Temporal Upscaling of KMC Simulation in Well-mixed Systems

Temporal Upscaling of KMC Simulation in Well-mixed Systems 

Recently, several approaches have been proposed to overcome the disparity of time scales for certain classes of problems. In order to overcome the problem of stiffness caused by rapid, partial equilibrated reactions in a living free-radical polymerization system, a hybrid analytical-KMC method was suggested (He 

The net-event KMC (NE-KMQ or lumping approach has been introduced by our group. The essence of the technique is that fast reversible events are lumped into an event with a rate equal to the net, i.e., the difference between forward and backward transition probabilities per unit time (Vlachos, 1998). The NE-KMC technique has recently been extended to spatially distributed systems (Snyder et al., 2005), and it was shown that savings are proportional to the separation of time scales between slow and fast events. The method is applicable to complex systems, and is robust and easy to implement. Furthermore, the method is self-adjusted, i.e., it behaves like a conventional KMC when there is no separation of time scales or at short times, and gradually switches to using the net-event construct, resulting in acceleration, only as PE is approached. A disadvantage of the method is that the noise is reduced. 

A comparison of the WP-KMC, NE-KMC, and conventional KMC is shown in Fig. 10. These acceleration approaches are successful regarding CPU. However, since the objective is often to study the role of noise, they do not provide the correct fluctuations. In a similar vein, use of simple rate expressions, such as the Michaelis-Menten or Hill kinetics, derived via PE and QSS approximations, are capable of accelerating KMC simulation since fast processes are eliminated. However, the noise of the resulting simulation, based on a reduced rate expression that lumps some of the reaction steps, is usually adversely affected (Bundschuh et al., 2003). 

Recently, Gillespie (2001) introduced an approximate approach, termed the r-leap method, for solving stochastic models. The main idea is the same as in the WP-KMC method. One selects a time increment r that is larger than the microscopic KMC time increment, and multiple molecular bundles of fast events occur. However, one now samples how many times each reaction will be executed from a Poisson rather than a uniform random number distribution. Prototype examples indicate that the r-leap method provides comparable noise with the microscopic KMC when the leap condition is satisfied, i.e., the time increments are such that the populations do not change significantly between time steps. 

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