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Switching-path distribution in multidimensional systems.

13 Nov 2008-Physical Review E (American Physical Society)-Vol. 78, Iss: 5, pp 051109-051109

TL;DR: The experiment provides the first demonstration of the lack of time-reversal symmetry in switching of systems far from thermal equilibrium and opens the possibility of efficient control of the switching probability based on the measured narrow path distribution.
Abstract: We explore the distribution of paths followed in fluctuation-induced switching between coexisting stable states. We introduce a quantitative characteristic of the path distribution in phase space that does not require a priori knowledge of system dynamics. The theory of the distribution is developed and its direct measurement is performed in a micromechanical oscillator driven into parametric resonance. The experimental and theoretical results on the shape and position of the distribution are in excellent agreement, with no adjustable parameters. In addition, the experiment provides the first demonstration of the lack of time-reversal symmetry in switching of systems far from thermal equilibrium. The results open the possibility of efficient control of the switching probability based on the measured narrow path distribution.

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Switching-path distribution in multidimensional systems
H. B. Chan,
1,
*
M. I. Dykman,
2,
and C. Stambaugh
1
1
Department of Physics, University of Florida, Gainesville, Florida 32611, USA
2
Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48823, USA
Received 27 June 2008; published 13 November 2008
We explore the distribution of paths followed in fluctuation-induced switching between coexisting stable
states. We introduce a quantitative characteristic of the path distribution in phase space that does not require a
priori knowledge of system dynamics. The theory of the distribution is developed and its direct measurement
is performed in a micromechanical oscillator driven into parametric resonance. The experimental and theoret-
ical results on the shape and position of the distribution are in excellent agreement, with no adjustable
parameters. In addition, the experiment provides the first demonstration of the lack of time-reversal symmetry
in switching of systems far from thermal equilibrium. The results open the possibility of efficient control of the
switching probability based on the measured narrow path distribution.
DOI: 10.1103/PhysRevE.78.051109 PACS numbers: 05.40.Ca, 05.45.a, 89.75.Da
I. INTRODUCTION
Fluctuation phenomena in systems with multiple stable
states have long been a topic of intense research interest.
When the fluctuation intensity is small, for most of the time
the system fluctuates about one of the stable states. Switch-
ing between the states requires a large fluctuation that would
allow the system to overcome the activation barrier. Such
large fluctuations are rare. However, they lead to large
changes in the system behavior. Fluctuational switching be-
tween coexisting states plays a crucial role in a variety of
systems and phenomena including nanomagnets 1, Joseph-
son junctions 2, protein folding 3, and chemical reactions.
A detailed theory of switching rates was first developed
by Kramers 4. The analysis referred to systems close to
thermal equilibrium, where the switching rate is determined
by the free energy barrier between the states. The concept of
this barrier is well-understood and the barrier height has been
found for many models. However, in recent years much at-
tention has been given to switching in systems far from ther-
mal equilibrium. Examples include electrons 5 and atoms
6,7 in modulated traps, rf-driven Josephson junctions 8,9,
and nano- and micromechanical resonators 1012. Apart
from fundamental interest, switching in modulated systems
is important for many applications, quantum measurements
being an example 8,9,13,14. Nonequilibrium systems gen-
erally lack detailed balance, and the switching rates may not
be found by a simple extension of the Kramers approach.
A basic, although somewhat counterintuitive, physical
feature of large infrequent fluctuations leading to switching
is that, in such fluctuations, the system is most likely to move
along a certain path in its phase space. This path is known as
the most probable switching path MPSP. For low fluctua-
tion intensity, the MPSP is obtained by solving a variational
problem. This problem also determines the switching activa-
tion barrier 1525. Despite its central role for the under-
standing of fluctuation-induced switching and switching
rates, the idea of the MPSP has not been tested experimen-
tally in multivariable systems. The question of how the paths
followed in switching are distributed in phase space has not
been asked either.
Perhaps closest to addressing the above issues was the
experiment on dropout events in a semiconductor laser with
optical feedback 26. In this experiment the switching-path
distribution in space and time was measured and calculated.
However, the system was characterized by only one dynamic
variable, and thus all paths lie on one line in phase space. In
another effort, electronic circuit simulations 27 compare
the distribution of fluctuational paths to and relaxational
paths from a certain point within one basin of attraction; the
data refer to the situation where switching does not occur.
The methods 26,27 do not apply to the switching-path dis-
tribution in multivariable systems, as explained below.
In the present paper we introduce the concept of
switching-path distribution in phase space and the quantity
that describes this distribution, calculate this quantity, and
report the direct observation of the tube of switching paths. A
brief account of the results was published in Ref. 28. The
experimental and theoretical results on the shape and posi-
tion of the path distribution are in excellent agreement, with
no adjustable parameters. The results open the possibility of
efficient control of the switching probability based on the
measured narrow path distribution.
In Sec. II we provide the qualitative picture of switching
and give a preview of the central theoretical and experimen-
tal results. Section III presents a theory of the switching-path
distribution in the basins of attraction to the initially occu-
pied and initially empty stable states as well as some simple
results for systems with detailed balance. In Sec. IV the sys-
tem used in the experiment, a micromechanical torsional os-
cillator, is described and quantitatively characterized. Section
V presents the results of the experimental studies of the
switching-path distribution for the micromechanical oscilla-
tor, with the coexisting stable states being the states of para-
metrically excited nonlinear vibrations. Generic features of
the distribution are discussed, and the lack of time-reversal
symmetry in switching of systems far from thermal equilib-
rium is revealed. Section VI contains concluding remarks.
*
hochan@phys.ufl.edu
dykman@pa.msu.edu
PHYSICAL REVIEW E 78, 051109 2008
1539-3755/2008/785/05110910 ©2008 The American Physical Society051109-1

II. QUALITATIVE PICTURE AND PREVIEW OF
THE RESULTS
We consider a bistable system with several dynamical
variables q =q
1
,...,q
N
. The stable states A
1
and A
2
are lo-
cated at q
A
1
and q
A
2
, respectively. A sketch of the phase
portrait for the case of two variables is shown in Fig. 1. For
low fluctuation intensity, the physical picture of switching is
as follows. The system prepared initially in the basin of at-
traction of state A
1
, for example, will approach q
A
1
over the
characteristic relaxation time t
r
and will then fluctuate about
q
A
1
. We assume the fluctuation intensity to be small. This
means that the typical amplitude of fluctuations about the
attractor the characteristic diffusion length l
D
is small com-
pared to the minimal distance q between the attractors and
from the attractors to the saddle point q
S
.
Even though fluctuations are small on average, occasion-
ally there occur large fluctuations, including those leading to
switching between the states. The switching rate W
12
from
state A
1
to A
2
is much less than the reciprocal relaxation time
t
r
−1
, that is, the system fluctuates about A
1
for a long time, on
the scale of t
r
, before a transition to A
2
occurs. In the transi-
tion the system most likely moves first from the vicinity of
q
A
1
to the vicinity of q
S
. Its trajectory is expected to be close
to the one for which the probability of the appropriate large
rare fluctuation is maximal. The corresponding trajectory is
illustrated in Fig. 1. From the vicinity of q
S
the system
moves to state A
2
close to the deterministic fluctuation-free
trajectory. These two trajectories comprise the MPSP.
For brevity, we call the sections of the MPSP from q
A
1
to
q
S
and from q
S
to q
A
2
the uphill and downhill trajectories,
respectively. The terms would literally apply to a Brownian
particle in a potential well, with A
1,2
corresponding to the
minima of the potential and S to the barrier top.
We characterize the switching-path distribution by the
probability density for the system to pass through a point q
on its way from A
1
to A
2
,
p
12
q,t =
2
dq
f
q
f
,t
f
;q,tq
0
,t
0
. 1
Here, the integrand is the three-time conditional probability
density for the system to be at points q
f
and q at times t
f
and
t, respectively, given that it was at q
0
at time t
0
. The point q
0
lies within distance l
D
of q
A
1
and is otherwise arbitrary.
Integration with respect to q
f
goes over the range
2
of
small fluctuations about q
A
2
; the typical linear size of this
range is l
D
.
We call p
12
q ,t the switching-path distribution. Of ut-
most interest is to study this distribution in the time range
W
12
−1
,W
21
−1
t
f
t,t t
0
t
˜
r
. 2
Here, t
˜
r
is the Suzuki time 29. It differs from t
r
by a loga-
rithmic factor logq/ l
D
. This factor arises because of the
motion slowing down near the saddle point. The time t
˜
r
is
much smaller than the reciprocal switching rates, and the
smaller the fluctuation intensity the stronger the difference
because the dependence of W
ij
on the fluctuation intensity is
of the activation type. If the noise causing fluctuations has a
finite correlation time,t
˜
r
is the maximum of the Suzuki time
and the noise correlation time.
For tt
0
t
r
, by time t the system has already “forgotten”
the initial position q
0
. Therefore the distribution
q
f
,t
f
;q ,t q
0
,t
0
, and thus p
12
, are independent of q
0
, t
0
.
On the other hand, if the system is on its way from A
1
to A
2
and at time t is in a state q far from the attractors, it will most
likely reach the vicinity of A
2
over time t
˜
r
and will then
fluctuate about q
A
2
. This will happen well before the time t
f
at which the system is observed near A
2
, and therefore p
12
is
independent of t
f
.
It is clear from the above arguments that, in the time
range 2, the distribution p
12
q ,t for q far from the attrac-
tors is formed by switching trajectories emanating from the
vicinity of A
1
. It gives the probability density for these tra-
jectories to pass through a given point q at time t. In other
terms, the distribution p
12
q ,t is formed by the probability
current from A
1
to A
2
and is determined by the current den-
sity.
A. Shape of the switching-path distribution
We show in Sec. III that p
12
q ,t peaks on the MPSP. The
peak is Gaussian transverse to the MPSP for q q
A
1,2
, q
q
S
l
D
,
p
12
q,t = W
12
v
−1
Z
−1
exp
1
2
Q
ˆ
, 3
where
and
are coordinates along and transverse to the
MPSP, and
v
is the velocity along the MPSP. The matrix
elements of matrix Q
ˆ
=Q
ˆ
are l
D
−2
, and Z
=关共2
N−1
/ det Q
ˆ
1/2
. It follows from Eq. 3 that the overall
probability flux along the MPSP is equal to the switching
rate,
d
p
12
q,t
v
= W
12
.
−2 −1 0 1
2
−1.5
0
1
.5
q
1
q
2
S
A
1
A
2
FIG. 1. Color online Phase portrait of a two-variable system
with two stable states A
1
and A
2
. The saddle point S lies on the
separatrix that separates the corresponding basins of attraction. The
thin solid lines show the downhill deterministic trajectories from
the saddle to the attractors. A portion of the separatrix near the
saddle point is shown as the dashed line. The thick solid line shows
the most probable trajectory that the system follows in a fluctuation
from A
1
to the saddle. The MPSP from A
1
to A
2
is comprised by this
uphill trajectory and the downhill trajectory from S to A
2
. The plot
refers to the system studied experimentally, see Sec. V.
CHAN, DYKMAN, AND STAMBAUGH PHYSICAL REVIEW E 78, 051109 2008
051109-2

We have observed a narrow peak of the switching-path
distribution in experiment. The results are shown in Fig. 2.
They were obtained using a microelectromechanical tor-
sional oscillator described in Sec. IV. The path distribution
displays a sharp ridge. We demonstrate that the cross-section
of the ridge has a Gaussian shape. As seen from Fig. 2, the
maximum of the ridge lies on the MPSP which was calcu-
lated for the studied system.
Equation 3 is written for a generally nonequilibrium sys-
tem, but the system is assumed to be stationary. In the ne-
glect of fluctuations its motion is described by equations with
time-independent coefficients. In this case p
12
q ,t is inde-
pendent of time t. A different situation may occur in periodi-
cally modulated systems. In such systems, attractors are pe-
riodic functions of time. If the typical relaxation time is
smaller than or of the order of the modulation period, the
MPSPs are well-synchronized and periodically repeat in
time. Then p
12
q ,t is also a periodic function of time. We
will not consider this case in the present paper.
B. Comparison with the prehistory distribution
The distribution of fluctuational paths was studied earlier
in the context of the “prehistory problem” 30. In this for-
mulation one is interested in the paths to a certain state q
f
that is far from the initially occupied attractor. The distribu-
tion of these paths p
h
is given by the probability density to
have passed a point q at time t given that the system is found
at q
f
at a later time t
f
whereas initially at time t
0
it was at
point q
0
near attractor A
1
,
p
h
q,tq
f
,t
f
=
q
f
,t
f
;q,tq
0
,t
0
q
f
,t
f
q
0
,t
0
. 4
The prehistory distribution 4 and its generalizations
were analyzed in a number of papers 26,27,3133. How-
ever, the problem of paths that lead to switching between the
states was addressed only for a stationary system with one
dynamical variable 26. In this case, the system must pass
through all the intermediate points between the two states
during a switch. For switching in systems with more than
one dynamical variable, the formulation 26,27,31 no longer
applies because it cannot be known in advance through what
points the system will pass. The aforementioned formulation
does not work even for one-variable periodically modulated
systems, since the distribution 4 depends not only on the
position of point q
f
, but also on the time t
f
when this point is
reached 34.
In contrast, the distribution p
12
q ,t is defined in such a
way that it is independent of the final point q
f
and of the time
t
f
of reaching it. The definition does not impose any con-
straint on paths except that they lead to switching between
the attractors. Therefore the introduction of the function
p
12
q ,t is essential in studying the switching-path distribu-
tion for multivariable systems.
III. THEORY OF THE SWITCHING-PATH DISTRIBUTION
A. Model of a fluctuating system
We derive Eq. 2 for a system described by the Langevin
equation of motion
q
˙
= Kq + ft, f
n
tf
m
t
兲典 =2D
nm
t t
. 5
Here, vector K determines the dynamics in the absence of
noise; K= 0 at the stable state positions q
A
1
, q
A
2
and at the
saddle point q
S
. We assume that q
S
lies on a smooth hyper-
surface that separates the basins of attraction of states A
1
and
A
2
, cf. Fig. 1. The function ft in Eq. 5 is white Gaussian
noise; the results can be also extended to colored noise. The
noise intensity D is assumed small. The dependence of the
switching rates W
nm
on D is given by the activation law,
log W
nm
D
−1
1521. This is also the case for noise-driven
extended systems, cf. Refs. 3537 and papers cited therein.
There exists extensive literature on numerical calculations of
the switching rate and switching paths, cf. Refs. 3842 and
papers cited therein.
In the model 5, the characteristic relaxation time t
r
and
the characteristic diffusion length l
D
are
t
r
= max
k
Re
k
−1
, l
D
= Dt
r
1/2
, 6
where
k
are the eigenvalues of the matrix
K
m
/
q
n
calcu-
lated at q
A
1
, q
A
2
, and q
S
.
For a white-noise driven system 5, the three-time prob-
ability distribution
q
f
,t
f
;q ,t q
0
,t
0
in Eq. 1 can be writ-
ten as a product of two-time transition probability densities,
q
f
,t
f
;q,tq
0
,t
0
=
q
f
,t
f
q,t
q,tq
0
,t
0
, 7
which simplifies further analysis. The analysis is done sepa-
rately for the cases where the observation point q lies within
the attraction basins of the initially empty attractor A
2
and
the initially occupied attractor A
1
.
B. switching-path distribution in the initially unoccupied basin
of attraction
We start with the case where the observation point q lies
in the basin of attraction of the initially empty state A
2
far
10
0
−10
5
0
−5
A
2
A
1
0
100
200
X (mrad)
(b)
(a)
P
12
(rad
−2
)
Y (mrad)
FIG. 2. Color兲共a Switching-path distribution in a parametri-
cally driven microelectromechanical oscillator. The probability dis-
tribution p
12
X ,Y is measured for switching out of state A
1
into
state A
2
. b The peak locations of the distribution are plotted as
black circles and the theoretical most probable switching path is
indicated by the red line. All trajectories originate from within the
green circle in the vicinity of A
1
and later arrive at the green circle
around A
2
. The portion of the distribution outside the blue lines is
omitted.
SWITCHING-PATH DISTRIBUTION IN PHYSICAL REVIEW E 78, 051109 2008
051109-3

from the stationary states, q q
S
, q q
A
1,2
l
D
. For weak
noise intensity, the system found at such q will most likely
approach q
A
2
over time t
r
moving close to the noise-free
trajectory q
˙
=K and will then fluctuate about q
A
2
. Therefore,
for q
f
not far from the attractor A
2
, i.e., q
f
q
A
2
l
D
,we
have
q
f
,t
f
q ,t兲⬇
2
q
f
. Here,
2
q
f
is the stationary
probability distribution in the attraction basin of A
2
in the
neglect of A
2
A
1
switching. In its central part it has the
form of a normalized Gaussian peak centered at q
A
2
, with
typical size l
D
. Then, from Eq. 1
p
12
q,t =
q,tq
0
,t
0
.
The analysis of the transition probability density
q ,t q
0
,t
0
in this expression is simplified by two observa-
tions. First, for time t in the range W
12
−1
t t
0
t
r
˜
, there is a
probability current from attractor A
1
to A
2
. This current gives
the switching rate W
12
, as found by Kramers 4. The current
density far from A
2
, i.e., for q q
A
2
l
D
, is independent of
time and is determined by the stationary Fokker-Planck
equation
q
K + D
q
2
q,tq
0
,t
0
=0. 8
The second observation is that, for both white and colored
Gaussian noise, in switching the system is most likely to go
close to the saddle point 20,43. Having passed through the
region near the saddle point the system moves close to the
deterministic downhill trajectory from S to A
2
, cf. Fig. 1.
This trajectory is described by equation q
˙
=K and gives the
MPSP in the basin of attraction of A
2
. We are interested in
finding
q ,t q
0
,t
0
for q close to this trajectory. The broad-
ening of the distribution is due to diffusion, which should
generally make it Gaussian in the transverse direction 44.
We parametrize the deterministic section of the MPSP by
its length
counted off from q
S
and introduce a unit vector
ˆ
along the vector K on the MPSP and N 1 vectors
perpendicular to it. The velocity on the MPSP is
v
v
=K
,
=0. Of interest for our analysis are the values of
of the order of the width of the path distribution trans-
verse to the MPSP, which is given by the diffusion length,
i.e.,
l
D
. We assume
to be small compared to the
radius of curvature d
ˆ
/ d
−1
.
Equation 8 can be solved near the MPSP by changing to
variables
,
, expanding K to first order in
, and replac-
ing
q
2
2
. One then obtains for
q ,t q
0
,t
0
= p
12
q ,t ex-
pression 8, with matrix Q
ˆ
given by the equation
v
dQ
ˆ
d
+
ˆ
Q
ˆ
+ Q
ˆ
ˆ
+2Q
ˆ
2
D =0. 9
Here,
ˆ
=
K
/
, with the derivatives evaluated for
=0; the subscripts
,
=1,...,N 1 enumerate the compo-
nents of
and the transverse components of K in the co-
moving frame. Equation 9 can be reduced to a linear equa-
tion for Q
ˆ
−1
. From Eq. 9, the matrix elements Q
1 / D.
Therefore the width of the switching-path distribution 2 is
l
D
, as expected from qualitative arguments.
C. switching-path distribution in the initially occupied basin
of attraction
The case where the observation point q lies in the basin of
attraction of the initially occupied state A
1
is somewhat more
complicated. Here, too, the two probability densities in the
right-hand side of Eq. 7 are independent of time t for q
q
A
1,2
, q q
S
l
D
; but in contrast to the previously studied
region, none of them is known in advance. They have to be
found from the Fokker-Planck equation 8 for
q ,t q
0
,t
0
and the backward equation for
q
f
,t
f
q ,t,
K
q
+ D
q
2
q
f
,t
f
q,t =0. 10
We seek the solutions of Eqs. 8 and 10 in the eikonal
form,
q,tq
0
,t
0
= exp S
F
q/D,
q
f
,t
f
q,t = expS
B
q/D
2
q
f
. 11
The functions S
F
and S
B
can be written as power series in the
noise intensity D, with S
F,B
=S
F,B
0
+DS
F,B
1
+¯ . To the lowest
order in D we have
Hq,
q
S
F,B
0
=0, Hq,p = p
2
+ pKq. 12
Equation 12 has the form of a Hamilton-Jacobi equation
for an auxiliary particle with coordinate q and momentum p.
This particle moves with energy H= 0. The functions S
F,B
0
q
are mechanical actions. Subscript F refers to motion of the
auxiliary particle to point q from the vicinity of A
1
,asitis
clear from Eq. 11. Using condition H=0, one can associate
S
F
0
q with the mechanical action for reaching q from q
A
1
;
the motion formally starts at t from q
A
1
, with momen-
tum p= 0 15.
Subscript B in Eq. 12 refers to the auxiliary Hamiltonian
particle that moves from q further away from attractor A
1
.In
this motion the original system goes close to the saddle
point, and so should the auxiliary particle, too. The perturba-
tion theory that underlies Eq. 12 applies where the particle
is approaching the saddle point, but has not gone beyond it.
Indeed, for H =0 the particle approaches the saddle point
asymptotically, for infinite time. Therefore S
B
0
is the me-
chanical action for reaching q
S
from q note that S
B
0
0.
From Eqs. 7 and 11, the MPSP inside the basin of
attraction of the initially occupied state corresponds to the
maximum of S
B
0
q S
B
0
q and thus is determined by equa-
tion
q
S
F
0
=
q
S
B
0
. 13
The MPSP is thus given by the heteroclinic Hamiltonian tra-
jectory that goes from the state q
A
1
,p =0 to q
S
,p =0.
To find S
F,B
0
q close to the MPSP it is convenient to
switch to a comoving frame on the MPSP
,
. From
Hamiltonian 12, the longitudinal direction
ˆ
and the veloc-
ity on the MPSP are given by expression
2
q
S
F
0
q + Kq =
v
ˆ
, 14
where the left-hand side is calculated for
=0. Equation
14 applies also if we use S
B
0
instead of S
F
0
. Note that the
CHAN, DYKMAN, AND STAMBAUGH PHYSICAL REVIEW E 78, 051109 2008
051109-4

MPSP direction
ˆ
is not along the velocity of the original
system in the absence of noise K, in the general case of a
system lacking detailed balance.
Close to the MPSP we can expand S
F,B
0
and K in
. From
Eqs. 1, 11, and 13, p
12
q ,t exp
Q
ˆ
/ 2,asinEq.
3. The matrix Q
ˆ
is expressed in terms of the actions S
F,B
0
close to the MPSP as
Q
ˆ
= Q
ˆ
F
Q
ˆ
B
,
Q
ˆ
F,B
= D
−1
2
S
F,B
0
/
, 15
with the derivatives calculated for
=0. From the condition
that p
12
q ,t be maximal on the MPSP it follows that matrix
Q
ˆ
is positive definite.
1. Prefactor
Interestingly, the prefactor in p
12
q ,t can be expressed
explicitly in terms of the velocity
v
and the matrix Q
ˆ
.
Formally, the prefactor is determined by the terms S
F,B
1
in Eq.
11. The equations for them follow from Eqs. 8 and 10,
2
q
S
F
0
+ K
q
S
F
1
q
K
q
2
S
F
0
=0,
2
q
S
B
0
+ K
q
S
B
1
+
q
2
S
B
0
=0. 16
From Eqs. 14 and 16, to leading order in
we have
v
S
1
v
−Tr
ˆ
+ DQ
ˆ
F
+ Q
ˆ
B
兲兴 =0,
S
1
= S
F
1
S
B
1
, 17
where, as before,
=
K
/
with the derivatives calcu-
lated for
=0.
On the other hand, by expanding in Hamilton-Jacobi
equations 12 S
F,B
0
near the MPSP to second order in
and
taking into account the relation between the derivatives of
S
F
0
and S
B
0
on the MPSP 13 and 14 we obtain an impor-
tant relation
v
Q
ˆ
+2DQ
ˆ
F
2
Q
ˆ
B
2
+
ˆ
Q
ˆ
+ Q
ˆ
ˆ
=0.
From this equation
Tr
ˆ
+ DQ
ˆ
F
+ Q
ˆ
B
兲兴 =−
1
2
v
Tr log Q
ˆ
.
By substituting this relation into Eq. 17 we obtain
S
1
= log
v
1
2
Tr log Q
ˆ
+ log C
1
, 18
where we explicitly indicate that S
1
,
v
, and Q
ˆ
are functions
of the distance,
, along the MPSP; C
1
is a constant of
integration.
Equations 1, 11, 15, and 18 lead to expression 3
for the switching-path distribution. Note that, from Eq. 15,
inside the initially occupied basin of attraction the width of
the peak of the distribution transverse to the MPSP is l
D
D
1/2
. The distribution describes a stationary probability
current. This current is the same in the basins of attraction of
states A
1
and A
2
. In obtaining Eq. 3 from Eq. 18 we found
C
1
from the condition
v
兲兰d
p
12
q ,t= W
12
.
From conservation of the stationary probability current it
follows that the distribution p
12
q ,t should sharply increase
near the saddle point. Indeed, the velocity
v
= 0 for q
=q
S
. The current close to q
S
is due to diffusion. In the gen-
eral case of nonequilibrium systems the shape of the
switching-path distribution near the saddle point is compli-
cated; its analysis is beyond the scope of this paper.
D. switching-path distribution for systems with detailed balance
An explicit solution for p
12
q ,t near the saddle point can
be obtained for systems with a gradient force K=−
q
Uq.
Such systems have detailed balance. The uphill section of the
MPSP is literally the uphill path that goes from the local
minimum of the potential Uq at A
1
to the saddle S and is
given by the equation q
˙
=
q
Uq兲关45兴关this can be seen from
Eq. 12兲兴. In contrast to systems without detailed balance
43,46, for smooth Uq the MPSP near the saddle point is
described by an analytic function of coordinates and
ˆ
is
perpendicular to the separating hypersurface.
The quasistationary solution of the forward Fokker-
Planck equation 8 near a saddle point has been known
since the work of Kramers 4 and Landauer and Swanson
47. The backward equation 10 can be solved similarly by
expanding K to first order in q q
S
and by using the condi-
tion that deep inside the basin of attraction of the initially
empty state A
2
we have
q
f
,t
f
q ,t兲⬇
2
q
f
. The solution
has the form
q
f
,t
f
q,t兲⬇
1
2
2
q
f
兲关1 + erf
˜
兲兴,
˜
=
/2D
1/2
S
. 19
Here, erfx is the error function,
S
is the position of the
saddle point on the MPSP, and
is the curvature of the
potential Uq at the saddle point in the steepest descent
direction
ˆ
, Uq兲⬇
S
2
/ 2 for
=0 and small
S
兩共we chose
S
0 inside the basin of attraction to
A
2
.
Equation 19 combined with the results 4,47 give ex-
pression 3 for p
12
q ,t near q
S
provided one replaces in this
expression
v
−1
/8
D
1/2
exp
˜
2
兲关1 erf
2
˜
兲兴. 20
Equation 20 goes over into
v
−1
for
S
l
D
.In
the opposite limit, that is very close to the saddle point, it
shows that
v
−1
is replaced by a factor
/ 8
D
1/2
t
r
/ l
D
.
This demonstrates that the distribution p
12
q ,t does not di-
verge at the saddle point, but it contains a large factor D
−1/2
.
IV. MICROMECHANICAL TORSIONAL OSCILLATOR
A. Device characteristics
We measure the switching-path distribution using a
high-Q microelectromechanical torsional oscillator Q
SWITCHING-PATH DISTRIBUTION IN PHYSICAL REVIEW E 78, 051109 2008
051109-5

Figures (7)
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TL;DR: A selfcontained survey of stochastic hybrid systems, large deviations and the Wentzel–Kramers–Brillouin method, adiabatic reductions, and queuing/renewal theory is provided.
Abstract: There has been a resurgence of interest in non-equilibrium stochastic processes in recent years, driven in part by the observation that the number of molecules (genes, mRNA, proteins) involved in gene expression are often of order 1–1000. This means that deterministic mass-action kinetics tends to break down, and one needs to take into account the discrete, stochastic nature of biochemical reactions. One of the major consequences of molecular noise is the occurrence of stochastic biological switching at both the genotypic and phenotypic levels. For example, individual gene regulatory networks can switch between graded and binary responses, exhibit translational/ transcriptional bursting, and support metastability (noise-induced switching between states that are stable in the deterministic limit). If random switching persists at the phenotypic level then this can confer certain advantages to cell populations growing in a changing environment, as exemplified by bacterial persistence in response to antibiotics. Gene expression at the single-cell level can also be regulated by changes in cell density at the population level, a process known as quorum sensing. In contrast to noise-driven phenotypic switching, the switching mechanism in quorum sensing is stimulus-driven and thus noise tends to have a detrimental effect. A common approach to modeling stochastic gene expression is to assume a large but finite system and to approximate the discrete processes by continuous processes using a systemsize expansion. However, there is a growing need to have some familiarity with the theory of stochastic processes that goes beyond the standard topics of chemical master equations, the system-size expansion, Langevin equations and the Fokker–Planck equation. Examples include stochastic hybrid systems (piecewise deterministic Markov processes), large deviations and the Wentzel–Kramers–Brillouin (WKB) method, adiabatic reductions, and queuing/renewal theory. The major aim of this review is to provide a selfcontained survey of these mathematical methods, mainly within the context of biological switching processes at both the genotypic and phenotypic levels. P C Bressloff Stochastic switching in biology: from genotype to phenotype Printed in the UK 133001 JPHAC5 © 2017 IOP Publishing Ltd 50 J. Phys. A: Math. Theor.

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