The evolution time from i to f can therefore be calculated as uni03C4 uni22EF

The evolution time from i to f can therefore be

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The evolution time from i to f can therefore be calculated as: uni03C4 = + + uni22EF + + uni22EF + + uni22EF + uni2192 uni2192 uni2192 t k k k t k k k t , i f i i i i n i f i n i i i n i n i f ,1 ,1 , ( ) 1 , ( ) ,1 , ( ) ( ) (3) and similarly, for other unknown variables such as uni2192 t f 1 , the definition follows: uni03C4 = + + uni22EF + + uni22EF + + uni22EF + uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni00A0uni22EE uni2192 uni2192 uni2192 t k k k t k k k t . f i i n i f n i n n f 1 1 1, 1, 1, (1) 1, (1) 1, 1, (1) (1) (4) We can rewrite the above equations into the matrix expression as below: uni239B uni239D uni239C uni239C uni239C uni239C uni239C uni239C uni239C uni239C uni239E uni23A0 uni239F uni239F uni239F uni239F uni239F uni239F uni239F uni239F uni239B uni239D uni239C uni239C uni239C uni239C uni239C uni239C uni239C uni239C uni239E uni23A0 uni239F uni239F uni239F uni239F uni239F uni239F uni239F uni239F uni239B uni239D uni239C uni239C uni239C uni239C uni239C uni239C uni239C uni239E uni23A0 uni239F uni239F uni239F uni239F uni239F uni239F uni239F uni03C4 uni03C4 - + uni22EF uni22EF - + uni22EF uni22EF + uni22EF + - uni22EF + uni22EF + uni22EF uni22EE uni22F1 uni22EE uni22EF uni2192 uni2192 uni22EE uni22EE = - uni22EE uni22EE k k k k k k k k k k k k t t 1 0 1 0 0 . i i i n i i n i i i n i i i n n i n i f f i ,1 ,1 , ( ) , ( ) ,1 , ( ) 1, 1, 1, (1) 1, (1) 1, 1, (1) 1 1 (5) The evolution time uni2192 t i f can be analytically derived by diagonalizing the n * n matrix above, where n is the number of nodes in the system. In reality, the dimension n can be very large, which makes the analytical solution very diffi- cult to obtain. Therefore, kMC [ 42 44 ] is widely employed to get the approximated numerical solution. The more tran- sition states are explored, the higher the accuracy of the kMC simulations [ 42 ]. Therefore, algorithms that can effi- ciently identify the important transition pathways and cor- responding energy barriers are desirable. As stated earlier, the aim of the present work is to improve the capabilities of the ABC algorithm. The original ABC method is inherently most likely to capture only the dominant transitions in a 1D chain (shown below). This approach was proven successful, especially in finding the governing mechanism in the evo- lution of non-equilibrium systems. On the other hand, in a system whose evolution is described by competing pro- cesses, the original ABC algorithm is not sufficient to describe the true kinetics. To overcome this challenge, we penalize the system from the same basin multiple times in the original ABC framework, while blocking the observed transitions. This new algorithm, ABC-E, is then able to cap- ture arbitrarily more transition states (seen below), thereby building the rate catalog as accurately as possible for kMC simulations.
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  • Summer '19
  • Transition state, KMC

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