Solving Master Equations Stochastically Monte Carlo Methods

Solving Master Equations Stochastically Monte Carlo Methods 

MC is also successful in far from equilibrium processes encountered in the areas of diffusion and reaction. It is precisely this class of non-equilibrium reaction/diffusion problems that is of interest here. Chemical engineering applications of MC include crystal growth (this is probably one of the first areas where physicists applied MC), catalysis, reaction networks, biology, etc. MC simulations provide the stochastic solution to a time-dependent master equation 

Direct solution of the master equation is impractical because of the huge number of equations needed to describe all possible states (combinations) even of relatively small-size systems. As one example, for a three-step linear pathway among 100 molecules, 104 such equations are needed. As another example, in biological simulation for the tumor suppressor p53, 211 states are estimated for the monomer and 244 for the tetramer (Rao et al., 2002). Instead of following all individual states, the MC method is used to follow the evolution of the system. For chemically reacting systems in a well-mixed environment, the foundations of stochastic simulation were laid down by Gillespie (1976, 1977). More 

The microscopic processes occurring in a system, along with their corresponding transition probabilities per unit time, are an input to a KMC simulation. This information can be obtained via the multiscale ladder using DFT, 

and/or MD simulations (the choice depends mainly on whether the process is activated or not). The creation of a database, a lookup table, or a map of transition probabilities for use in KMC simulation emerges as a powerful modeling approach in computational materials science and reaction arenas (Maroudas, 2001 Raimondeau et al., 2001). This idea parallels tabulation efforts in computationally intensive chemical kinetics simulations (Pope, 1997). In turn, the KMC technique computes system averages, which are usually of interest, as well as the probability density function (pdf) or higher moments, and spatiotemporal information in a spatially distributed simulation. 

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B. Solving Master Equations Stochastically Monte Carlo Methods... 

As was mentioned earlier the master equation (5.37) generally cannot be solved. To get some experience of the behaviour of chemical systems we might do stochastic simulation experiments using Monte-Carlo techniques (Introductions to Monte-Carlo methods are given in Hammersby Hand-scomb 1964, and Srejder 1965. Their applications in chemical physics are discussed in Binder (1979.)... 

We have solved numerically the above Master equations and, in addition, we simulated the underlying stochastic processes using a Monte-Carlo type method. A first result is shown in Fig. 5 for the thermal explosion case. 

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