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
共Received 27 June 2008; published 13 November 2008
兲
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 number共s兲: 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 关1兴, Joseph-
son junctions 关2兴, protein folding 关3兴, and chemical reactions.
A detailed theory of switching rates was first developed
by Kramers 关4兴. 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 关5兴 and atoms
关6,7兴 in modulated traps, rf-driven Josephson junctions 关8,9兴,
and nano- and micromechanical resonators 关10–12兴. Apart
from fundamental interest, switching in modulated systems
is important for many applications, quantum measurements
being an example 关8,9,13,14兴. 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 共MPSP兲. For low fluctua-
tion intensity, the MPSP is obtained by solving a variational
problem. This problem also determines the switching activa-
tion barrier 关15–25兴. 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 关26兴. 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 关27兴 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 关26,27兴 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. 关28兴. 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 共2008兲
1539-3755/2008/78共5兲/051109共10兲 ©2008 The American Physical Society051109-1
II. QUALITATIVE PICTURE AND PREVIEW OF
THE RESULTS
We consider a bistable system with several dynamical
variables q =共q
1
,...,q
N
兲. 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 共the characteristic diffusion length兲 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
共q,t兲 =
冕
⍀
2
dq
f
共q
f
,t
f
;q,t兩q
0
,t
0
兲. 共1兲
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
共q ,t兲 the switching-path distribution. Of ut-
most interest is to study this distribution in the time range
W
12
−1
,W
21
−1
Ⰷ t
f
− t,t − t
0
Ⰷ t
˜
r
. 共2兲
Here, t
˜
r
is the Suzuki time 关29兴. It differs from t
r
by a loga-
rithmic factor ⬃log关⌬q/ l
D
兴. 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 t−t
0
Ⰷt
r
, by time t the system has already “forgotten”
the initial position q
0
. Therefore the distribution
共q
f
,t
f
;q ,t 兩q
0
,t
0
兲, 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 ⱗt
˜
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 共2兲, the distribution p
12
共q ,t兲 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
共q ,t兲 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
共q ,t兲 peaks on the MPSP. The
peak is Gaussian transverse to the MPSP for 兩q− q
A
1,2
兩, 兩q
−q
S
兩Ⰷ l
D
,
p
12
共q,t兲 = W
12
v
−1
共
储
兲Z
−1
exp
冉
−
1
2
⬜
Q
ˆ
⬜
冊
, 共3兲
where
储
and
⬜
are coordinates along and transverse to the
MPSP, and
v
共
储
兲 is the velocity along the MPSP. The matrix
elements of matrix Q
ˆ
=Q
ˆ
共
储
兲 are ⬀l
D
−2
, and Z
=关共2
兲
N−1
/ det Q
ˆ
兴
1/2
. It follows from Eq. 共3兲 that the overall
probability flux along the MPSP is equal to the switching
rate,
冕
d
⬜
p
12
共q,t兲
v
共
储
兲 = W
12
.
−2 −1 0 1
2
−1.5
0
1
.5
q
1
q
2
S
A
1
A
2
FIG. 1. 共Color online兲 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 共2008兲
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 共3兲 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
共q ,t兲 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
共q ,t兲 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” 关30兴. 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
共q,t兩q
f
,t
f
兲 =
共q
f
,t
f
;q,t兩q
0
,t
0
兲
共q
f
,t
f
兩q
0
,t
0
兲
. 共4兲
The prehistory distribution 共4兲 and its generalizations
were analyzed in a number of papers 关26,27,31–33兴. How-
ever, the problem of paths that lead to switching between the
states was addressed only for a stationary system with one
dynamical variable 关26兴. 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 关26,27,31兴 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 共4兲 depends not only on the
position of point q
f
, but also on the time t
f
when this point is
reached 关34兴.
In contrast, the distribution p
12
共q ,t兲 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
共q ,t兲 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. 共2兲 for a system described by the Langevin
equation of motion
q
˙
= K共q兲 + f共t兲, 具f
n
共t兲f
m
共t
⬘
兲典 =2D
␦
nm
␦
共t − t
⬘
兲. 共5兲
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 f共t兲 in Eq. 共5兲 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
关15–21兴. This is also the case for noise-driven
extended systems, cf. Refs. 关35–37兴 and papers cited therein.
There exists extensive literature on numerical calculations of
the switching rate and switching paths, cf. Refs. 关38–42兴 and
papers cited therein.
In the model 共5兲, the characteristic relaxation time t
r
and
the characteristic diffusion length l
D
are
t
r
= max
k
兩Re
k
兩
−1
, l
D
= 共Dt
r
兲
1/2
, 共6兲
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 共5兲, the three-time prob-
ability distribution
共q
f
,t
f
;q ,t 兩q
0
,t
0
兲 in Eq. 共1兲 can be writ-
ten as a product of two-time transition probability densities,
共q
f
,t
f
;q,t兩q
0
,t
0
兲 =
共q
f
,t
f
兩q,t兲
共q,t兩q
0
,t
0
兲, 共7兲
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. 共Color兲共a兲 Switching-path distribution in a parametri-
cally driven microelectromechanical oscillator. The probability dis-
tribution p
12
共X ,Y兲 is measured for switching out of state A
1
into
state A
2
. 共b兲 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 共2008兲
051109-3
from the stationary states, 兩q −q
S
兩, 兩q −q
A
1,2
兩Ⰷ 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., 兩 q
f
−q
A
2
兩ⱗ l
D
,we
have
共q
f
,t
f
兩q ,t兲⬇
2
共q
f
兲. Here,
2
共q
f
兲 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. 共1兲
p
12
共q,t兲 =
共q,t兩q
0
,t
0
兲.
The analysis of the transition probability density
共q ,t 兩q
0
,t
0
兲 in this expression is simplified by two observa-
tions. First, for time t in the range W
12
−1
Ⰷt −t
0
Ⰷt
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 关4兴. The current
density far from A
2
, i.e., for 兩 q − q
A
2
兩Ⰷ l
D
, is independent of
time and is determined by the stationary Fokker-Planck
equation
关−
q
K + D
q
2
兴
共q,t兩q
0
,t
0
兲 =0. 共8兲
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 关20,43兴. 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
共q ,t 兩q
0
,t
0
兲 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 关44兴.
We parametrize the deterministic section of the MPSP by
its length
储
counted off from q
S
and introduce a unit vector
ˆ
储
along the vector K on the MPSP and N −1 vectors
⬜
perpendicular to it. The velocity on the MPSP is
v
⬅
v
共
储
兲
=K共
储
,
⬜
=0兲. Of interest for our analysis are the values of
兩
⬜
兩 of the order of the width of the path distribution trans-
verse to the MPSP, which is given by the diffusion length,
i.e., 兩
⬜
兩ⱗ l
D
. We assume 兩
⬜
兩 to be small compared to the
radius of curvature 兩d
ˆ
储
/ d
储
兩
−1
.
Equation 共8兲 can be solved near the MPSP by changing to
variables
储
,
⬜
, expanding K to first order in
⬜
, and replac-
ing
q
2
→
⬜
2
. One then obtains for
共q ,t 兩q
0
,t
0
兲= p
12
共q ,t兲 ex-
pression 共8兲, with matrix Q
ˆ
given by the equation
v
dQ
ˆ
d
储
+
ˆ
†
Q
ˆ
+ Q
ˆ
ˆ
+2Q
ˆ
2
D =0. 共9兲
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 共9兲 can be reduced to a linear equa-
tion for Q
ˆ
−1
. From Eq. 共9兲, the matrix elements Q
⬀1 / D.
Therefore the width of the switching-path distribution 共2兲 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. 共7兲 are independent of time t for 兩q
−q
A
1,2
兩, 兩q −q
S
兩Ⰷ 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 共8兲 for
共q ,t 兩q
0
,t
0
兲
and the backward equation for
共q
f
,t
f
兩q ,t兲,
共K
q
+ D
q
2
兲
共q
f
,t
f
兩q,t兲 =0. 共10兲
We seek the solutions of Eqs. 共8兲 and 共10兲 in the eikonal
form,
共q,t兩q
0
,t
0
兲 = exp关− S
F
共q兲/D兴,
共q
f
,t
f
兩q,t兲 = exp关S
B
共q兲/D兴
2
共q
f
兲. 共11兲
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
共0兲
+DS
F,B
共1兲
+¯ . To the lowest
order in D we have
H共q,
q
S
F,B
共0兲
兲 =0, H共q,p兲 = p
2
+ pK共q兲. 共12兲
Equation 共12兲 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
共0兲
共q兲
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. 共11兲. Using condition H=0, one can associate
S
F
共0兲
共q兲 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 关15兴.
Subscript B in Eq. 共12兲 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. 共12兲 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
共0兲
is the me-
chanical action for reaching q
S
from q 共note that S
B
共0兲
艋0兲.
From Eqs. 共7兲 and 共11兲, the MPSP inside the basin of
attraction of the initially occupied state corresponds to the
maximum of S
B
共0兲
共q兲− S
B
共0兲
共q兲 and thus is determined by equa-
tion
q
S
F
共0兲
=
q
S
B
共0兲
. 共13兲
The MPSP is thus given by the heteroclinic Hamiltonian tra-
jectory that goes from the state 共q
A
1
,p =0兲 to 共q
S
,p =0兲.
To find S
F,B
共0兲
共q兲 close to the MPSP it is convenient to
switch to a comoving frame on the MPSP 共
储
,
⬜
兲. From
Hamiltonian 共12兲, the longitudinal direction
ˆ
储
and the veloc-
ity on the MPSP are given by expression
2
q
S
F
共0兲
共q兲 + K共q兲 =
v
共
储
兲
ˆ
储
, 共14兲
where the left-hand side is calculated for
⬜
=0. 关Equation
共14兲 applies also if we use S
B
共0兲
instead of S
F
共0兲
.兴 Note that the
CHAN, DYKMAN, AND STAMBAUGH PHYSICAL REVIEW E 78, 051109 共2008兲
051109-4
MPSP direction
ˆ
储
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
共0兲
and K in
⬜
. From
Eqs. 共1兲, 共11兲, and 共13兲, p
12
共q ,t兲⬀ exp共−
⬜
Q
ˆ
⬜
/ 2兲,asinEq.
共3兲. The matrix Q
ˆ
is expressed in terms of the actions S
F,B
共0兲
close to the MPSP as
Q
ˆ
= Q
ˆ
F
− Q
ˆ
B
,
共Q
ˆ
F,B
兲
= D
−1
2
S
F,B
共0兲
/
⬜
⬜
, 共15兲
with the derivatives calculated for
⬜
=0. From the condition
that p
12
共q ,t兲 be maximal on the MPSP it follows that matrix
Q
ˆ
is positive definite.
1. Prefactor
Interestingly, the prefactor in p
12
共q ,t兲 can be expressed
explicitly in terms of the velocity
v
共
储
兲 and the matrix Q
ˆ
.
Formally, the prefactor is determined by the terms S
F,B
共1兲
in Eq.
共11兲. The equations for them follow from Eqs. 共8兲 and 共10兲,
共2
q
S
F
共0兲
+ K兲
q
S
F
共1兲
−
q
K −
q
2
S
F
共0兲
=0,
共2
q
S
B
共0兲
+ K兲
q
S
B
共1兲
+
q
2
S
B
共0兲
=0. 共16兲
From Eqs. 共14兲 and 共16兲, to leading order in
⬜
we have
v
储
S
共1兲
−
储
v
−Tr关
ˆ
+ D共Q
ˆ
F
+ Q
ˆ
B
兲兴 =0,
S
共1兲
= S
F
共1兲
− S
B
共1兲
, 共17兲
where, as before,
=
K
/
⬜
with the derivatives calcu-
lated for
⬜
=0.
On the other hand, by expanding in Hamilton-Jacobi
equations 共12兲 S
F,B
共0兲
near the MPSP to second order in
⬜
and
taking into account the relation between the derivatives of
S
F
共0兲
and S
B
共0兲
on the MPSP 共13兲 and 共14兲 we obtain an impor-
tant relation
v
储
Q
ˆ
+2D共Q
ˆ
F
2
− Q
ˆ
B
2
兲 +
ˆ
†
Q
ˆ
+ Q
ˆ
ˆ
=0.
From this equation
Tr关
ˆ
+ D共Q
ˆ
F
+ Q
ˆ
B
兲兴 =−
1
2
v
储
Tr log Q
ˆ
.
By substituting this relation into Eq. 共17兲 we obtain
S
共1兲
共
储
兲 = log
v
共
储
兲 −
1
2
Tr log Q
ˆ
共
储
兲 + log C
1
, 共18兲
where we explicitly indicate that S
共1兲
,
v
, and Q
ˆ
are functions
of the distance,
储
, along the MPSP; C
1
is a constant of
integration.
Equations 共1兲, 共11兲, 共15兲, and 共18兲 lead to expression 共3兲
for the switching-path distribution. Note that, from Eq. 共15兲,
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. 共3兲 from Eq. 共18兲 we found
C
1
from the condition
v
共
储
兲兰d
⬜
p
12
共q ,t兲= W
12
.
From conservation of the stationary probability current it
follows that the distribution p
12
共q ,t兲 should sharply increase
near the saddle point. Indeed, the velocity
v
共
储
兲= 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
共q ,t兲 near the saddle point can
be obtained for systems with a gradient force K=−
q
U共q兲.
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 U共q兲 at A
1
to the saddle S and is
given by the equation q
˙
=
q
U共q兲关45兴关this can be seen from
Eq. 共12兲兴. In contrast to systems without detailed balance
关43,46兴, for smooth U共q兲 the MPSP near the saddle point is
described by an analytic function of coordinates and
ˆ
储
is
perpendicular to the separating hypersurface.
The quasistationary solution of the forward Fokker-
Planck equation 共8兲 near a saddle point has been known
since the work of Kramers 关4兴 and Landauer and Swanson
关47兴. The backward equation 共10兲 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
共q
f
,t
f
兩q ,t兲⬇
2
共q
f
兲. The solution
has the form
共q
f
,t
f
兩q,t兲⬇
1
2
2
共q
f
兲关1 + erf共
˜
储
兲兴,
˜
储
= 共
储
/2D兲
1/2
共
储
−
储
S
兲. 共19兲
Here, erf共x兲 is the error function,
储
S
is the position of the
saddle point on the MPSP, and
储
is the curvature of the
potential U共q兲 at the saddle point in the steepest descent
direction
ˆ
储
, U共q兲⬇−
储
共
储
−
储
S
兲
2
/ 2 for
⬜
=0 and small 兩
储
−
储
S
兩共we chose
储
−
储
S
⬎0 inside the basin of attraction to
A
2
兲.
Equation 共19兲 combined with the results 关4,47兴 give ex-
pression 共3兲 for p
12
共q ,t兲 near q
S
provided one replaces in this
expression
v
−1
共
储
兲 → 共
/8
储
D兲
1/2
exp共
˜
储
2
兲关1 − erf
2
共
˜
储
兲兴. 共20兲
Equation 共20兲 goes over into
v
−1
共
储
兲 for 兩
储
−
储
S
兩Ⰷ l
D
.In
the opposite limit, that is very close to the saddle point, it
shows that
v
−1
is replaced by a factor 共
/ 8
储
D兲
1/2
⬃t
r
/ l
D
.
This demonstrates that the distribution p
12
共q ,t兲 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 共Q
SWITCHING-PATH DISTRIBUTION IN… PHYSICAL REVIEW E 78, 051109 共2008兲
051109-5
