Bryan Eisenhower and Igor Mezic- Targeted Activation in Deterministic and Stochastic Systems

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Targeted Activation in Deterministic and Stochastic Systems

Bryan Eisenhower and Igor Mezic

Department of Mechanical Engineering,

University of California Santa Barbara

(Dated: March 6, 2009)

Abstract

Metastable escape and transitions are ubiquitous in many physical systems and are becoming

a concern in engineering design as these designs become more and more inspired by dynamics of

biological, molecular, or other natural systems (eg. swarms of vehicles, coupled building energetics,

nanoengineering, etc). We study an example of a chain of coupled bistable oscillators which has

two global conformation states. Specifically we investigate re-conformation between these two

states and how specialized or targeted disturbance effects the transition process. We derive a

multi-phase averaged approximation to these dynamics which illustrates the influence of actions in

modal coordinates on the coarse behavior of this process. An activation condition that characterizes

temporal rate in relation to initial energy is then derived. This result emphasizes spatial sensitivity

and the impact of targeted activation on reconformation. We also find an unexpected inverse energy

cascade through these actions, funneling energy to one coarse coordinate leading to the global

transition. The prediction tools are derived for deterministic dynamics while we also present

analogous behavior in the stochastic setting and an ultimate divergence of Kramers activation

behavior under targeted activation conditions.

PACS numbers: Valid PACS appear here

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A chain of strongly coupled oscillators, each of which oscillates in a bistable potential, is a

coarse representation of many physical systems from macromolecules to Josephson junctions

which have multiple conformations. Switching between these conformations can occur by

coordinated motion of the entire chain over a global energetic barrier [1]. Characterizing this

global barrier with respect to local barrier heights is essential if this structure is to be used

for engineering design. We present such results for a deterministic harmonically coupled

oscillator chain with a strong backbone.

The class of nonlinear systems of coupled oscillators that we study are close to a coupled

chain of linear harmonic oscillators. Such near-integrable systems have been studied in [2]

where equipartition of energy and dynamic properties related to integrable instability theory

of partial differential equations were investigated [3].

The Hamiltonian for an N oscillator system which has a strong linear backbone and local

nonlinearity takes the form

H =Ni=1

1

2q2i +

1

2B(qi1 qi)2 +U(qi) (1)

where qi R, B is the linear coupling strength andUis a local bistable nonlinear potential.We have studied many such potentials, including a Morse potential acting on pendula (see [1],

[4], [5]) which is affiliated with conformation change in molecular systems but its exponential

form makes analytical progress challenging. In this study we investigate a 2-4 potential

U(qi) =

1

2q2i

1

4q4i

(2)

where is a constant that separates the two nonzero equilibria (qeq = {0,}) to control thecurvature of the saddle to better capture dynamics of proteins, see [6]. This potential has

the same behavior as the Morse potential while being easier to analyze. The backbone has

periodic boundary (qN+1 = q1), and is much stronger than the local nonlinearity (B ).

When this is the case the transition process is collective and coordinated as shown in Figure1 (and as described in [1]).

In systems where the strength of the coupling is of the same order as the strength of

the local nonlinearity, neighboring oscillators do not restrain each other from excursions

across local barriers. These excursions or breathers initiate the transition process of the

entire chain. In [7] it was shown that the transition process occurs when combination of

small breathers merge into larger breathers with enough strength to pull the entire oscillator

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5 0 50

20

40

60

80

100T = 0

OscillatorNumber

2 0 20

20

40

60

80

100T = 72.5

2 0 20

20

40

60

80

100T = 109

2 0 20

20

40

60

80

100T = 145.5

OscillatorNumber

Position2 0 20

20

40

60

80

100T = 182

Position2 0 20

20

40

60

80

100T = 254.5

Position

FIG. 1: Snapshots of the evolution of 100 oscillators from one equilibrium to the other in the

potential (2) ( = 1.0e 3, = 1, B = 1). As an initial condition, two oscillators are perturbedin the potential and because the coupling is much stronger than the nonlinearity the transition is

collective, closely following the mean (dashed line).

array over the barrier. In [8] the same authors developed an analytical calculation for the

transition process considering a single oscillator diffusing through a separatrix layer and

subsequently pulling the other oscillators along. This theory does not work in our setting

since the coupling strength is much greater than the local nonlinearity and excursions of a

few oscillators and formation of breathers is discouraged. We find that it is not excursions

of individual or groups of oscillators that drives the transition process but rather dynamics

of the mean position of all oscillators. In fact, we show that in the time-averaged sense,

the dynamics of the mean position and velocity are sufficient to predict properties of the

activation process.

Since the empirical evidence of Arrhenius, noise assisted activation has been studied to a

great extent and has led to many predictive tools [9]. In this setting, external thermal energy

lifts the oscillators over the barrier. On the other hand, limited tools exist for the micro-canonical example where energy is conserved and escape is only realized by redistribution of

initial energy. Such behavior is remarkable as energy is appropriately distributed throughout

the system leading to the activation process without any external influence from a thermal

bath. We show that such behavior is caused by an interaction between incommensurate

linear modes due to the local nonlinearities.

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We perform two canonical coordinate transformations to identify the coarse behavior and

energy transfer dynamics of the transition process. Because of the translational symmetry

on the backbone the fundamental normal basis is Fourier modes. We perform the canonical

transformation into these coordinates using

qi =

2

N

N21

k=1

[q0

2+ cos (2ik/N)qk (3)

+cos(k)

2qN

2

+ sin (2ik/N)qN2+k]

where qk are modal amplitudes and k is the wavenumber (time differentiation of (3)

gives transform rules for velocities qi to modal velocities qk) . The frequencies k =

2cos(2k/N) 2 are irrationally related and range from 0.0 to 2.0 with linear spacing

at low frequencies and decreasing spacing as they approach 2.0. The zeroth mode q0 is theaveraged coordinate (scaled by

N) and the coordinates q

1...N21

are Fourier modes.

Projecting the nonzero modal dynamics onto canonical action-angle variables takes the

system to a series of nearly integrable oscillators which interact with one highly nonlinear

mode (we rename q0 0 and q0 J0 for notational convenience)0 = J0

J0 = g0( J, , ),

k = k + fk( J, , )

Jk = gk( J, , )(4)

k = 1, 2, . . . (N 1)where k are angle variables and Jk are the conjugate actions. To obtain a coarse repre-

sentation we average over the angles of M nonzero modes (M N as the analysis can beperformed on a reduced order model since the transformations are canonical). The averaged

Hamiltonian becomes

H( J) = 1(2)M

2

0

2

0

H( J, )d1 . . . d M (5)

For the specific 2-4 potential the resulting averaged ODE derived from Hamiltons equationsis

0 = J0 (6)

J0 =

3

N

Mk=1

Jkk

0

3

0

N

Jk = 0, k = 1, 2, . . . (M 1)

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In an averaged sense, action in the higher modes are stationary in time and feed into

the zeroth mode dynamics. In this sense, (6) represents a parametric 2-4 potential for the

planar variables (0,J0) where higher order actions (Jk, k = 1, 2, . . . , M ) alter the existence

and location of the equilibria branches.

The rate of activation is dependent on energy in both the microcanonical and thermally

activated settings. In the deterministic setting, there exist a dependency between the ac-

tivation rate and the amount of initial energy (and in particular, the spatial character of

the initial energy), and this dependency can be captured quantitatively using the averaged

model (6). In the averaged sense, the zeroth mode has the typical figure eight phase space of

the 2-4 potential with equilibria at 0eq =

N in modal coordinates. With an initial con-

dition with the zeroth mode at this location, and no energy in higher modes, no oscillations

will occur. As soon as energy from a higher mode is introduced the equilibrium of the zeroth

mode begins to shift, and since the oscillators are no longer on the original equilibrium they

begin to oscillate (in a state of libration). In fact, the equilibria shift toward the origin

bringing the separatrix closer to the initial location of the zeroth mode. With sufficient

energy in any one of these higher order modes, the separatrix of the zeroth mode will be

driven closer to the origin and eventually the original location of the oscillators will breach

it. When this occurs the zeroth mode will be forced into the rotation state and transition

to the second equilibrium will occur. This concept is illustrated in Figure 2.

4

2

0

2

4

0.4

0.2

0

0.2

0.40

2

4

6

8

10

12

0

J0

J5

FIG. 2: Impact of energy in higher order modes (eg J1) on the zeroth mode illustrating how the

original equilibria are moved outside of the libration state to the rotation state.

With this in mind, the minimum activation energy becomes the amount of energy needed

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from the higher mode(s) to pull the separatrix over the original equilibrium location 0. The

activation condition is

U(0 = 0.0, J) =U(0 = 0eq , J) (7)which defines the minimum activation energy for transition. Performing the necessary cal-

culations, the resulting activation condition for the 2-4 potential becomes

Mi=1

Jii

=N

6(8)

which describes an affine hyperplane with a dimension equal to the number of nonzero

actions. It is evident in Figure 2 that with further increase of action in the higher modes,

the concept of having two wells disappears, and the zeroth mode will wildly circle both

original equilibria in a state of rotation (with dissipation the masses will settle in one of the

wells). Indeed a pitchfork bifurcation exists for these high energy orbits at

Mi=1

Jii

= N3

which is twice the minimum activation energy.

To calculate the full activation time vs. activation energy profile we note that the when

the oscillation of the zeroth mode is in its rotation state (which is needed for activation), the

activation time is the time in which the trajectory crosses the origin (ie 0 = 0.0). In fact,

since the phase space for the rotation state is symmetrical this becomes 1/4 of the period of

a rotation state. We can analytically derive the rate of activation by calculating the period

of rotation oscillations of the zeroth mode in (6) as it is parametrically altered by higher

wavelength actions. To calculate the period of rotation we generalize the Hamiltonian of the

averaged dynamics as

H = J20

2

a202

+b40

4

(9)

a =

3Ji

iN

, b =

N

This system admits an analytical solution that is different inside and outside of the figureeight separatrix, each of which uses Jacobi elliptic functions. On the outside of the separatrix

we calculate the period as

Tout =4K()

out(10)

where =0(0)

b

2(a + b0(0)2)

, out =

a

22 1

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where is the elliptic modulus and K(k) is the complete elliptic integral of the first kind.

The variable 0(0) is the amplitude of the initial condition of the zeroth mode.

For comparison, numerical data was generated using N = 100 oscillators, B = 1, =

0.001, and = 1. The dynamics were integrated in time using a 4th order symplectic method

[10] using a fixed stepsize of 0.005. The analytical expression captures numerical data well

when the activation rates exceed the modal frequencies. The discrepancy in lowest mode at

high activation rates is because the modal frequency (of which we average over) approaches

the activation rate itself.

0 100 200 300 400 500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

U(0) / U

Activation

Rate[1/s]

Mode 1Mode 2

Mode 3Mode 4

Mode 5Mode 6

Mode 7Mode 8

Mode 9Mode 10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

1

2

3

4

5

6

7

Mode Frequency

MinimumA

ctivationEnergy

Data

Analytical

FIG. 3: Activation energy vs rate (top) and minimum activation energy (bottom). Analytical -

solid, numerical simulation - discrete points. U = N4

is the sum of the barrier heights and U(0)

is the initial energy.

The activation behavior discussed above is clearly impacted by selective modal energies.

The same dynamics were numerically analyzed in a stochastic setting using the Langevin

framework to determine if the phenomenon persists. In Figure 4 we illustrate the stochastic

rates using the same parameters as before with a Langevin damping value of 0 .05. In this case

we impose an initial energy at a constant value of 5U in different spatial patterns including

a randomly selected series of positions. The data is simulated for different temperatures

(Boltzmanns constant k = 1.0) and compared to Kramers estimate for low damping (see

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equation 4.49 of [9]). We find that at higher temperatures, Kramers theory predicts the

behavior well, while at lower temperatures inertial effects dominate and the numerical data

diverges from Kramers estimates. In addition, we find that this divergence is related to the

spatial character of the perturbation.

1 2 3 4 5 6 7 8

103

102

U/ (kT)

ActivationRate[1/s]

0th Mode

1st Mode

2nd Mode

3rd Mode

Random

Kramers

FIG. 4: Stochastic simulation with targeted perturbation in the first 4 Fourier modes compared to

a random initial condition. These rates are compared to Kramers estimate.

Above we have been able to predict a static relationship between initial energy and

activation time. However, the transfer of energy from modes which contain initial energy

(higher modes) to the zeroth mode is a dynamical process. The equipartition of energythroughout modes in high dimensional systems is expected due to their ergodic nature but

is not always the case. As shown initially by Fermi, Pasta, and Ulam (FPU) [11] this

balancing of energy may not occur for even relatively simple dynamics. In fact, Ford later

illustrated that this lack of energy balance is due to a lack of commensurate temporal

frequencies in the set of spatial modes [12]. As we have mentioned, the frequencies of the

linear normal modes in our model are incommensurate. Without any nonlinear perturbation

to the backbone, we expect there to be no energy transfer at all. However as we have shown,

there is no doubt that energy transfer occurs because activation can be observed by global

motions of the zeroth mode and this can be realized by imposing initial energy on modes of

high wavelength. In fact, the energy does not lead to equipartition immediately, it follows

a unidirectional route to the lowest mode.

By calculating the Jacobian of Hamiltons equations of (4) we obtain a matrix that reveals

the change in action variables as influenced by other actions and angles. It turns out that

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the change in action is mostly dependent on other actions (as opposed to angles) and so

we present this submatrix in Figure 5. We see in this figure that the change in action in

FIG. 5: Absolute value of the time averaged Jacobian of action variables illustrating the funneling

of energy towards the lowest mode.

any particular mode is dominated by influences of modes of equal or higher wavenumber

only. This natural funneling of energy is remarkable and strengthens the robustness of the

transition process. In fact, the funneling concept is associated with the horizontal-verticalgraph theoretic structure of the dynamics which has been shown to be a way to characterize

the robustness of a dynamical system (see reference [13]).

We have found that even though this high dimensional system uses a complex nonlinear

process to achieve global transition between different states, the qualitative nature is easily

understood as a funneling of energy from higher modes to the lowest mode whos phase

space is manipulated by these higher modes. In fact the dynamics in the averaged sense

are enough to predict the activation process. These results were compared in the stochastic

setting and it is shown that the effect of targeted perturbation is an accelerant with respect

to Kramers theory and the spatial character of the perturbation effects the divergence from

Kramers estimate.

This work was supported in part by DARPA DSO under AFOSR contract FA9550-07-C-

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0024 and AFOSR Grant FA9550-06-1-0088

[1] I. Mezic, PNAS 103 (2006).

[2] M. G. Forest, C. G. Goedde, and A. Sinha, Physical Review Letters pp. 27225 (1992).

[3] N. Ercolani, M. G. Forest, and D. W. McLaughlin, Physica D Nonlinear Phenomena pp.

34984 (1990).

[4] B. Eisenhower and I. Mezic, Proc. of the 46th IEEE CDC (2007).

[5] P. DuToit, I. Mezic, and J. Marsden, Physica D (in print) (2009).

[6] S. Lee and W. Sung, Physical Review E 6302 (2001), ISSN 1063-651X.

[7] S. Fugmann, D. Hennig, L. Schimansky-Geier, and P. Hanggi, Physical Review E 77 (2008),

ISSN 1539-3755.

[8] D. Hennig, S. Fugmann, L. Schimansky-Geier, and P. Hanggi, Acta Physica Polonica B 39

(2008).

[9] P. Hanggi, P. Talkner, and M. Borkovec, Reviews of Modern Physics 62, 251 (1990).

[10] H. Yoshida, Physics Letters A 150 (1990).

[11] E. Fermi, J. Pasta, and S. Ulam, Los Alamos report LA-1940 (1955).

[12] J. Ford, Journal of Mathematical Physics 2 (1961).

[13] I. Mezic, 43rd IEEE Conference on Decision and Control (2004).

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Bryan Eisenhower and Igor Mezic- Targeted Activation in Deterministic and Stochastic Systems