# Switching-path distribution in multidimensional systems.

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.

Topics: Phase space (53%), Multidimensional systems (52%), Parametric oscillator (51%)

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

##### Citations

More filters

••

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.

105 citations

### Cites background from "Switching-path distribution in mult..."

...Following Chan et al [61], one can define a switching path-distribution that gives the probability density of passing a given point in state space during switching....

[...]

••

James M. L. Miller

^{1}, Azadeh Ansari^{2}, David B. Heinz^{1}, Yunhan Chen^{1}+4 more•Institutions (3)Abstract: Quality factor (Q) is an important property of micro- and nano-electromechanical (MEM/NEM) resonators that underlie timing references, frequency sources, atomic force microscopes, gyroscopes, and mass sensors. Various methods have been utilized to tune the effective quality factor of MEM/NEM resonators, including external proportional feedback control, optical pumping, mechanical pumping, thermal-piezoresistive pumping, and parametric pumping. This work reviews these mechanisms and compares the effective Q tuning using a position-proportional and a velocity-proportional force expression. We further clarify the relationship between the mechanical Q, the effective Q, and the thermomechanical noise of a resonator. We finally show that parametric pumping and thermal-piezoresistive pumping enhance the effective Q of a micromechanical resonator by experimentally studying the thermomechanical noise spectrum of a device subjected to both techniques.

71 citations

••

Imran Mahboob

^{1}, Mickael Mounaix^{1}, Katsuhiko Nishiguchi^{1}, Akira Fujiwara^{1}+1 more•Institutions (1)TL;DR: This work identifies 75 harmonic vibration modes in a single electromechanical resonator of which 7 can also be parametrically excited and exploits this array to execute a mechanical byte memory, a shift-register and a controlled-NOT gate thus vividly illustrating the availability and functionality of an electromech mechanical resonator array by simply utilising higher order vibration modes.

Abstract: Electromechanical resonators have emerged as a versatile platform in which detectors with unprecedented sensitivities and quantum mechanics in a macroscopic context can be developed. These schemes invariably utilise a single resonator but increasingly the concept of an array of electromechanical resonators is promising a wealth of new possibilities. In spite of this, experimental realisations of such arrays have remained scarce due to the formidable challenges involved in their fabrication. In a variation to this approach, we identify 75 harmonic vibration modes in a single electromechanical resonator of which 7 can also be parametrically excited. The parametrically resonating modes exhibit vibrations with only 2 oscillation phases which are used to build a binary information array. We exploit this array to execute a mechanical byte memory, a shift-register and a controlled-NOT gate thus vividly illustrating the availability and functionality of an electromechanical resonator array by simply utilising higher order vibration modes.

57 citations

••

TL;DR: These results suggest that an electromechanical simulator could be built for the Ising Hamiltonian in a nontrivial configuration, namely, for a large number of spins with multiple degrees of coupling.

Abstract: Solving intractable mathematical problems in simulators composed of atoms, ions, photons, or electrons has recently emerged as a subject of intense interest. We extend this concept to phonons that are localized in spectrally pure resonances in an electromechanical system that enables their interactions to be exquisitely fashioned via electrical means. We harness this platform to emulate the Ising Hamiltonian whose spin 1/2 particles are replicated by the phase bistable vibrations from the parametric resonances of multiple modes. The coupling between the mechanical spins is created by generating two-mode squeezed states, which impart correlations between modes that can imitate a random, ferromagnetic state or an antiferromagnetic state on demand. These results suggest that an electromechanical simulator could be built for the Ising Hamiltonian in a nontrivial configuration, namely, for a large number of spins with multiple degrees of coupling.

52 citations

### Cites background from "Switching-path distribution in mult..."

...When the parametric resonance is deactivated, the oscillating mode returns to the origin via another trajectory, which does not overlap with the upward trajectory (21)....

[...]

••

Abstract: We demonstrate squeezing of a strongly interacting optoelectromechanical system using a parametric drive. By employing real-time feedback on the phase of the pump at twice the resonance frequency the thermomechanical noise is squeezed beyond the 3 dB instability limit. Surprisingly, this method can also be used to generate highly nonlinear states. We show that using the parametric drive with feedback on, classical numberlike and catlike states can be prepared. This presents a valuable electro-optomechanical state-preparation protocol that is extendable to quantum regime.

29 citations

### Cites background from "Switching-path distribution in mult..."

...The resonator dynamically switches between them [41] resulting in the pdf that resembles the cat state in Fig....

[...]

##### References

More filters

••

Abstract: A particle which is caught in a potential hole and which, through the shuttling action of Brownian motion, can escape over a potential barrier yields a suitable model for elucidating the applicability of the transition state method for calculating the rate of chemical reactions.

6,870 citations

•

01 Jan 1984-

Abstract: 1.Random Perturbations.- 2.Small Random Perturbations on a Finite Time Interval.- 3.Action Functional.- 4.Gaussian Perturbations of Dynamical Systems. Neighborhood of an Equilibrium Point.- 5.Perturbations Leading to Markov Processes.- 6.Markov Perturbations on Large Time Intervals.- 7.The Averaging Principle. Fluctuations in Dynamical Systems with Averaging.- 8.Random Perturbations of Hamiltonian Systems.- 9. The Multidimensional Case.- 10.Stability Under Random Perturbations.- 11.Sharpenings and Generalizations.- References.- Index.

3,873 citations

••

Peter G. Bolhuis

^{1}, David Chandler^{2}, Christoph Dellago^{3}, Phillip L. Geissler^{4}•Institutions (4)TL;DR: This article reviews the concepts and methods of transition path sampling, which allow computational studies of rare events without requiring prior knowledge of mechanisms, reaction coordinates, and transition states.

Abstract: This article reviews the concepts and methods of transition path sampling. These methods allow computational studies of rare events without requiring prior knowledge of mechanisms, reaction coordinates, and transition states. Based upon a statistical mechanics of trajectory space, they provide a perspective with which time dependent phenomena, even for systems driven far from equilibrium, can be examined with the same types of importance sampling tools that in the past have been applied so successfully to static equilibrium properties.

1,688 citations

••

Abstract: The probability of a given succession of (nonequilibrium) states of a spontaneously fluctuating thermodynamic system is calculated, on the assumption that the macroscopic variables defining a state are Gaussian random variables whose average behavior is given by the laws governing irreversible processes.This probability can be expressed in terms of the dissipation function; the resulting relation, which is an extension of Boltzmann's principle, shows the statistical significance of the dissipation function. From the form of the relation, the principle of least dissipation of energy becomes evident by inspection.

1,411 citations

••

TL;DR: A mechanical degenerate parametric amplifier has been devised which greatly increases the motional response of a microcantilever for small harmonic force excitations and can improve force detection sensitivity for measurements dominated by sensor noise or backaction effects.

Abstract: A mechanical degenerate parametric amplifier has been devised which greatly increases the motional response of a microcantilever for small harmonic force excitations. The amplifier can improve force detection sensitivity for measurements dominated by sensor noise or backaction effects and can also produce mechanical squeezed states. In an initial squeezing demonstration, the thermal noise (Brownian motion) of the cantilever was reduced in one phase by 4.9 dB.

529 citations