A
Phase
Transition
in a
System
Driven
by
Coloured
Noise
M.
ArrayasW,
M.I.
DykmanW,
R.
Mannella^,
P.V.E.
McClintockW,
and
N.D.
Stein^
^Department
of
Physics, Lancaster
University,
Lancaster
LA1
4YB?
UK
^Department
of
Physics
and
Astronomy,
Michigan State University, East Lansing,
MI
^
Dipartimento
di
Fisica, Universita
di
Pisa
and and
INFM
UdR
Pisa,
Piazza Torricelli
2,
56100
Pisa, Italy
Abstract.
For a
system driven
by
coloured noise,
we
investigate
the
activation
en
ergy
of
escape,
and the
dynamics during
the
escape.
We
have performed analogue
experiments
to
measure
the
change
in
activation energy
as the
power spectrum
of the
noise varies.
An
adiabatic approach based
on
path integral theory allows
us to
calcu
late analytically
the
critical value
at
which
a
phase transition
in the
activation energy
occurs.
INTRODUCTION
Gaussian
white noise
is a
mathematical
idealization which applies
to a
limited
number
of
physical situations where noise
has a
broad power spectrum. More
realistic
models
of
random
forces
have also been proposed,
and one of
them,
which
is
in some sense opposite to the white noise model, is quasimonochromatic noise
(QMN)
[1,2].
In
QMN, most
of the
power
is
concentrated within
a
relatively narrow
range
of
frequencies.
The
problem
of
activated escape
and
diffusion
in far
from
equilibrium systems
under
the
influence
of
different
kinds
of
noise
is of
general interest.
It has
been
attracting increasing
attention
in
diverse
scientific
contexts,
from
crystal growth
and
currentinduced desorption
from
crystal
surfaces,
to
current switching
in mi
crostructures,
and
biological applications.
For an
overdamped system
in a
double well potential driven
by
QMN,
a
bifur
cation
in the
optimal path (the optimal trajectory
for
departure
from
an
initial
attractor)
was
found
theoretically [3], leading
to the
prediction
of a
marked reduc
tion
in the
mean escape time [1].
As we
show below, related
effects
which involve
the
onset
of
focusing
singularities
and
caustics, were observed
in
models with more
than
one
degree
of
freedom acted upon
by
white noise [4].
CP502, Stochastic
and
Chaotic Dynamics
in the
Lakes:
STOCHAOS,
edited
by D. S.
Broomhead,
E. A.
Luchinskaya, P.V.
E.
McClintock,
and T.
Mullin
©
2000 American Institute
of
Physics l563969157/00/$17.00
42
0.35
0.3
0.25
0.2
3
0.15
0.1
0.05
0.25
0.5
0.75
r
1.25
1.5
FIGURE
1.
Measured
and
calculated
values
of the
action
5* as a
function
of
F
for the
symmet
rical
potential
with
CJQ
= 10. The
squares
represent
experimental
data
from
analogue
simulations,
the
diamonds
are
derived
by
solution
of the
nonadiabatic
Hamiltonian,
and the
circles
are ob
tained
by
solution
of the
adiabatic
theory.
The
horizontal
line
at S =
0.25
indicates
the
white
noise
solution,
and the
line
through
the
origin
represents
a
leastsquares
fit to the
adiabatic
theory
for
small
F.
In the
present
paper
we
develop
an
adiabatic
theory
for a
QMNdriven
system
in
order
to
study
the
bifurcation
in the
activation
energy
which
is, in
many
respects,
closely
analogous
to a
phase
transition.
We
show
that
this
is a
"continuous"
tran
sition
which
corresponds
to
spontaneous
breaking
of
time
symmetry
of the
most
probable
paths
for
escape
from
a
metastable
state.
We
present
the
first
experimen
tal
demonstration
of
this
transition
through
use of
analogue
electronic
simulations
[5].
THE
MODEL
A
simple
picture
of QMN is the
noise
which
results
from
filtering
white
noise
through
a
harmonic
oscillator
filter
with
natural
frequency
u;
0
and
damping
F
s).
(1)
Here,
£(t)
represents
Gaussian
white
noise
of
zero
mean
and
intensity
D, and we
assume
that
F
<C
^o
43
The
stochastic
dynamics
of the
overdamped
QMNdriven
system
that
we
consider
is
given
by
i +
U'(x)
=
f(t)
(2)
where
U(x)
is an
arbitrary
potential
that
we
will
assume
to
have
more
than
one
minimum.
For
weak
noise
/(t),
the
system
mostly
performs
small
fluctuations
about
one of the
potential
minima
but,
occasionally,
a
large
fluctuation
occurs
in
which
the
system
switches
from
one
potential
well
to
another.
The
probability
P
nm
of a
transition
n
—}
m,
where
n and m are two
minima,
can be
obtained
[1]
through
use of a
pathintegral
technique
[6,7].
The
escape
rate
when
D is
small
can be
found
up to a
prefactor
as
P[x
n
^
m
]
=
Nexp(~S[x
n
^
m
(t)}/D)
(3)
The
idea
is to
find
the
path
x(i)
which
minimises
the
action
5'.
This
path
is
the
solution
of a
variational
problem
which
relates
the
optimal
realization
of the
force
£(£)
(with
a
probability
density
functional
which
can be
immediately
found
from
(1))
and the
trajectory
of the
coordinate
(2),
subject
to the
constraint
that
this
trajectory
starts
at the
potential
minimum
and
ends
at the
saddle
[1].
The
action
is the
minimal
value
of the
corresponding
variational
functional.
We
will
call
the
minimum
action
the
activation
energy
of
escape,
by
analogy
with
white
noise
driven
systems.
The
variational
equations
can be
reduced
to a
sixth
order
ordinary
differential
equation
for
a?(t),
the
EulerLagrange
equation.
Einchcomb
and McKane [3]
showed
that
the
problem
could
be
recast
in
Hamiltonian
form,
which
proves
to be
convenient
both
for
accurate
numerical
calculation
of the
action
and
also
for
visualisation
purposes.
The
Hamiltonian
is
H(x,p)
=
{woW
x
2
U(x
1
}")
+
x
3
U"(x
1
)
+
z
2
2
t/'"(zi)}
+
Piz
2
+
P2Z
3
(4)
The
phase
space
is six
dimensional,
because
the
EulerLagrange
equation
is
sixth
order.
The
components
of x
turn
out to be
rather
simple
in
form
x
=
(xi,
x
2
,
x
3
]
=
(x,
x,
x]
(5)
The
expressions
for the
generalised
momenta
are
more
complicated
and we
refer
the
interested
reader
to
[3].
If
F
is
small,
the
motion
of a
particle
driven
by QMN
consists
of
rapid
oscillations
superimposed
on a
slow
motion
of the
centre
of
oscillations.
This
suggests
[1]
that
we
write
x(t)
= x
0
+
x
+
e*
wot
+
xe~
iwot
(6)
and
replace
U(x)
by an
effective
potential
V(;TO,
#+,#)
defined
as
44
V = —
/
2
V(>o
+
x+e**
+
x.e^di?
(7)
Z7T
JO
In
this
picture
we
obtain
an
effective
adiabatic
Hamiltonian
l
pl^xl,
(8)
and the
phase
space
is
reduced
from
six to
four
dimensions.
As the
phase
of the
oscillations
is
arbitrary,
we can set
x+
=
x^
in
deriving
Eq.
(8).
A
remarkable
feature
of the
adiabatic
Hamiltonian
is its
symmetry
with
respect
to the
transformation
x+
—>
—
x+,
p+
—±
—
p+.
This
symmetry
follows
from
the
form
of
the
potential
V,
which
depends
only
on the
product
x
+
x_
=
xjj_.
Therefore
there
always
exists
an
extreme
trajectory
of (8)
with
x+
= p+ = 0.
However,
besides
this
trajectory,
there
may
exist
trajectories
with
broken
symmetry,
where
x
+
^
0.
Clearly
such
trajectories
emerge
in
pairs.
From
(6),
they
correspond
to
fast
oscillating
solutions
x(i)
of the
original
variational
problem.
When
they
provide
the
minimum
to the
action
functional,
this
signals
breaking
of the
time
symmetry.
Since
the
phase
of
fast
oscillations
is
arbitrary,
this
bifurcation
is
analogous
to a
continuous
symmetrybreaking
transition.
THE
DOUBLE
WELL
POTENTIAL
We
consider
the
simple
doublewell
potential
U(x)
= x + x (9)
which
has
minima
at x = ±1 and a
saddle
at x = 0.
For
this
potential,
we
were
able
to
confirm
the
numerical
results
reported
by
Einchcomb
and
McKane.
We
have
performed
the
first
experimental
tests
of the
theory
by
measurement
of the
activation
energy
using
an
analogue
circuit.
The
results
for
this
symmetrical
case
are
plotted
in
Figure
1.
There
is a
critical
value
of
F
below
which
our
adiabatic
equations
yield
two
solutions
that
provide
extrema
for the
action.
One of
them
gives
a
value
for the
action
identical
with
that
for
white
noise;
the
other
corresponds
to
oscillatory
motion.
We
will
refer
to
them
as the
white
solution
and the
coloured
solution
respectively.
Above
the
critical
value
of
F,
we can see
only
the
white
solution.
Einchcomb
and
McKane
gave
an
estimate
of
F
c
~
0.46
for the
critical
value
at
which
the
bifurcation
in the
optimal
path
occurs.
The
analytical
result
which
follows
from
(8)
is
F
c
= 0.5
[9],
a
value
which
we
also
found
in
analogue
simulations
[10].
Our
numerical
solution
of
Hamilton's
equations
utilised
a
shooting
technique.
We
followed
a
large
number
of
trajectories
emanating
from
a
small
region
of
phase
space
surrounding
the
potential
minimum
at x =
—
1.
Paths
which
passed
through
the
saddle
at x = 0
were
identified
as
optimal
paths
and the
corresponding
action
45
FIGURE
2.
Optimal
trajectories
for
T
= 0.3
was
calculated.
Three
such
paths
can be
seen
in
Figure
2: the
xaxis
and the two
bounding
curves.
The
majority
of
initial
conditions
give
rise
to
paths
which
do not
reach
x = 0 and
therefore
do not
result
in
interwell
transitions.
Figure
2
shows
the
rather
complex
topology
of
these
paths,
with
a
cusp
being
clearly
visible
at
(so,
s+)«
(0.22,0).
Similar
behaviour
was
found
in the
system
which
Maier
and
Stein
introduced
in
their
analysis
of the
escape
problem
in
nonequilibrium
systems
[4].
They
describe
the
motion
of an
overdamped
particle
in a
twodimensional
field.
The
particle
is
subject
to
additive
isotropic
white
noise
and its
position
on the
(x,
y)
plane
satisfies
the
coupled
Langevin
equations
x
= x — x
3
—
axy
3
+
f
x
(t)
y
=
y

X
2
y
+
f
y
(t)
(10)
Since
the
field
is not
potential
(unless
a = 1) the
dynamics
do not
satisfy
detailed
balance.
Clearly,
the
system
(10)
has the
symmetry
y
^
—
y,
which
makes
it
similar
to the
QMNdriven
system.
We
note
that
the
QMNdriven
system
provides
a
natural
physical
realization
of the
system
discussed
by
Maier
and
Stein.
46