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If c2 =0 and c3 c1, Eq. gives λ+ = c3 and λ- = c1. Setting those results into Eq. gives a formula that is indeterminate when c3 = c1. But applying L'Hospital's rule to that indeterminate form, taking derivatives with respect to c3 yields the pdf c12 t e- c1 t. This nonexponential form, which goes to zero as t→0, is the pdf of the gamma random variable (c1,2), which is defined as the sum of two statistically independent exponentials with the same mean c1 -1. And this is exactly what we should expect for the time for an S1 - S3 conversion via reactions when c2 =0 and c3 = c1.
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If c2 =0 and c3 c1, Eq. gives λ+ = c3 and λ- = c1. Setting those results into Eq. gives a formula that is indeterminate when c3 = c1. But applying L'Hospital's rule to that indeterminate form, taking derivatives with respect to c3 yields the pdf c12 t e- c1 t. This nonexponential form, which goes to zero as t→0, is the pdf of the gamma random variable (c1,2), which is defined as the sum of two statistically independent exponentials with the same mean c1 -1. And this is exactly what we should expect for the time for an S1-S3 conversion via reactions when c2 =0 and c3 = c1.
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12
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0037444724
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In Ref., it was stated that the condition for applying the ssSSA to reactions is (c1 + c2) 2 c1 c3 x12. That is incorrect, as it arises from comparing a single-walker timescale with a many-walker timescale. The correct condition is simply c2 c3, as can be seen not only from the result but also from the argument at Eq. The reason why it is not necessary to supplement the condition c2 c3 with the condition c1 c3 is explained in the second paragraph of Sec.
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In Ref., it was stated that the condition for applying the ssSSA to reactions is (c1 + c2) 2 c1 c3 x12. That is incorrect, as it arises from comparing a single-walker timescale with a many-walker timescale. The correct condition is simply c2 c3, as can be seen not only from the result but also from the argument at Eq. The reason why it is not necessary to supplement the condition c2 c3 with the condition c1 c3 is explained in the second paragraph of Sec.
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