Stochastic Simulation of Chemical Reactions

For near and supercritical conditions, combustion gas-phase data are often used as the point of reference to assess solvent effects. The gas-phase values of kig, available for temperatures 800-2500 K, show the activation energy 90 kJ mol In condensed phase, stabilization of H2O molecules via H-bonding may increase the activation barrier, but on the other hand the reaction can be promoted by the solvent cage effect. Diffusion-kinetic modelling and stochastic simulation of chemical reactions in radiation tracks have shown that the occurrence of reaction (15.19) is consistent with the anomalous increase in H2 yield observed in water radiolysis at temperatures above 523 K, if kig is of the order of 1-2x10 s (4-8x10 s ) at 573 K. Considering the two... 

Figure 18.1 Regimes of the problem space for multiscale stochastic simulations of chemical reaction kinetics. The r-axis represents the number of molecules of reacting species, x, and the ) -axis measures the frequency of reaction events, A. The threshold variables demarcate the partitions of modeling formalisms. In area I, the number of molecules is so small and the reaction events are so infrequent that a discrete-stochastic simulation algorithm, like the SSA, is needed. In contrast, in area V, which extends to infinity, the thermodynamic limit assumption becomes vahd and a continuous-deterministic modehng formalism becomes valid. Other areas admit different modehng formalisms, such as ones based on chemical Langevin equations, or probabilistic steady-state assumptions.

Not being able to solve the master equation in the more general cases we are often satisfied by the determination of the first and second moments. Furthermore, different techniques can be applied to approximate the jump processes by continuous processes, which are more easily solvable. The clear structure of the stochastic model of chemical reactions allows the possibility of simulating the reaction. By simulation procedures realisations of the processes can be obtained. The methods for obtaining solutions will be illustrated by discussing particular examples. 

Reaction kinetic models can be simulated not only by solving the kinetic system of differential equations but also via simulating the equivalent stochastic models. Computer codes are available that solve the stochastic kinetic equations. One of these is the Chemical Kinetics Simulator (CKS) program that was developed at IBM s Almaden Research Centre. It provides a rapid, interactive method for the accurate simulation of chemical reactions. CKS is a good tool for teaching the principles of stochastic reaction kinetics to students and trainees. 

D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem. 81, 2340 (1977). 

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. 

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. 

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. 

D. T. Gillespie. Exact stochastic simulation of coupled chemical reactions. 

Quantum chemistry has estabhshed itself as a valuable tool in the studies of polymerization processes [25,26]. However, direct quantum chemical studies on the relationship between the catalyst structure and the topology of the resulting polymer, as well as on the influence of the reaction conditions, are not practical without the aid of statistical methods. We have to this end proposed a combined approach in which quantum chemical methods are used to provide information on the microscopic energetics of elementary reactions in the catalytic cycle, that is required for a mesoscopic stochastic simulations of polymer growth [25]. A stochastic approach makes it possible to discuss the effects of temperature and olefin pressure. 

Erdi, P., Sipos, T. Toth, J. (1973). Stochastic simulation of complex chemical reactions by computer. Magy. Kem. Foly., 79, 97-108 (in Hungarian). 

Sipos, T., Toth, J. Erdi, P. (1974a). Stochastic simulation of complex chemical reactions by digital computer, I. The model. React. Kinet. Catal. Lett., 1, 113-17. Sipos, T., T6th, J. Erdi, P. (1974b). Stochastic simulation of complex chemical reactions by digital computer, II. Applications. React. Kinet. Catal. Lett., 1, 209-13. 

In a continuous-flow chemical reactor, the concern is not only with probabilistic transitions among chemical species but also with probabilistic liansitions of each chemical species between the interior and exterior of the reactor. Pippel and Philipp [8] used Markov chains for simulating the dynamics of a chemical system. In their approach, the kinetics of a chemical reaction are treated deterministically and the flow through the system are treated stochastically by means of a Markov chain. Shinnar et al. [9] superimposed the kinetics of the first order chemical reactions on a stochastically modeled mixing process to characterize the performance of a continuous-flow reactor and compared it with that of the corresponding batch reactor. Most stochastic approaches to analysis and modeling of chemical reactions in a flow system have combined deterministic chemical kinetics and stochastic flows. 

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