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Efficient computation of the first passage time distribution of the generalized master equation by steady-state relaxation
1.N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1992).
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15.The term “generalized master equation” has been used in multiple contexts. It is often used to describe the time evolution of the quantum density matrix (Ref. 1), but here we are only interested in the classical form (i.e., corresponding to a diagonal density matrix). The form of the generalized master equation presented in Eq. (8) of Ref. 14 only defines for . Equation (3) here is a minor extension that defines for all while being equivalent for .
17.R. Zwanzig, in Lectures in Theoretical Physics:, edited by W. E. Brittin, B. W. Downs, and J. Downs (Interscience, New York, 1961).
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21.H. Scher and M. Lax, Phys. Rev. B 7, 4491 (1973). Equation (8) are a simple extension of the on-lattice equations of this reference to the case of general network connectivity. While Eq. (8a) is commonly written as an integral equation, we use the integrodifferential form to highlight the relationship to Eq. (3). For convenience, we also extend the upper limits of the integrals from to . This has no effect since the boundary conditions and homogeneity of the equations for imply that .
23.L. Farkas, Z. Phys. Chem. (Leipzig) 125, 236 (1927).
24.Using Eq. (12) instead of Eq. (14) would yield instead of Eq. (17). This expression has a removable singularity at that slightly complicates the analysis.
25.E. Vanden-Eijnden (private communication) has noted that Eq. (18) can also be derived by using the Laplace transforms of Eqs. (8) rather than the Laplace transform of Eq. (3).
26. is not fully independent of the because the mean waiting time for leaving may depend on the state to which a transition is made. We can account for this by replacing in Eq. (27) with , where is the mean waiting time for the subset of first transitions out of that go to , and replace Eq. (28) withHowever, this requires estimation of the and the variances of the , which will require larger amounts of simulation data than estimation of and alone. We expect that this formula will be slightly better for optimizing the efficiency of high-accuracy results where enough simulations will be available to estimate the additional parameters, but it could be less effective than Eq. (28) for optimizing inexpensive low-accuracy results if not enough simulations were available to accurately estimate them. This is probably splitting hairs since we expect that, in most cases, the overall error will be dominated by the error in estimating which will not be affected by this change.
27.The sums over and need only include the states to which state makes transitions. In the special case where the states are connected in a linear order (i.e., is tridiagonal), the expression can be simplified by replacing the multinomial parameters with a single binomial parameter for each .
28.In principle, a better procedure might be to adaptively adjust Eq. (28) as more samples are gathered. However, in the example of Sec. IV B, efficiency did not depend sensitively on the size of the pilot, so an adaptive procedure may not give much further improvement.
29.When the maximum likelihood estimator of Eq. (25b) was used instead of the Bayes-Laplace estimator, the geometric rms error for cost=maxcost/256 increased from 52% to 62% and the error for cost=maxcost/128 increased from 37% to 41%. The change in the other values was insignificant.
30. is a positive quantity and we expect that the errors in its calculation will be roughly lognormally distributed (this was empirically verified for the example), so the rms geometric error is an appropriate error measure. For exact value and -computed values , the geometric rms error is .
31.To reduce computational time and eliminate quantization errors, the CTRW values for infinitesimal quantization length were actually computed using the equivalent Eqs. (24) and (25) with .
32. depends on and , so only one additional set of moments of needs to be computed for each additional FPT moment.
33.We have found empirically that a slightly better fit for the same computational cost is obtained by determining and by matching the MFPT and the second exponential moment , rather than using Eqs. (29). Moreover, this procedure is slightly simpler to use since it is not even necessary to expand Eq. (18) to second order.
34.Since it is only the magnitude of , not the number of interpolation points, that limits statistical accuracy, the interpolation points could be made denser with no significant increase in computational cost. However, this is not expected to greatly improve accuracy.
35.Matching the constrained rational function is numerically equivalent to matching an unconstrained rational function at the points , where is a small number. Matching at ensures that and matching at ensures Eq. (33). This can be done using standard rational interpolation software or algorithms.
36.A. B. Gelman, J. S. Carlin, H. S. Stern, and D. B. Rubin, Bayesian Data Analysis (Chapman and Hall, London/CRC, Boca Raton, FL, 1997).
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