Stochastic Nature of Chemical Kinetics

If we deal with an individual particle, then we cannot predict whether this particle will react in the next instant of time or not. In contrast, we can make statements for a large number of particles, e.g., how the concentration of a certain species in a chemical reaction changes in time. 

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In several theoretical models for small systems [77,79,80-81] the stochastic nature of the biochemical processes under certain circumstances reveals itself as the apparent violation of the Second Law of Thermodynamics. We consider below some of these models, demonstrating that the description of such processes in small systems based on averaging (nonlocal) formalism can often be misleading. These models demonstrate several unusual thermodynamic and kinetic properties that contradict the conventional laws of chemical thermodynamics and kinetics, described as a rule in terms of the average concentrations of the reagents and chemical intermediates. 

Note that a classification of the surface reaction mechanisms can be done either on the base of the nature of limiting stages, or on the base of dynamical models of elementary acts. The first way of classification is conditional, depending strongly on the relative values of different terms in the equations of chemical kinetics (6.1.19) or (6.3.1). Classification on the base of dynamical models (non-adiabatic, adiabatic, collineai-, impact, stochastic, etc.) needs the detailed study of the physical nature of reactive interactions. Such study is at the very beginning now both in theoretical and experimental (molecular beams) directions, and it should lead to detailed information on mechanisms of surface reactions. 

Thus far, we have described the time-dependent nature of polymerizing environments both through stochastic [49-51] and lattice [52,53] models capable of addressing this kind of dynamics in a complex environment. The current article focuses on the former approach, but now rephrases the earlier justification of the use of the irreversible Langevin equation, iGLE, to the polymerization problem in the context of kinetic models, and specifically the chemical stochastic equation. The nonstationarity in the solvent response due to the collective polymerization of the dense solvent now appears naturally. This leads to a clear recipe for the construction of the requisite terms in the iGLE. Namely the potential of mean force and the friction kernel as described in Section 3. With these tools in hand, the iGLE is used... 

Gillespie s algorithm numerically reproduces the solution of the chemical master equation, simulating the individual occurrences of reactions. This type of description is called a jump Markov process, a type of stochastic process. A jump Markov process describes a system that has a probability of discontinuously transitioning from one state to another. This type of algorithm is also known as kinetic Monte Carlo. An ensemble of simulation trajectories in state space is required to accurately capture the probabilistic nature of the transient behavior of the system. 

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