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Building a kinetic Monte Carlo model with a chosen accuracy
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10.1063/1.4812319
/content/aip/journal/jcp/138/24/10.1063/1.4812319
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/24/10.1063/1.4812319
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

(a) Schematic of a basin denoted B in the potential energy surface (PES). Consider the case where four atomic processes from the basin B are found by performing basin escape pathway search (BEPS). (b) By repeating this procedure for other basins, we obtain a “kinetic map” of the PES, which we will term the KMC model. This KMC model consists of a list of states and atomic processes found. In addition, the process rates, the time for which the system resides in each basin, and the number of times each process has been witnessed in the BEPS calculations are also stored.

Image of FIG. 2.
FIG. 2.

Schematic of number of times processes are observed when BEPS calculations are performed in a particular basin. The x-axis denotes the time t spent in the basin with BEPS. Numbers inside the colored vertical bands denote the process index; multiple sightings of a process are represented by multiple circles. It is likely that processes belonging to the inaccessible timescales and some of the processes from the accessible timescales are missing from the catalog.

Image of FIG. 3.
FIG. 3.

Missing rate from a catalog C for a basin that contains one process with rate constant 10 s. The catalog C is initially empty. Processes from the basin are found by sampling escape pathways from the basin. The rate estimate for the inaccessible timescales (dashed line; Eq. (11) ) decreases as the time t spent in the basin increases. Circles denote the time at which the process was first observed (shown for 100 independent catalog generation calculations). The solid line denotes the correct unknown rate for one such catalog.

Image of FIG. 4.
FIG. 4.

Missing rate for a basin with N = 100 processes. Each process has a rate constant 10 s. Missing processes are observed as time t elapsed in the basin increases. The rate estimate from the inaccessible timescales (dashed line; Eq. (11) ) is smaller than the correct unknown rate k. When contributions from the accessible timescale spectral band are added to the ones from the inaccessible timescales the rate estimate (blue line) is very close to k. Symbols denote the times when the rates were computed. Here, n denotes the number of known processes.

Image of FIG. 5.
FIG. 5.

Number of processes observed m times when the catalog was generated in Fig. 4 after a total of m = 50 escapes from the basin. The value of that results in the least sum of squared error with respect to the data from the histogram is regarded as an estimate for the number of processes in a spectral band.

Image of FIG. 6.
FIG. 6.

Sum of squared errors (SSE) obtained from a catalog being generated. Three complete catalogs with the number of processes N being (a) 10, (b) 100, and (c) 1000 processes were considered (N is indicated by the dashed vertical line). All processes have identical rate constant given by k = 10 s. SSE is plotted for different values of the parameter in Eq. (17) . The value of that gives the smallest SSE is the best estimate for the number of processes in the spectral band.

Image of FIG. 7.
FIG. 7.

Histogram for from different catalogs generated for a basin that contains N = 100 processes, each having a rate constant k = 10 s. Histogram was obtained after m number of escapes occurred.

Image of FIG. 8.
FIG. 8.

Estimate for the missing rate for a catalog generated with BEPS. The catalog contains (a) N = 100 and (b) 1000 processes. Each process has a rate constant k = 10 s. Black line in panels (a) and (b) denotes the correct unknown rate k. Red dashed and blue solid lines denote rate estimate from Eqs. (11) and (22) , respectively. Symbols denote the time when the rate estimate was obtained. The estimate is non-zero even though k becomes zero after some time. (c) Validity times for catalog generated in panels (a) and (b) using δ = 0.1.

Image of FIG. 9.
FIG. 9.

Estimate for missing rate from a catalog C generated for a basin using BEPS. Two spectral bands are present. The first band contains N = 100 processes with individual rate constant k = 10 s, while the second band contains N = 100 processes with rate constant (a) k = 10 s, (b) k = 10 s1, and (c) k = 10 s. Black line denotes the correct unknown rate. Dashed red and solid blue lines denote estimate from Eqs. (11) and (23) . Symbols denote the times where the unknown rate was computed.

Image of FIG. 10.
FIG. 10.

Estimate for missing rate for a particular basin with (a) N = 100 and (b) N = 1000 processes. Rate constants and the three spectral bands to which the rates belong for panel (a) are shown in the inset by the vertical lines and the shaded area, respectively. Black line denotes the correct unknown rate. Dashed red and solid blue lines denote estimate from Eqs. (11) and (23) .

Image of FIG. 11.
FIG. 11.

(a) Number of processes observed for catalog in Fig. 10 with time t spent in the basin. All processes known once times in the shaded area are reached. (b) Validity time for the catalog as it is being generated. (c) Average observed error δ is less than the target error δ = 0.1. Averaging was performed over 100 independent KMC runs.

Image of FIG. 12.
FIG. 12.

Histogram for average observed error δ for different values of target error δ (a) 0.001, (b) 0.01, and (c) 0.1 (shown as percentage in figure) for the catalog in Fig. 10 . The shaded area shows cases where the error does not exceed δ. Out of 100 different catalogs generated, the number of catalogs that resulted in more than δ was 14, 7, and 6 times for panels (a), (b), and (c), respectively.

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/content/aip/journal/jcp/138/24/10.1063/1.4812319
2013-06-28
2014-04-24
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Building a kinetic Monte Carlo model with a chosen accuracy
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/24/10.1063/1.4812319
10.1063/1.4812319
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