Escape Problems &
Stochastic Resonance
18.S995 - L05
dunkel@mit.edu
1.4
Escape problem
Escape problems are ubiquitous in biological, biophysical and biochemical processes. Prominent examples include, but are not restricted to,
• unbinding of molecules from receptors,
• chemical reactions,
• transfer of ion through through pores,
• evolutionary transitions between di↵erent fitness optima.
Their mathematical treatment typically involves models that are structurally very similar
to the one-dimensional examples discussed in this section13 .
1.4.1
Generic minimal model
Consider the over-damped SDE
dx(t) =
@x U dt +
p
2D ⇤ dB(t)
(1.68a)
with a confining potential U (x)
lim U (x) ! 1
(1.68b)
x!±1
that has two (or more) minima and maxima. A typical example is the bistable quartile
double-well
U (x) =
a 2 b 4
x + x ,
2
4
a, b > 0
(1.68c)
Examples
Illustration by J.P. Cartailler. Copyright 2007, Symmation LLC.
transport & evolution:
stochastic
escape problems
http://evolutionarysystemsbiology.org/intro/
dunkel@math.mit.edu
Arrhenius law
Stochastic resonance
Gammaitoni et al.:
FIG. 1. Stochastic resonance in a symmetric double well. (a)
Sketch of the double-well potential V(x)5(1/4)bx 4
2(1/2)ax 2 . The minima are located at 6x m , where
x m 5(a/b) 1/2. These are separated by a potential barrier with
the height given by DV5a 2 /(4b). The barrier top is located at
dunkel@math.mit.edu
https://www.youtube.com/watch?v=4pdiAneMMhU
Freund et al (2002) J Theor Biol
FIG. 1. (a) A juvenile paddle"sh. (b) Close-up view of clusters of electroreceptor pores on the rostrum.
Paddle Fish
juvenile paddle"sh. (b) Close-up view of clusters of electroreceptor pores on the rostrum.
FIG. 2. (a) A single Daphnia. (b) Electric potential from a tethered Daphnia measured at a distance of 0.5 cm. The oscill
sult from higher frequency feeding motions of the legs and lower frequency swimming motions of the antennae. (c) P
.
ss
ision but
o adapt to
creatures
retaceous
mis, 1991),
ctrorecepd rostrum
Fig. 1(a).
mpullae of
the water
minate in
Fig. 1(b).
ected and
n entirely
rostral aral., 2000;
Freund et al (2002) J Theor Biol
$%&&%'( &) *+&,'%
Paddle Fish
demic Press
right
above
60 left
0
n=312
–60
–60
0
60
below
Vertical distance (mm)
juvenile
(b) Close-up view
clustersnoise
of electroreceptor
pores
on the rostrum.
a paddle"sh.
c
b ofOptimal
Control
High
noise
(0.5 µV cm–1 r.m.s.)
(20 µV cm–1 r.m.s.)
60
60
0
0
n=373
–60
–60
0
n=46
–60
60
–60
0
60
Horizontal distance (mm)
Russel et al (1999) Nature
.
ss
ision but
o adapt to
creatures
retaceous
mis, 1991),
ctrorecepd rostrum
Fig. 1(a).
mpullae of
the water
minate in
Fig. 1(b).
ected and
n entirely
rostral aral., 2000;
Freund et al110,
(2002)
J Theor Biol
nstitut fuK r Physik, Humboldt-;niversitaK t zu Berlin, Invalidenstr.
D-10115
Berlin, German
enter for ¸imnology, ;niversity of =isconsin-Madison, Madison, =I 53706, ;.S.A. and A Center f
Neurodynamics, ;niversity of Missouri at St. ¸ouis, St. ¸ouis, MO 63121, ;.S.A.
demic Press
(Received on 21 May 2001, Accepted in revised form on 18 August 2001)
Paddle Fish
Zooplankton emit weak electric "elds into the surrounding water that originate from their
own paddle"sh.
muscular activities
associatedview
with of
swimming
Juvenile paddle"sh
prey
juvenile
(b) Close-up
clustersand
of feeding.
electroreceptor
pores on
theupon
rostrum.
single zooplankton by detecting and tracking these weak electric signatures. The passive
electric sense in this "sh is provided by an elaborate array of electroreceptors, Ampullae of
Lorenzini, spread over the surface of an elongated rostrum. We have previously shown that the
"sh use stochastic resonance to enhance prey capture near the detection threshold of their
sensory system. However, stochastic resonance requires an external source of electrical noise in
order to function. A swarm of plankton, for example Daphnia, can provide the required noise.
We hypothesize that juvenile paddle"sh can detect and attack single Daphnia as outliers in the
vicinity of the swarm by using noise from the swarm itself. From the power spectral density of
the noise plus the weak signal from a single Daphnia, we calculate the signal-to-noise ratio,
Fisher information and discriminability at the surface of the paddle"sh's rostrum. The results
predict a speci"c attack pattern for the paddle"sh that appears to be experimentally testable.
! 2002 Academic Press
Introduction
ddle"sh, Polyodon spathula, are among the
and muddy water obscure normal vision b
where plankton are plentiful. In order to adapt
to the one-dimensional examples discussed in this section13 .
1.4.1
Generic minimal model
Consider the over-damped SDE
dx(t) =
@x U dt +
p
2D ⇤ dB(t)
(1.68a)
with a confining potential U (x)
lim U (x) ! 1
(1.68b)
x!±1
that has two (or more) minima and maxima. A typical example is the bistable quartile
double-well
U (x) =
a 2 b 4
x + x ,
2
4
a, b > 0
(1.68c)
p
with minima at ± a/b.
Generally, we are interested in characterizing the transitions between neighboring minima in terms of a rate k (units of time 1 ) or, equivalently, by the typical time required for
escaping from one of the minima. To this end, we shall first dicuss the general structure of
the time-dependent solution of the FPE14 for the corresponding PDF p(t, x), which reads
@t p =
@x j ,
j(t, x) =
[(@x U )p + D@x p],
(1.68d)
and has the stationary zero-current (j ⌘ 0) solution
ps (x) =
13
14
e
U (x)/D
Z
,
Z=
Z
+1
dx e
U (x)/D
.
1
Although things usually get more complicated in higher-dimensions.
FPEs for over-damped processes are sometimes referred to as Smoluchowski equations.
(1.69)
Generally, we are interested in characterizing the transitions between neighboring minima in terms of a rate k (units of time 1 ) or, equivalently, by the typical time required for
escaping from one of the minima. To this end, we shall first dicuss the general structure of
the time-dependent solution of the FPE14 for the corresponding PDF p(t, x), which reads
General time-dependent solution
@t p =
@x j ,
j(t, x) =
[(@x U )p + D@x p],
(1.68d)
and has the stationary zero-current (j ⌘ 0) solution
To find the time-dependent solution, we can make the ansatz
Z +1
U (x)/D
e
ps (x) =
dx ,e U (x)/D .
p(t, x), = %(t,Zx)=e U (x)/(2D)
Z
1
13 which leads to a Schrödinger equation in imaginary time
Although things usually get more complicated in higher-dimensions.
14
⇥
⇤
FPEs for over-damped processes are sometimes
2 referred to as Smoluchowski equations.
@t % = D@x + W (x) % =: H%,
(1.69)
(1.70)
(1.71a)
16
with an e↵ective potential
1
W (x) =
(@x U )2
4D
1 2
@ U.
2 x
(1.71b)
Assuming the Hamilton operator H has a discrete non-degenerate spectrum,
the general solution p(t, x) may be written as
p(t, x) = e
U (x)/(2D)
1
X
cn
n (x) e
nt
,
0
<
1
< . . .,
(1.72a)
n=0
where the eigenfunctions
n
of H satisfy
Z
dx ⇤n (x)
m (x)
=
nm ,
and the constants cn are determined by the initial conditions
Z
cn = dx ⇤n (x) eU (x)/(2D) p(0, x).
(1.72b)
(1.72c)
Generally, we are interested in characterizing the transitions between neighboring minima in terms of a rate k (units of time 1 ) or, equivalently, by the typical time required for
escaping from one of the minima. To this end, we shall first dicuss the general structure of
the time-dependent solution of the FPE14 for the corresponding PDF p(t, x), which reads
General time-dependent solution
@t p =
@x j ,
j(t, x) =
[(@x U )p + D@x p],
(1.68d)
and has the stationary zero-current (j ⌘ 0) solution
To find the time-dependent solution, we can make the ansatz
Z +1
U (x)/D
e
ps (x) =
dx ,e U (x)/D .
p(t, x), = %(t,Zx)=e U (x)/(2D)
Z
1
13 which leads to a Schrödinger equation in imaginary time
Although things usually get more complicated in higher-dimensions.
14
⇥
⇤
FPEs for over-damped processes are sometimes
2 referred to as Smoluchowski equations.
@t % = D@x + W (x) % =: H%,
(1.69)
(1.70)
(1.71a)
16
with an e↵ective potential
1
W (x) =
(@x U )2
4D
1 2
@ U.
2 x
(1.71b)
Assuming the Hamilton operator H has a discrete non-degenerate spectrum,
the general solution p(t, x) may be written as
p(t, x) = e
U (x)/(2D)
1
X
cn
n (x) e
nt
,
0
<
1
< . . .,
(1.72a)
n=0
where the eigenfunctions
n
of H satisfy
Z
dx ⇤n (x)
m (x)
=
nm ,
and the constants cn are determined by the initial conditions
Z
cn = dx ⇤n (x) eU (x)/(2D) p(0, x).
(1.72b)
(1.72c)
p(t, x)
e
W=(x)
=
cn) n (x) e @ U.
,
(@x U
x
4D n=0
2
(1.72a)
General time-dependent solution
wherethe
the Hamilton
eigenfunctions
satisfy
n of HH
Assuming
operator
has a discrete non-degenerate spectrum,
the general solution p(t, x) mayZ be written
as
⇤
dx
n (x)
m (x)
=
nm ,
<
(x)/(2D)
and the constants cn are
the initial conditions
p(t,determined
x) = e Uby
cn n (x) e
Z
n=0
⇤
U (x)/(2D)
cn = dx n (x) e
p(0, x).
nt
,
< ..
(1.72a
(1.72c)
of H satisfy
At large times, t ! 1, the solution
(1.72a) must approach the stationary solution (1.69),
Z
implying that
dx ⇤n (x) m (x) = nm ,
and
1
(1.72b)
1
X
where the eigenfunctions
0
(1.71b
n
1
e U (x)/(2D)
p
c0 = p ,
.
0 = 0 ,
0 (x) =
Z the initial conditions
Z
the constants cn are determined by
Note that 0 = 0 in particular Z
means that the first non-zero eigenvalue
⇤
the relaxation dynamics cat large
= times
dx and,
(x)therefore,
eU (x)/(2D) p(0, x).
n
(1.73)
1
> 0 dominates
(1.72
n
⌧⇤ = 1/
(1.72b
(1.74)
1
At large times, t ! 1, the solution (1.72a) must approach the stationary solution (1.69
is a natural measure of the escape time. In practice, the eigenvalue 1 can be computed
implying
that standard methods (WKB approximation, Ritz method, techniques exploiting
by various
supersymmetry, etc.) depending on the specifics of the e↵ective potential W .
0
Note that
0
=0,
1
c0 = p ,
Z
0 (x)
=
e
U (x)/(2D)
p
Z
.
= 0 in particular means that the first non-zero eigenvalue
(1.73
1
> 0 dominate
1.4.2
Two-state approximation
We next illustrate a commonly used simplified description of escape problems, which can
be related to (1.74). As a specific example, we can again consider the escape of a particle
from the left well of a symmetric quartic double well-potential
U (x) =
a 2 b 4
x + x ,
2
4
p(0, x) = (x
x )
(1.75a)
where
x =
p
a/b
(1.75b)
is the location of the left minimum, but the general approach is applicable to other types
of potentials as well.
The basic idea of the two-state approximation is to project the full FPE dynamics onto
simpler set of master equations by considering the probabilities P± (t) of the coarse-grained
particle-states ‘left well’ ( ) and ‘right well’ (+), defined by
Z 0
P (t) =
dx p(t, x),
(1.76a)
1
Z 1
P+ (t) =
dx p(t, x).
(1.76b)
_
0
+
If all particles start in the left well, then
P (0) = 1 ,
P+ (0) = 0.
(1.77)
Whilst the exact dynamics of P± (t) is governed by the FPE (1.68d), the two-state approximation assumes that this dynamics can be approximated by the set of master equations15
Ṗ =
k+ P + k P+ ,
Ṗ+ = k+ P
k P+ .
(1.78)
The basic idea of the two-state approximation is to project the full FPE dynamics onto
simpler set of master equations by considering the probabilities P± (t) of the coarse-grained
t=0
particle-states ‘left well’ ( ) and ‘right well’ (+), defined by
Z 0
P (t) =
dx p(t, x),
(1.76a)
1
Z 1
P+ (t) =
dx p(t, x).
(1.76b)
_
+
0
If all particles start in the left well, then
P (0) = 1 ,
P+ (0) = 0.
(1.77)
Whilst the exact dynamics of P± (t) is governed by the FPE (1.68d), the two-state approximation assumes that this dynamics can be approximated by the set of master equations15
Ṗ =
k+ P + k P+ ,
Ṗ+ = k+ P
k P+ .
(1.78)
For a symmetric potential, U (x) = U ( x), forward and backward rates are equal, k+ =
k = k, and in this case, the solution of Eq. (1.78) is given by
1 1
P± (t) = ⌥ e
2 2
2k t
.
(1.79)
For comparison, from the FPE solution (1.72a), we find in the long-time limit
U (x)/2D
p(t, x) ' ps (x) + c1 e
1 (x)
e
1t
,
(1.80)
Due to the symmetry of ps (x), we then have
1
P (t) ' + C1 e
2
15
1t
Note that Eqs. (1.78) conserve the total probability, P (t) + P (t) = 1.
18
(1.81a)
The basic idea of the two-state approximation is to project the full FPE dynamics onto
simpler set of master equations by considering the probabilities P± (t) of the coarse-grained
t=0
particle-states ‘left well’ ( ) and ‘right well’ (+), defined by
Z 0
P (t) =
dx p(t, x),
(1.76a)
1
Z 1
P+ (t) =
dx p(t, x).
(1.76b)
_
+
0
If all particles start in the left well, then
P (0) = 1 ,
P+ (0) = 0.
(1.77)
Whilst the exact dynamics of P± (t) is governed by the FPE (1.68d), the two-state approximation assumes that this dynamics can be approximated by the set of master equations15
Ṗ =
k+ P + k P+ ,
Ṗ+ = k+ P
k P+ .
(1.78)
For a symmetric potential, U (x) = U ( x), forward and backward rates are equal, k+ =
k = k, and in this case, the solution of Eq. (1.78) is given by
1 1
P± (t) = ⌥ e
2 2
2k t
.
(1.79)
For comparison, from the FPE solution (1.72a), we find in the long-time limit
U (x)/2D
p(t, x) ' ps (x) + c1 e
1 (x)
e
1t
,
(1.80)
Due to the symmetry of ps (x), we then have
1
P (t) ' + C1 e
2
15
1t
Note that Eqs. (1.78) conserve the total probability, P (t) + P (t) = 1.
18
(1.81a)
1 1
P± (t) = ⌥ e
2 2
2k t
.
(1.79)
arison, from the FPE solution (1.72a), we find in the long-time limit
p(t, x) ' ps (x) + c1 e
U (x)/2D
1 (x)
e
1t
,
_
+
(1.80)
Solution
he symmetry
of ps (x), we then have
1
P (t) ' + C1 e
2
1t
(1.81a)
hat Eqs. (1.78) conserve the total probability, P (t) + P (t) = 1.
where
C 1 = c1
Z
0
18U (x)/2D
e
1 (x)
,
c1 =
1
⇤
1 (x
) eU (x
)/(2D)
.
(1.81b)
Since Eq. (1.81a) neglects higher-order eigenfunctions, C1 is in general not exactly equal
but usually close to 1/2. But, by comparing the time-dependence of (1.81a) and (1.79), it
is natural to identify
1
k'
=
.
2
2⌧⇤
1
(1.82)
We next discuss, by considering in a slightly di↵erent setting, how one can obtain an
explicit result for the rate k in terms of the parameters of the potential U .
1.4.3
Constant-current solution
Consider a bistable potential as in Eq. (1.75), but now with a particle source at x0 < x < 0
Since Eq. (1.81a) neglects higher-order eigenfunctions, C1 is in general not exactly equal
but usually close to 1/2. But, by comparing the time-dependence of (1.81a) and (1.79), it
is natural to identify
_
1
k'
=
.
2
2⌧⇤
1
(1.82)
We next discuss, by considering in a slightly di↵erent setting, how one can obtain an
explicit result for the rate k in terms of the parameters of the potential U .
source
1.4.3
Constant-current solution
Consider a bistable potential as in Eq. (1.75), but now with a particle source at x0 < x < 0
and a sink16 at x1 > xb = 0. Assuming that particles are injected at x0 at constant flux
j(t, x) ⌘ J = const, the escape rate can be defined by
k :=
J
,
P
(1.83)
with P denoting the probability of being in the left well, as defined in Eq. (1.76a) above.
To compute the rate from Eq. (1.83), we need to find a stationary constant flux solution
pJ (x) of Eq. (1.68d), satisfying pJ (x1 ) = 0 and
J=
(@x U )pJ
D@x pJ
(1.84)
for some constant J. This solution is given by [HTB90]
Z
J U (x)/D x1
pJ (x) = e
dy eU (y)/D ,
D
x
(1.85)
as one can verify by di↵erentiation
(@x U )pJ
D@x pJ =
(@x U )pJ
=
(@x U )pJ
= J.

Z
J U (x)/D x1
D@x
e
dy eU (y)/D
D
x

Z x1
(@x U ) U (x)/D
J
e
dy eU (y)/D
D
x
1
(1.86)
+
sink
Therefore, the inverse rate k
k
1
1
from Eq. (1.83) can be formally expressed as
Z
Z x1
P
1 x1
=
=
dx e U (x)/D
dy eU (y)/D ,
J
D 1
x
and a partial integration yields the equivalent representation
Z x1
Z x
1
k 1=
dx eU (x)/D
dy e U (y)/D .
D 1
1
(1.87)
(1.88)
Assuming a sufficiently steep barrier, the integrals in Eq. (1.88) may be evaluated by
adopting steepest descent approximations near the potential minimum at x and near the
maximum at the barrier position xb . More precisely, taking into account that U 0 (x ) =
U 0 (xb ) = 0, one can replace the potentials in the exponents by the harmonic approximations
1
(x
2⌧b
1
U (y) ' U (x ) +
(y
2⌧
U (x) ' U (x )
x )2 ,
(1.89a)
x )2 ,
(1.89b)
where
⌧ =
U 00 (x0 ) > 0 ,
⌧b = U 00 (xb ) > 0
(1.90)
carry units of time. Inserting (1.89) into (1.88) and replacing the upper integral boundaries
by +1, one thus obtains the so-called Kramers rate [HTB90, GHJM98]
e U/D
k' p
=: kK ,
2⇡ ⌧ ⌧b
U = U (xb )
U (x ).
(1.91)
This result agrees with the well-known empirical Arrhenius law. Note that, because typically D / kB T for thermal noise, binding/unbinding rates depend sensitively on temperature – this is one of the reasons why many organisms tend to function properly only within
a limited temperature range.
Therefore, the inverse rate k
k
1
1
from Eq. (1.83) can be formally expressed as
Z
Z x1
P
1 x1
=
=
dx e U (x)/D
dy eU (y)/D ,
J
D 1
x
and a partial integration yields the equivalent representation
Z x1
Z x
1
k 1=
dx eU (x)/D
dy e U (y)/D .
D 1
1
(1.87)
(1.88)
Assuming a sufficiently steep barrier, the integrals in Eq. (1.88) may be evaluated by
adopting steepest descent approximations near the potential minimum at x and near the
maximum at the barrier position xb . More precisely, taking into account that U 0 (x ) =
U 0 (xb ) = 0, one can replace the potentials in the exponents by the harmonic approximations
1
(x
2⌧b
1
U (y) ' U (x ) +
(y
2⌧
U (x) ' U (x )
x )2 ,
(1.89a)
x )2 ,
(1.89b)
where
⌧ =
U 00 (x0 ) > 0 ,
⌧b = U 00 (xb ) > 0
(1.90)
carry units of time. Inserting (1.89) into (1.88) and replacing the upper integral boundaries
by +1, one thus obtains the so-called Kramers rate [HTB90, GHJM98]
e U/D
k' p
=: kK ,
2⇡ ⌧ ⌧b
U = U (xb )
U (x ).
_(1.91) +
This result agrees with the well-known empirical Arrhenius law. Note that, because typically D / kB T for thermal noise, binding/unbinding rates depend sensitively on temperature – this is one of the reasons why many organisms tend to function properly only within
source
a limited temperature range.
Arrhenius law
sink
k
1
P
1
=
=
J
D
Z
x1
dx e
U (x)/D
1
Z
x1
dy eU (y)/D ,
x
Stochastic resonance
and a partial integration yields the equivalent representation
Z x1
Z x
1
1
U (x)/D
U (y)/D
k
=
dx
e
dy
e
.
Gammaitoni et al.: Stochastic resonance
D 1
1
17
of a bistable string
c resonance in
(1.88)
1. a nonlinear measurement device ,
d On
nce
stochastic resonance
nance in the deep
tially extended
(1.87)
261
261
262
Assuming
a sufficiently
steep
the integrals
in Eq. (1.88)
2. a periodic
signal weaker
thanbarrier,
the threshold
of measurement
device,may be evaluated by
adopting steepest descent approximations near the potential minimum at x and near the
263
3. additional
input
noise,position
uncorrelated
with the
signal oftaking
interest.
maximum
at the
barrier
xb . More
precisely,
into account that U 0 (x ) =
Gammaitoni et al.: Stochastic resonance
U 0 (x267
b ) = 0, one can replace the potentials in the exponents by the harmonic approximations
267
To provide some intuition, assume that a weak periodic signal (frequency ⌦) is detected
escape rate r K . H
268
by
a
particle
that
can
move
move
in
the
bistable
double
well-potential
(1.75).
For
weak
tried to reproduce
, and crisis
270
(including here th
noise,272the particle will remain trapped in one of the
we
will
be
unable
toa synchroni
1 minima and
2
that
274
U (x) motion.
' U (xSimilarly,
)
(xnot xmuch
) , information abouttime
(1.89a)
the conditio
infer 274
the signal from the particle’s
the
2⌧
b
ther D or V. In F
d the dithering effect
274
underlying
signal
can
be
gained
if
the
noise
is
too
strong,
for
in
this
case
the
particle
will
input-output
sync
1
upled systems
274
2
tem (1.89b)
Eqs. (2.1)–(2
stems
274
U (y)
' U
(xminima.
)+
(y
x ) , the noise strength
jump
randomly
back
and
forth
between
the
If,
however,
is
increased from low
obally coupled
2⌧
very large values,
tuned275such that the Kramers escape rate (1.91) is of the order of the driving frequency,
n models
chastic resonance
resonance
onance
ed topics
es
ecular motors
ally driven systems
y
275
275
275
276
277
277
277
278
279
279
281
281
283
where
⌧ =
kK ⇠ ⌦,
U 00 (x0 ) > 0 ,
⌧b = U 00 (xb ) > 0
Eqs. (2.8). In the
comes tightly lock
(1.92)
the noise intensit
quency V is incre
alternate asymme
correlated
positive or negativ
input signal. How
rections within an
of V, the effect o
and the symmetry
restored. Finally,
nism is establishe
In the following
resonance as a
enon, resulting fr
periodic forcing in
(1.93a)
to this purpose is
(1.90)
then it is plausible to expect that the particle’s escape dynamics will be closely
with
the
driving
frequency,
thus(1.89)
exhibiting
SR. and replacing the upper integral boundaries
carry
units
of time.
Inserting
into (1.88)
FIG. 1. Stochastic resonance in a symmetric double well. (a)
Sketch of the double-well potential V(x)5(1/4)bx 4
2(1/2)ax 2 . The minima are located at 6x m , where
x m 5(a/b) 1/2. These are separated by a potential barrier with
the height given by DV5a 2 /(4b). The
barrier top is located at
U/D
x b 50. In the presence of periodic driving, the double-well potential V(x,t)5V(x)2A 0 x cos(Vt) is tilted back K
and forth,
b
thereby raising and lowering successively the potential barriers
of the right and the left well, respectively, in an antisymmetric
by +1, one thus obtains the so-called Kramers rate [HTB90, GHJM98]
1.5.1
Generic model
e
To illustrate SR morek quantitatively,
periodically
' p
=:consider
k , the U
= U (xb ) driven
U (x SDE
).
(1.91)
2⇡ ⌧ ⌧
p
dunkel@math.mit.edu
dX(t) =
@x U dt + A cos(⌦t) dt +
2D ⇤ dB(t),
kK ⇠ ⌦,
(1.92)
then it is plausible to expect that the particle’s escape dynamics will be closely correlated
with the driving frequency, thus exhibiting SR.
1.5
Stochastic resonance
1.5.1 typically
Generic impairs
model signal transduction, but under certain conditions an
Noise
of
randomness
actuallyconsider
help the
to periodically
enhance driven
weak SDE
signals [GHJM98]. Th
To illustrate
SR moremay
quantitatively,
p
phenomenon isdX(t)
known
as
stochastic
resonance
(SR). Whilst SR
was originall
= @x U dt + A cos(⌦t) dt + 2D ⇤ dB(t),
(1.93a)
a possible explanation for periodically recurring climate cycles [NN81, BPS
where A is the signal amplitude
and
ments
suggest [FSGB+ 02]
that some organisms like juvenile paddle-fish mig
a 2foraging
b 4
to enhance signal detection
while
for food.
U (x) =
x + x
(1.93b)
2
4
The occurrence of SR requires three main ‘ingredients’
p
a symmetric double-well potential with minima at ±x⇤ = ±
U = a2 /(4b). Introducing rescaled variables
a/b and barrier height
20
x0 = x/x⇤ ,
t0 = at ,
A0 = A/(ax⇤ ) ,
D0 = D/(ax2⇤ ) ,
⌦0 = ⌦/a.
and dropping primes. we can rewrite (1.93a) in the dimensionless form
p
3
dX(t) = (x x ) dt + A cos(⌦t) dt + 2D ⇤ dB(t),
with a rescaled barrier height
@t p =
(1.93c)
U = 1/4. The associated FPE reads
@x {[ (@x U ) + A cos(⌦t)]p
D@x p}.
(1.94)
For our subsequent discussion, it is useful to rearrange terms on the rhs. as
@t p = @x [(@x U )p + D@x p]
17
A cos(⌦t)@x p.
(1.95)
That is, the input-output relationship between the input signal and the observable must be nonlinear
dX(t) = (x
with a rescaled barrier height
3
x ) dt + A cos(⌦t) dt +
p
2D ⇤ dB(t),
U = 1/4. The associated FPE reads
Perturbation
theory
@ {[ (@ U ) + A cos(⌦t)]p D@ p}.
@t p =
(1.93c)
x
x
x
(1.94)
For our subsequent discussion, it is useful to rearrange terms on the rhs. as
@t p = @x [(@x U )p + D@x p]
17
A cos(⌦t)@x p.
(1.95)
That is, the input-output relationship between the input signal and the observable must be nonlinear
To solve Eq. (1.95) perturbatively, we insert the series ansatz
p(t, x) =
21
1
X
An pn (t, x),
(1.96)
n=0
which gives
1
X
n=0
An @t pn =
1
X
An @x [(@x U )pn + D@x pn ]
An+1 cos(⌦t)@x pn
(1.97)
n=0
Focussing on the liner response regime, corresponding to powers A0 and A1 , we find
@t p0 = @x [(@x U )p0 + D@x p0 ]
@t p1 = @x [(@x U )p1 + D@x p1 ]
cos(⌦t)@x p0
(1.98a)
(1.98b)
Equation (1.98a) is just an ordinary time-independent FPE, and we know its stationary
solution is just the Boltzmann distribution
Z
e U (x)/D
p0 (x) =
,
Z0 = dx e U (x)/D
(1.99)
Z0
To obtain a formal solution to Eq. (1.98b), we make use of the following ansatz
dX(t) = (x
with a rescaled barrier height
3
x ) dt + A cos(⌦t) dt +
p
2D ⇤ dB(t),
U = 1/4. The associated FPE reads
Perturbation
theory
@ {[ (@ U ) + A cos(⌦t)]p D@ p}.
@t p =
(1.93c)
x
x
x
(1.94)
For our subsequent discussion, it is useful to rearrange terms on the rhs. as
@t p = @x [(@x U )p + D@x p]
17
A cos(⌦t)@x p.
(1.95)
That is, the input-output relationship between the input signal and the observable must be nonlinear
To solve Eq. (1.95) perturbatively, we insert the series ansatz
p(t, x) =
21
1
X
An pn (t, x),
(1.96)
n=0
which gives
1
X
n=0
An @t pn =
1
X
An @x [(@x U )pn + D@x pn ]
An+1 cos(⌦t)@x pn
(1.97)
n=0
Focussing on the liner response regime, corresponding to powers A0 and A1 , we find
@t p0 = @x [(@x U )p0 + D@x p0 ]
@t p1 = @x [(@x U )p1 + D@x p1 ]
cos(⌦t)@x p0
(1.98a)
(1.98b)
Equation (1.98a) is just an ordinary time-independent FPE, and we know its stationary
solution is just the Boltzmann distribution
Z
e U (x)/D
p0 (x) =
,
Z0 = dx e U (x)/D
(1.99)
Z0
To obtain a formal solution to Eq. (1.98b), we make use of the following ansatz
dX(t) = (x
with a rescaled barrier height
3
x ) dt + A cos(⌦t) dt +
p
2D ⇤ dB(t),
U = 1/4. The associated FPE reads
Perturbation
theory
@ {[ (@ U ) + A cos(⌦t)]p D@ p}.
@t p =
(1.93c)
x
x
x
(1.94)
For our subsequent discussion, it is useful to rearrange terms on the rhs. as
@t p = @x [(@x U )p + D@x p]
17
A cos(⌦t)@x p.
(1.95)
That is, the input-output relationship between the input signal and the observable must be nonlinear
To solve Eq. (1.95) perturbatively, we insert the series ansatz
p(t, x) =
21
1
X
An pn (t, x),
(1.96)
n=0
which gives
1
X
n=0
An @t pn =
1
X
An @x [(@x U )pn + D@x pn ]
An+1 cos(⌦t)@x pn
(1.97)
n=0
Focussing on the liner response regime, corresponding to powers A0 and A1 , we find
@t p0 = @x [(@x U )p0 + D@x p0 ]
@t p1 = @x [(@x U )p1 + D@x p1 ]
cos(⌦t)@x p0
(1.98a)
(1.98b)
Equation (1.98a) is just an ordinary time-independent FPE, and we know its stationary
solution is just the Boltzmann distribution
Z
e U (x)/D
p0 (x) =
,
Z0 = dx e U (x)/D
(1.99)
Z0
To obtain a formal solution to Eq. (1.98b), we make use of the following ansatz
n=0
n=0
@t p0 = @x [(@x U )p0 + D@x p0 ]
@t p1 = @x [(@x U )p1 + D@x p1 ]
(1.98a)
(1.98b)
First-order
correction
Equation (1.98a) is just an ordinary time-independent FPE, and we know its stationary
cos(⌦t)@x0p0
Focussing on the liner response regime, corresponding to powers A and A1 , we find
@t p0the=Boltzmann
@x [(@x Udistribution
)p0 + D@x p0 ]
solution is just
Zcos(⌦t)@x p0
@t p1 = @x [(@xeU )p
U (x)/D
1 + D@x p1 ]
p0 (x) =
,
Z0
Z0 =
dx e
U (x)/D
(1.98a)
(1.98b)
(1.99)
Equation (1.98a) is just an ordinary time-independent FPE, and we know its stationary
To the
obtain
a formal solution
to Eq. (1.98b), we make use of the following ansatz
solution is just
Boltzmann
distribution
1
Z
X
U (x)/D
e p1 (t, x) = e U (x)/(2D)
a (t) m (x),
(1.100)
p0 (x) =
,
Z0 = m=1dx1me U (x)/D
(1.99)
Z0
where
m (x)
are the eigenfunctions of the unperturbed e↵ective Hamiltonian, cf. Eq. (1.71),
To obtain a formal solution to Eq. (1.98b), we make use of the following ansatz
H0 =
p1 (t, x) = e
1
(@x U )2
X4D
D@x2 +1
U (x)/(2D)
Inserting (1.100) into Eq. (1.98b) gives
a1m (t)
1 2
@ U.
2 x
m (x),
(1.101)
(1.100)
m=1
where
m (x)
1
X
1
X
are the eigenfunctions
e↵ective
Hamiltonian,
cf. Eq.(1.102)
(1.71),
ȧ1m m of
= the unperturbed
cos(⌦t)
eU (x)/(2D)
@ x p0 .
m a1m m
m=1
m=1
1
1
2
Multiplying thisHequation
by x2 +
1 to +1 while exploiting
the
D@
(@xintegrating
U )2
@from
(1.101)
n (x), and
0 =
x U.
orthonormality of the system { m4D
}, we obtain the2 coupled ODEs
Inserting (1.100) into Eq. (1.98b) gives
ȧ1m =
1
1
X
X
with
‘transition matrix’
elements
m=1
ȧ1m
m
=
m=1
Mm0 cos(⌦t),
cos(⌦t) eU (x)/(2D) @x p0 .
Zm
= dx m eU (x)/(2D) @x p0 .
m a1m
Mm0
m a1m
(1.103)
(1.102)
(1.104)
Multiplying this equation by n (x), and integrating from 1 to +1 while exploiting the
orthonormality of the system { }, we obtain the22coupled ODEs
n=0
n=0
@t p0 = @x [(@x U )p0 + D@x p0 ]
@t p1 = @x [(@x U )p1 + D@x p1 ]
(1.98a)
(1.98b)
First-order
correction
Equation (1.98a) is just an ordinary time-independent FPE, and we know its stationary
cos(⌦t)@x0p0
Focussing on the liner response regime, corresponding to powers A and A1 , we find
@t p0the=Boltzmann
@x [(@x Udistribution
)p0 + D@x p0 ]
solution is just
Zcos(⌦t)@x p0
@t p1 = @x [(@xeU )p
U (x)/D
1 + D@x p1 ]
p0 (x) =
,
Z0
Z0 =
dx e
U (x)/D
(1.98a)
(1.98b)
(1.99)
Equation (1.98a) is just an ordinary time-independent FPE, and we know its stationary
To the
obtain
a formal solution
to Eq. (1.98b), we make use of the following ansatz
solution is just
Boltzmann
distribution
1
Z
X
U (x)/D
e p1 (t, x) = e U (x)/(2D)
a (t) m (x),
(1.100)
p0 (x) =
,
Z0 = m=1dx1me U (x)/D
(1.99)
Z0
where
m (x)
are the eigenfunctions of the unperturbed e↵ective Hamiltonian, cf. Eq. (1.71),
To obtain a formal solution to Eq. (1.98b), we make use of the following ansatz
H0 =
p1 (t, x) = e
1
(@x U )2
X4D
D@x2 +1
U (x)/(2D)
Inserting (1.100) into Eq. (1.98b) gives
a1m (t)
1 2
@ U.
2 x
m (x),
(1.101)
(1.100)
m=1
where
m (x)
1
X
1
X
are the eigenfunctions
e↵ective
Hamiltonian,
cf. Eq.(1.102)
(1.71),
ȧ1m m of
= the unperturbed
cos(⌦t)
eU (x)/(2D)
@ x p0 .
m a1m m
m=1
m=1
1
1
2
Multiplying thisHequation
by x2 +
1 to +1 while exploiting
the
D@
(@xintegrating
U )2
@from
(1.101)
n (x), and
0 =
x U.
orthonormality of the system { m4D
}, we obtain the2 coupled ODEs
Inserting (1.100) into Eq. (1.98b) gives
ȧ1m =
1
1
X
X
with
‘transition matrix’
elements
m=1
ȧ1m
m
=
m=1
Mm0 cos(⌦t),
cos(⌦t) eU (x)/(2D) @x p0 .
Zm
= dx m eU (x)/(2D) @x p0 .
m a1m
Mm0
m a1m
(1.103)
(1.102)
(1.104)
Multiplying this equation by n (x), and integrating from 1 to +1 while exploiting the
orthonormality of the system { }, we obtain the22coupled ODEs
1
X
ȧ1m
m
1
X
=
m a1m
cos(⌦t) eU (x)/(2D) @x p0 .
m
(1.102)
First-order
correction
Multiplying this equation by (x), and integrating from 1 to +1 while exploiting the
m=1
m=1
n
orthonormality of the system {
m },
ȧ1m =
we obtain the coupled ODEs
m a1m
Mm0 cos(⌦t),
(1.103)
eU (x)/(2D) @x p0 .
(1.104)
with ‘transition matrix’ elements
Z
Mm0 =
dx
m
22
The asymptotic solution of Eq. (1.103) reads

⌦
a1m (t) = Mm0 2
sin(⌦t) +
2
+
⌦
m
m
2
m
+ ⌦2
cos(⌦t) .
(1.105)
Note that, because @x p0 is an antisymmetric function, the matrix elements Mm0 vanish18
for even values m = 0, 2, 4, . . ., so that only the contributions from odd values m = 1, 3, 5 . . .
are asymptotically relevant.
Focussing on the leading order contribution, m = 1, and noting that p0 (x) = p0 ( x),
we can estimate the position mean value
Z
E[X(t)] = dx p(t, x) x
(1.106)
from
E[X(t)] ' A
' A
=
Z
Z
dx p1 (t, x) x
dx e

AM10
U (x)/(2D)
⌦
a11 (t)
1 (x) x
sin(⌦t) +
1
cos(⌦t)
Z
dx e
U (x)/(2D)
1 (x) x
a1m (t) =
Mm0

⌦
sin(⌦t) +
2 + ⌦2
m
m
cos(⌦t) .
Linear response
2
m
+ ⌦2
(1.105)
Note that, because @x p0 is an antisymmetric function, the matrix elements Mm0 vanish18
for even values m = 0, 2, 4, . . ., so that only the contributions from odd values m = 1, 3, 5 . . .
are asymptotically relevant.
Focussing on the leading order contribution, m = 1, and noting that p0 (x) = p0 ( x),
we can estimate the position mean value
Z
E[X(t)] = dx p(t, x) x
(1.106)
from
E[X(t)] ' A
' A
=
Z
dx p1 (t, x) x
Z
dx e

AM10
U (x)/(2D)
a11 (t)
1 (x) x
⌦
sin(⌦t) +
2
2
+
⌦
1
1
2
1
+
⌦2
cos(⌦t)
Z
dx e
U (x)/(2D)
1 (x) x
Using 1 = 2kK , where kK is the Kramers rate from Eq. (1.91), we can rewrite this more
compactly as
E[X(t)] = X cos(⌦t
')
(1.107a)
with phase shift
✓
⌦
' = arctan
2kK
◆
(1.107b)
and amplitude
X=
M10
A
2
(4kK
+ ⌦2 )1/2
Z
dx e
U (x)/(2D)
1 (x) x.
(1.107c)
The amplitude X depends on the noise strength D through kK , through the integral factor
and also through the matrix element
Z
M10 =
dx 1 eU (x)/(2D) @x p0 .
(1.108)
with phase shift
✓
◆
Linear
response
⌦
' = arctan
2kK
(1.107b)
and amplitude
X=
M10
A
2
(4kK
+ ⌦2 )1/2
Z
dx e
U (x)/(2D)
1 (x) x.
(1.107c)
The amplitude X depends on the noise strength D through kK , through the integral factor
and also through the matrix element
Z
M10 =
dx 1 eU (x)/(2D) @x p0 .
(1.108)
18
potential
(x) is
symmetric
and,determine
therefore, the
Hamiltonian of
commutes
parity
To The
compute
X, Uone
first
needs to
the e↵ective
eigenfunction
H0 as with
defined
in
1
operator, implying that the eigenfunctions 2k are symmetric under x 7! x, whereas eigenfunctions
Eq. (1.101).
For the quartic double-well potential (1.93b), this cannot be done analytically
2k+1 are antisymmetric under this map.
but there exist standard techniques
p (e.g., Ritz method) for approximating 1 by functions
that are orthogonal to 0 = p0 /Z0 . Depending on the method employed, analytical
23
estimates for X may vary quantitatively but always show a non-monotonic dependence on
the noise strength D for fixed potential-parameters (a, b). As discussed in [GHJM98], a
reasonably accurate estimate for X is given by
✓
◆1/2
2
Aa
4kK
X'
,
(1.109)
2
Db 4kK
+ ⌦2
which exhibits a maximum for a critical value D⇤ determined by
✓
◆
U
2
4kK
= ⌦2
1 .
D⇤
(1.110)
That is, the value D⇤ corresponds to the optimal noise strength, for which the mean
value E[X(t)] shows maximal response to the underlying periodic signal – hence the name
‘stochastic resonance’ (SR).
1.5.2
Master equation approach
That is, the value D⇤ corresponds to the optimal noise strength, for which the mean
value E[X(t)] shows maximal response to the underlying periodic signal – hence the name
‘stochastic resonance’ (SR).
1.5.2
Master equation approach
Similar to the case of the escape problem, one can obtain an alternative description of
SR by projecting the full FPE dynamics onto a simpler set of master equations for the
probabilities P± (t) of the coarse-grained particle-states ‘left well’ ( ) and ‘right well’ (+),
as defined by Eq. (1.76). This approach leads to the following two-state master equations
with time-dependent rates
Ṗ (t) =
Ṗ+ (t) =
k+ (t) P + k (t) P+ ,
k+ (t) P
k (t) P+ .
The general solution of this pair of ODEs is given by [GHJM98]

Z t
k± (s)
P± (t) = g(t) P± (t0 ) +
ds
g(s)
t0
(1.111a)
(1.111b)
(1.112a)
where
g(t) = exp
⇢ Z
t
ds [k+ (s) + k (s)] .
(1.112b)
t0
To discuss SR within this framework, it is plausible to postulate time-dependent Arrheniustype rates,

Ax⇤
k± (t) = kK exp ±
cos(⌦t) .
(1.113)
D
24
Adopting these rates and considering the asymptotic limit t0 ! 1, one can Taylorexpand the exact solution (1.112) for Ax⇤ ⌧ D to obtain
"
#
✓
◆2
Ax⇤
Ax⇤
P± (t) = kK 1 ±
cos(⌦t) +
cos2 (⌦t) ± . . . .
(1.114)
D
D
These approximations are valid for slow driving (adiabatic regime), and they allow us to
compute expectation values to leading order in Ax⇤ /D. In particular, one then finds for
the mean position the asymptotic linear response result [GHJM98]
E[X(t)] = X cos(⌦t
')
(1.115a)
where
X=
Ax2⇤
D
✓
2
4kK
2
4kK
+ ⌦2
◆1/2
,
' = arctan
✓
⌦
2kK
◆
(1.115b)
with kK denoting Kramers rate as defined in Eq. (1.91). Note that Eqs. (1.115) are consistent with our earlier result (1.107).
1.6
Brownian motors
Many biophysical processes, from muscular contractions to self-propulsion of microorganisms or intracellular transport, rely on biological motors. These are, roughly speaking,