CRITICAL EXPONENTS IN ACTIVATED ESCAPE OF NONEQUILIBRIUM SYSTEMS

Mark DykmanDepartment of Physics and Astronomy

Michigan State University

In collaboration withB. Golding (MSU)O. Kogan (Caltech)D. Ryvkine (MSU)I. Schwartz (NRL)M. Shapiro (MSU)

universal features

onset of soft modes leading to scaling of the escape rate

large susceptibility where coexisting stable states are almost equally occupied

experimental accessibility + applications

comparatively large escape probability for weak noise

high sensitivity to system parameters

Motivation

Noise-induced escape in dynamical systems far from thermal equilibrium:

bifurcation points

noise-induced frequency mixing, stochastic resonance, …

Near the bifurcation point one of the motions is slow, a soft mode universal behavior of the escape rate

Kurkijarvi (1972), MD & Krivoglaz (1979, 1980), Graham & Tél (1987), Victora (1989)

Stationary multivariable systems:

No noise: the relaxation time is large for small2/)(/1 2/12 −=∂−= ηaqr qUt ||η

,)( 331 qqqUUq q η+−=−∂=&

noise D=kBT in thermal equilibrium

)'(2)'()(),( ττδτττ −=+ Dfff

ζξ ηνην ∝=Δ=−= ,),/exp( 34URDRW

Noise: delta-correlated in slow time.

Saddle-node bifurcation

U

q

UΔ

2/1,2/3 == ζξ

The escape rate

Adiabatic bifurcation amplitude : the states touch each other, once per period, for

Slow driving: or

The system follows the driving field adiabatically (?)

Adiabatic periodic states:

[local minimum and maximum of the potential U(q,t)]

),(),(),(),( FttttA periodically modulated dynamical system:

τ+=+= qKqKfqKq&

rFF t>>= ωπτ /21

The barrier is at its lowest once per period, for

the effective 1D potential is a cubic parabola,

In the fully adiabatic picture, escape rate is determined by the instantaneous barrier height

adadad331 ,),( cF AAAqAqntqU −=−−≈= δδτ

ζξ δνδ )(,)()0( adadad

34

ad AAtUR −∝−==Δ= 4/1,2/3

Adiabatic scaling:

== ζξ

Relaxation time . The adiabaticity condition0for ad →∞→ Atr δbreaks down at the bifurcation point.

1

The true saddle-node bifurcation occurs for where the stable and the unstable states merge for all times,

cAA =

)()()( ttt bac qqq ==

cc AAA /)( −=ε

Avoided crossing of stable and unstable states where adiabaticity is broken

Overall evolution of periodic states

A step back…

tFω

adAqq δ+= 2& 1~, γ

The adiabatic relaxation time

2/])[(|)(2|)( 2/121 −− −== adFaadr Attqtt δγω

The adiabaticity conditions:

1

Shift of the bifurcation point due to crossing avoidance

Fadc

slc AA γω+=

Control parameter )/()( Fslc AA γωη −=

Scaled coordinate and time ll ttqtQ /, == τ

Locally nonadiabatic regime

:1~η locally nonadiabatic regime,

1

Shift of the bifurcation point due to crossing avoidance

Fcc AA γω+=adsl

Control parameter )/()( Fslc AA γωη −=

Scaled coordinate and time ll ttqtQ /, == τ

Locally nonadiabatic regime

)(~122 τηττ

fQddQ

+−+−=

1D time-dependent local Langevin equation

optQ

Close to the bifurcation point, , the variational problem for the optimal escape path can be linearized and solved:

[ ]∫ ∞− −− −−=τ ττττηττ

21

21

2

21)( 1 eedQopt

1

The escape activation energy and the prefactor in the escape rate are

ζξ ηνηπ ∝= ,)8/( 2/1R

locally nonadiabatic adiabatic crossover:

2/32,)( ⇔=−∝ ξξAAR c

Slow driving, :1r

Results for a model system

1,cos),( (0)r413

31 =−+−= ttAqqqtqU Fω

Underdamped systems close to a bifurcation point

)(),();,(

)(),();,(

)()(

)()(

tfqpqpHdtdp

tfqpqpHdtdq

ppq

qqp

+−−∂=

+−∂=

ευη

ευη

friction noise

)'(),(2)'()( )()( ttqpDtftf ijji −= δ

A Hamiltonian system with weak damping and noise, close but not too close to the saddle-node bifurcation point:

Hamiltonian q

U

Frequency of vibrations about the center is small, but friction is still smaller,

cω 1,1 −∝>> εω rrc ttq

p

*

Assume:

where the control parameter (“Hamiltonian bifurcation point”)

is analytic in for small | |

for );,( ηqpH

0→cω 0→η

0=ηη η

0≠∂ Hη

Nonlocal scenario: squeezing of the homoclinic orbit as , with no change in

U

q

The scenarios of approaching the bifurcation

Local scenario: shrinking of the homoclinic orbit as ,with q

p

*

0→η

ηcq sq

0→η

sc qq →

Generic Hamiltonian, to leading order in

qqpH η+−= 3312

21

The energy barrier );,();,( ηη ccss qpHqpHE −=Δ2/3

34η=ΔE

U

q

sc qq ,

qp

*cq sq

)(221 qUpH η+=

)()(, cs qUqUCCE −==Δ η

crossover to

Friction: drift over energy towards a metastable stateNoise: energy diffusion, ultimately leading to escape

Escape rate

)/exp( DRW −∝

Crossover for a resonantly driven oscillator

2/3η∝R

)/()( 212 BBB FFFF −− )/()( 212 BBB FFFF −−

R~ R~scaled friction 01.0=Γ

wscaled friction 1.0=Γ

w

(Kogan et al., 2006)

Local scenario: ,2/3εηlocCR = the same scaling as in the overdamped region

Non-local scenario: ,εηnlCR = with decreasing η

Modulated nonlinear oscillator

Eigenfrequency depends on vibration amplitude, 200~aγωω +→

Cubic equation for the amplitude of forced vibrations, 2220

222

)~(4/

Γ+−−=

aF

aF

F

γωωω

tFqqqq Fωγω cos232

0 =++Γ+ &&&

φ andBoth a display hysteresis

Periodic state: )cos( φω += taq F

F2

a2

F2B1F2

B2

Hysteresis in a modulated Josephson junction

tFqqqq Fωγω cos232

0 =++Γ+ &&&

Periodic state: )cos( φω += taq F

φ andBoth a display hysteresis

SS I0 , JI C

δ phasedifference

A Josephson junction based nonlinear oscillator(Siddiqi et al. PRL 2004, 2005)

F2

a2

F2B1F2

B2

Hysteresis and switching in nanooscillators

Resonantly modulated MEMS

(Stambaugh & Chan, PRL, PRB 2006)

Resonantly modulated nanoelectromechanical resonators

(Schwab & Roukes, Phys. Today 2005)

(Aldridge & Cleland, PRL 2005)

3μm14.02.03 ××

High sensitivity to a system parameter (e.g., oscillator eigenfrequency): a small change in the parameter leads to a shift of the bifurcation point

1μμ =

2μμ =

Quantum readout with a bifurcation amplifier

(Siddiqi et al., 2004-2006)

Real switching:fluctuation-induced smearing

Strong signal amplification at a bifurcation point

∞→)input()output(

dd

-3.5 -2.5 -1.5

-7

-6

-5

log Δω (rad s-1)

log

Ea

A modulated oscillator does not have detailed balance. Switchingoccurs between periodic states. For resonant modulation, the activation energy scales as close to bifurcation points. For small damping it may also display scaling

Scaling for a classical oscillator

2/3η∝R

(MD and Krivoglaz, 1979, 1980)

MEMS (Stambaugh & Chan, 2005/2006)(ln

W)2

/3

2RFi

Josephson junctions (Siddiqi et al., 2005)

η∝R

The predicted scaling has been seen near a cusp on the bifurcation curve (Aldridge & Cleland, 2005) and for a parametrically modulated oscillator (Stambaugh & Chan, 2006)

2η∝R

Quantum scaling for resonant driving

)12/( ph2/3 κη ++∝ nRA

12/31 ,)12( BA FFnR −=+∝

− ηη

Still closer to motion becomes overdamped and 2,1BF

4/5tun η∝S

Underdamped regime:

small-amplitude

large-amplitude

small-amplitude

large-amplitude 21 ,)12( BA FFnR −=+∝

− ηη

small-amplitude: tunneling decay, (Dmitriev & Dyakonov, 1986)

scaled dephasing rate (MD, 2006)

F2

a2

F2B1F2

B2

Activation energy and the prefactor in the escape rate scale as powers of the distance to the bifurcation point

Systems lacking detailed balance display scaling behavior that does not arise in systems in thermal equilibrium, with new scaling exponents.

Scaling crossovers: in slowly modulated systems and in underdamped systems

Critical exponents near bifurcation points have been observed in modulated nonlinear oscillators.

2/322/3 →→=ξ

Conclusions

2/31→=ξ