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The slow-scale stochastic simulation algorithm
1.D. T. Gillespie, J. Comput. Phys. 22, 403 (1976);
1.D. T. Gillespie, J. Phys. Chem. 81, 2340 (1977);
1.D. T. Gillespie, Physica A 188, 404 (1992).
2.E. L. Haseltine and J. B. Rawlings, J. Chem. Phys. 117, 6959 (2002).
3.C. V. Rao and A. P. Arkin, J. Chem. Phys. 118, 4999 (2003).
4.The concepts of temporal and ensemble averages in are discussed in some detail in F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), see Sec. 15.14. Briefly, the argument is that for any stable process and sufficiently large T and N, we can write the “temporal average” of as The first equality invokes a partitioning of the interval into N subintervals of size according to The second equality follows from the fact that, when T is sufficiently large, the random variable can be replaced almost everywhere by its asymptotic form so the values become, in aggregate, just N random samples of The final expression approximates, for sufficiently large N, the “ensemble average” of the random variable that average can also be computed as
5.M. Rathinam, L. R. Petzold, Y. Cao, and D. T. Gillespie, J. Chem. Phys. 119, 12784 (2003).
6.D. T. Gillespie, J. Chem. Phys. 115, 1716 (2001);
6.D. T. Gillespie and L. R. Petzold, J. Chem. Phys. 119, 8229 (2003).
7.A. Arkin, J. Ross, and H. H. McAdams, Genetics 149, 1633 (1998).
8.That the components of a multivariate Markov process usually are not individually Markovian is widely recognized. For example, in Brownian motion the Brownian particle’s velocity and position together form a bivariate Markov process however, is not by itself a Markov process, and will be Markovian only if there is no position-dependent force field. It should therefore come as no surprise that our fast and slow processes and which together form the Markov process are not individually Markovian. The reason can be understood heuristically as follows: Each of these two processes supplies to the other, through the connections indicated by Eqs. (3) and (4b), information about the other’s past values that goes beyond what is conveyed by the other’s present value. Since that information gets used in advancing the two processes in time, the processes by themselves are not “past-forgetting” in the Markov sense. More formally, a process will be Markovian if and only if it satisfies the Chapman–Kolmogorov condition—an integral relation of which the familiar master equation is a special differential form for processes of the jump type. It follows that a jump process cannot be Markovian unless it satisfies a canonical form master equation, like our Eq. (5). and separately do not satisfy such an equation, so they are not Markovian. References 2 and 3 take for their fast process the (real) fast process conditioned on the slow process. That process is shown in Ref. 3 to satisfy its Eq. (5)—an equation that almost has the canonical form of our Eq. (5), but not quite; it falls short in that the slow variables in the two propensity functions on the right-hand side have different values. This implies that the fast process conditioned on the slow process is not Markovian. To carry out a proper analysis of a non-Markovian process on its own terms—i.e., without first “embedding” it in a larger Markov process and then analyzing that larger process—is horrendously difficult, largely because one must ensure the simultaneous satisfaction of infinitely many coupled integral equations;
8.see D. Gillespie, W. Alltop, and J. Martin, Chaos 11, 548 (2001).
9.H. Kurata, H. El-Samad, T.-M. Yi, J. Khammash, and J. Doyle, in Proceedings of the 40th IEEE Conference on Decision and Control, 2001 (unpublished).
10.J. Doyle (private communication).
11.D. T. Gillespie, Markov Processes: An Introduction for Physical Scientists (Academic, San Diego, 1992). See especially Sec. 6.4 and Appendix D; in the latter it is shown that the “effective width” of a Gaussian peak is times the standard deviation.
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