Biology Reference
In-Depth Information
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