic theory of chemical kinetics without dependence on the deterministic rate equations. 

Although the collision and transition state theories represent two important methods of attacking the theoretical calculation of reaction rates, they are not the only approaches available. Alternative methods include theories based on nonequilibrium statistical mechanics, stochastic theories, and Monte Carlo simulations of chemical dynamics. Consult the texts by Johnson (62), Laidler (60), and Benson (59) and the review by Wayne (63) for a further introduction to the theoretical aspects of reaction kinetics. 

Ray Kapral came to Toronto from the United States in 1969. His research interests center on theories of rate processes both in systems close to equilibrium, where the goal is the development of a microscopic theory of condensed phase reaction rates,89 and in systems far from chemical equilibrium, where descriptions of the complex spatial and temporal reactive dynamics that these systems exhibit have been developed.90 He and his collaborators have carried out research on the dynamics of phase transitions and critical phenomena, the dynamics of colloidal suspensions, the kinetic theory of chemical reactions in liquids, nonequilibrium statistical mechanics of liquids and mode coupling theory, mechanisms for the onset of chaos in nonlinear dynamical systems, the stochastic theory of chemical rate processes, studies of pattern formation in chemically reacting systems, and the development of molecular dynamics simulation methods for activated chemical rate processes. His recent research activities center on the theory of quantum and classical rate processes in the condensed phase91 and in clusters, and studies of chemical waves and patterns in reacting systems at both the macroscopic and mesoscopic levels. 

Simpler BGK kinetic theory models have, however, been applied to the study of isomerization dynamics. The solutions to the kinetic equation have been carried out either by expansions in eigenfunctions of the BGK collision operator (these are similar in spirit to the discussion in Section IX.B) or by stochastic simulation of the kinetic equation. The stochastic trajectory simulation of the BGK kinetic equation involves the calculation of the trajectories of an ensemble of particles as in the Brownian dynamics method described earlier. 

Norman, G.E., Stegailov, V.V. Stochastic theory of the classical molecular d5mamics method. Math. Models Comput. Simul. 5, 305-333 (2013)... 

Encouragingly, the acceleration methods described by Luo et al. are appHcable to many software packages and could improve many PB solvers. Although many applications can benefit from faster PB methods, perhaps the most profound advantage will be in the realm of stochastic dynamics simulations by enabling the efficient implicit solvent simulations of protein dynamics at the PB level of electrostatics theory. 

This article describes stochastic computer simulations of the local segmental dynamics of synthetic polymers. Particular attention is given to local dynamics in solution. Related work involving experimental methods, analytical theory, and molecular dynamics simulations will also be discussed. An introduction to the concepts involved in stochastic simulations will be presented. Methods of characterizing local segmental dynamics will also be described. The main portion of the article describes various features of the simulated dynamics. The approximations inherent in stochastic approaches and their influence on the observed dynamics will be discussed. Issues of cooperativity in conformational transitions will be highlighted. 

We can also relate these two approximations through the stochastic theory of the line shape developed by Kubo [11] and applied to molecular line shapes by Saven and Skinner [10]. As shown by Kubo, the overlap function given by Eq. (13) is a general result for a Gaussian-dis-tributed random variable in a Markovian process [11]. In the limit of a very slow decay of the time-correlation function of this random variable, the overlap function reduces to Eq. (9) and the line has a Lorentzian shape. In the limit of a very fast decay of the time-correlation function, the overlap function reduces to Eq. (15) and the line has a Gaussian shape. Employing molecidar dynamics simulations of chromophores within non-polar fluids. 

In this section, Redfield s theory of spin relaxation in the presence of a fluctuating transverse field is formulated in such a way that it can be included in a simulation using stochastic theory which utilises the wavefunction of the chemical system. Although the method of simulating relaxation as a disaete event on f is plausible (as described in the previous section), it is nevertheless important to compare this method with more usual methods to ensure no errors have been intfoduced. 

In conclusion, we have developed a Monte Carlo simulation in order to obtain the intensity correlation function in the multiple scattering regime in a magneto-optically active medium. For the diffusion regime, the results predicted by the simple stochastic theory are qualitatively verified. In the intermediate regime, the correlation function can be described by the one-dimensional random walk model, which explains the origin of the unexpected oscillations of the correlation function. 

Finally, in Chapters 14-18, I introduce elements of Monte Carlo, molecular dynamics and stochastic kinetic simulations, presenting them as the natural, numerical extension of statistical mechaiucal theories. 

Chapter 7 discusses a variety of topics all of which are related to the class of probabilistic CA (PCA) i.e. CA that involve some elements of probability in their state and/or time-evolution. The chapter begins with a physicist s overview of critical phenomena. Later sections include discussions of the equivalence between PCA and spin models, the critical behavior of PCA, mean-field theory, CA simulation of conventional spin models and a stochastic version of Conway s Life rule. 

In this chapter, we give a brief account of two related aspects of chemical reactions in solution the so-called "stochastic" theoretical approach to the rates of reactions and related features, and Molecular Dynamics (MD) computer simulations designed to test such theories and to otherwise provide insight on the reaction dynamics. 

Hewson, J. and A. R. Kerstein (2001). Stochastic simulation of transport and chemical kinetics in turbulent CO/H2/N2 flames. Combustion Theory and Modelling 5, 669-697. 

The methods of simulated annealing (26), genetic algorithms (27), and taboo search (29) are three of the most popular stochastic optimization techniques, inspired by ideas from statistical mechanics, theory of evolutionary biology, and operations research, respectively. They are applicable to our current problem and have been used by researchers for computational library design. Because SA is employed in this chapter, a more-detailed description of the (generalized) SA is given below. 

Monte Carlo simulations [54], analytical effective medium theory [64], and stochastic hopping theory [46] predict a dependence of the charge carrier mobility as a function of temperature and electric field given in (3) ... 

Real catalytic reactions upon solid surfaces are of great complexity and this is why they are inherently very difficult to deal with. The detailed understanding of such reactions is very important in applied research, but rarely has such a detailed understanding been achieved neither from experiment nor from theory. Theoretically there are three basic approaches kinetic equations of the mean-field type, computer simulations (Monte Carlo, MC) and cellular automata CA, or stochastic models (master equations). 

In spite of simple theoretical formalism (for example, mean-field descriptions of certain aspects) structural aspects of the systems are still explicitly taken into account. This leads to the results which are in a good agreement with computer simulations. But the stochastic model avoids the main difficulty of computer simulations the tremendous amount of computer time which is needed for obtaining good statistics for the reliable results. Therefore more complex systems can be studied in detail which may eventually lead to better understanding of real systems. In the theory discussed below we deal with a disordered surface. This additional complication will be handled in terms of the stochastic approach. This is also a very important case in catalytic reactions. 

The lowest-lying potential energy surfaces for the 0(3P) + CH2=C=CH2 reaction were theoretically characterized using CBS-QB3, RRKM statistical rate theory, and weak-collision master equation analysis using the exact stochastic simulation method. The results predicted that the electrophilic O-addition pathways on the central and terminal carbon atom are dominant up to combustion temperatures. Major predicted end-products are in agreement with experimental evidence. New H-abstraction pathways, resulting in OH and propargyl radicals, have been identified.254... 

A common and exact theory of sampling of inhomogeneous materials with stochastic composition is presented by BRANDS [1983]. Because experimental evaluation of this theory involves some difficulties, it was verified by simulation experiments. 

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