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 nonequilibrium 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, ABCE, is then able to cap
ture arbitrarily more transition states (seen below), thereby
building the rate catalog as accurately as possible for kMC
simulations.