Individual Hopping Is Random in Nature
•
Residence time show broad distributions
staying in a certain
configuration
when hopping happens

** Subscribe** to view the full document.

•
Previous model is over-simplified, each timestep
∆?
is non-uniform
in reality
Individual Hopping Is Random in Nature
0
~
exp[
]
−
b
B
E
k
k
k T
Transition State Theory
(Arrhenius Law)
Frequently trying with a success rate of
k
, the system
can jump out the current basin at any time
0<
t
<∞
Define the survival probability at time
t
:
P
s
(t)
( there must be
P
s
(t=0)=1, P
s
(t=
∞
)=0
)
𝑃
?
? + ∆?
= 𝑃
?
?
∙ (1 − ? ∙ ∆?)
𝑑𝑃
𝑠
?
𝑑?
=
𝑃
𝑠
?+∆? −𝑃
𝑠
?
∆?
= −? ∙ 𝑃
?
?
֜ 𝑃
?
?
= ?
−𝑘∙?
probability density function (
PDF
) of the first jump happening at time
t
is therefore:
?
?
= −
?𝑃
?
?
??
= ? ∙
?
−𝑘∙?
(??? ?𝑎? ?𝑎???? ?ℎ???: න
0
∞
?
?
∙ ?? = 1)
the average residence time is therefore:
?
= න
0
∞
?
?
∙ ? ∙ ?? = 1/?
therefore the earlier model is only
a first order approximation

7
How to Sample Arbitrary
p(x) by
Monte Carlo
•
Define the cumulative distribution function (
CDF
) of
p(x)
:
𝐶
?
≡ න
?𝑖?
𝑥
?
?
∙ ??
𝐶
?
0
1
ξ
2
generate a uniformly distributed random number
ξ
2
~(0,1)
the corresponding
x
would then be selected so that
C(x)=
ξ
2
.
?
?
?
?
the high probability positions in
p(x)
correspond to larger cross-sections in
C(x)

** Subscribe** to view the full document.