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average amplitude but with occasional large excursions, the perturbed dynamic system
will behave differently. Freidlin and Wentzell 26 proposed a large deviation theory of
stochastic process as a theoretical framework to analyze the behavior of the dynamical
system with multiple attractors. Supposing that our system satisfies the Langevin dynamics,
the governing equations are described by the following ODEs:
<
x 1
_
f 1
ð
x 1
; ...;
x n
Þ 1 ξ 1 ð
t
Þ
5
^
x i 5
_
f i ð
x 1 ; ...;
x n Þ 1 ξ i ð
t
Þ
(5.15)
:
^
x n 5
_
f n 1 ð
x 1 ; ...;
x n Þ 1 ξ n ð
t
Þ
Suppose now that a ball is perturbed to go from attractor state x A -
x B , one defines an
action function V AB to measure the
'
energy
'
barrier to be overcome for this transition:
(
"
# dt
)
ð tB
X
n
1
2 min
2
V AB 5
1 jj_
x i 2
f i ð
x
Þjj
(5.16)
tA
i
5
Here the action function V AB is defined as a time integral of the square of the
'
remainder
'
of the dynamic equations (deviation from deterministic trajectory) over the whole
trajectory
X
( t ) from attractor
x A
to
x B
. If a ball is only driven by the
'
forces
'
specified in
the deterministic part of the ODEs ( Eq. 5.14 ), it will correspond to
'
free fall
'
in the
'
gravity
field
'
an d the action is zero. If the ball is perturbed against the
'
forces,
'
the remainder term
2 will not be zero (in all of this metaphoric picture one is reminded that we are
in an inertia-free world). We integrate the forces over the whole trajectory and obtain the total
action when a ball switches from attractor
jj_
x i 2
f i ðxÞjj
x B . Based on the variational principle, there
exists a unique minimum integral, namely the action function V AB , which is an objective
measure of the difficulty for a state switching in a nonequilibrium dynamic system.
x A to
90
The Relationship Between the Freidlin-Wentzell Action Function and the
Normal Decomposition
Although the Freidlin-Wentzell potential V and the normal potential U norm are defined in
different ways, they are mathematically related. For a dynamical system, F
U norm
52 r
F
,
1
\
U norm , F \ )
(
2 r
0. We can rewrite the Freidlin-Wentzell potential as:
5
ð tB
tA : X
2 dt
1
2 min
V AB 5
F
:
2
ð tB
tA : X
2 dt
1
2 min
U norm
1 r
F \ :
5
2
ð tB
tA : X
4 ð tB
tA
2 dt
r
dt
1
2 min
X
U norm
U norm
F \ 2 r
:
F \
5
2
1
2
(5.17)
ð tB
tA X
4 ð tB
2 dt
dt
1
2 min
tA X
U norm
U norm
F \ 2 r
r
5
2
1
ð tB
tA : X
h
i dt
1
2 min
2
U norm
U norm
B
U norm
A
F \ 2 r
:
4
ð
Þ
5
2
1
2
U norm
B
U norm
A
2
ð
Þ
5
2
During the process of exiting an attractor, the least action path (LAP) of a ball follows the
governing equation:
U norm
F
5 r
F \
(5.18)
1
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