Biology Reference
In-Depth Information
If the gradient of
U
norm
is normal to the remainder force
F
\
, i.e.:
U
norm
ðr
;
F
\
Þ
5
0
;
(5.8)
then the function
U
norm
will decrease monotonically with time during the process of
reaching the attractors, thus satisfying Lyapunov
'
s condition for metastability:
2
2
dU
norm
dt
U
norm
@
U
norm
@
52
@
@
0
(5.9)
2
,
x
y
We can therefore see that the condition for the normal decomposition has an obvious
physical meaning in that for (
0,
U
norm
corresponds to a Lyapunov function
U
norm
of dynamical systems and can represent the global (meta)stability as opposed to local
(linear) stability.
U
norm
,F
\
)
r
5
Thus, we can decompose any sufficiently smooth vector field into a conservative potential
field
U
norm
and the remaining forces F
\
:
U
norm
F
52
r
F
\
1
(5.10)
U
norm
ð
2
r
;
F
\
Þ
5
0
If the physical interpretation is that if
U
norm
(x) represents a landscape over the state space
region
x
, then for a ball in an attractor state that is perturbed to exit with least
'
energy
'
against the system
F(
x
) that keeps it in the attractor, we can decompose the
field such that F
\
will NOT contribute to this process, but only forces from the gradient
field
U
norm
can contribute. Based on Freidlin-Wentzell
'
s
'
driving force
'
s large deviation theory of a
stochastic process discussed below,
26
U
norm
can thus be used to compute the transition rate
in this nonequilibrium dynamic system.
Importantly, the normal potential
U
norm
can also be directly
'
89
'
'
the system equations
without time-stepping solution. We can calculate the potential field
U
norm
as follows:
read off
U
norm
U
norm
ðr
; F
1
r
Þ
5
0
(5.11)
This can be written in a component format called the Hamilton-Jacob equation:
U
norm
@
U
norm
@
U
norm
@
U
norm
@
U
norm
@
U
norm
@
@
F
x
1
1
@
1?1
@
F
x
i
1
@
1?1
@
F
x
n
1
@
0
5
x
1
x
1
x
i
x
i
x
n
x
n
(5.12)
The Hamilton-Jacob equation is a nonlinear partial differential equation, which usually has
no analytical solutions. However,
U
norm
can be solved numerically using the iterative
Newton-Raphson method after boundary conditions are specified for a real problem.
28,29
The Freidlin-Wentzell Theory of Large Deviation in Multistable Systems
For a dynamic system governed by deterministic forces F(
x
,
t
):
d
0
dt
5
x
F
ðx
0
;
t
Þ
(5.13)
Let us now consider that the system is under a stochastic perturbation
ξ
(
t
):
d
x
dt
5
F
ð
x
;
t
Þ
1
ε ξð
t
Þ
(5.14)
If
is sufficiently small, the perturbed system will converge to the original dynamical
system, i.e.
ε
jjx
2
x
0
jj
-
0. However, if the perturbation is a random process with small
