Multiscale algorithms for chemical kinetics

Briefly, the algorithm first determines when the next reaction will occur based on the sum of probabilistic reaction rates, using Eq. 18.2. The method then establishes which of the M reaction channels will indeed occur, given the relative propensity values. The fxth reaction occurs when the cumulative sum of the x first terms becomes greater than r2a (Eq. 18.3). 

Implementation of Eqs. 18.2 and 18.3 samples the probability distribution of these events, P(t, fx), which is expressed as follows  

Subsequently the reaction propensities are updated, since the system has jumped to a neighboring state and the process is repeated until the system reaches the desired end time point. 

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. 

Gibson and Bruck improved the performance of SSA by resourcefully managing the need for random numbers, creating the Next Reaction variant of SSA. Cao and co-workers optimized the Direct Reaction variant of the SSA, proving that for certain systems this approach is more efficient than the Next Reaction variant. 

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