Relationship between dynamical instability and transport ...wchaos11/abstracts/akimoto_wchaos11.pdf · Introduction In nite Ergodic Theory Deterministic Subdi usion Conclusion Equilibrium

. . . . . .

Introduction Infinite Ergodic Theory Deterministic Subdiffusion Conclusion

Relationship between dynamicalinstability and transport coefficient in

anomalous subdiffusion

Takuma Akimoto

Keio university, 3-14-1 Hiyoshi Kohoku-ku Yokohama, Kanagawa223-8522, Japan

July, 15, 2011

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Outline

Introduction

Infinite Ergodic Theory

Deterministic Subdiffusion

Conclusion

. . . . . .


Equilibrium and non-equilibrium states

• Equilibrium state (dT = 0)

• Non-equilibrium steady state (0 < dT < Tc)- conduction

• Non-equilibrium non-steady state (Tc < dT )- turbulence- approach to an equilibrium (dT = 0, T → T ′)

T + dT

T

fluid

. . . . . .


Key fact in statistical mechanics

• Equilibrium state: intensive macroscopic observables,such as temperature and density, do not depend on a partof a system

• Non-equilibrium steady state: intensive macroscopicobservables, such as temperature and density, depend on apart of system

• Non-equilibrium non-steady state: macroscopicobservables are intrinsically random

Key fact

Macroscopic observables result from the time averages ofmicroscopic observation functions

. . . . . .


Equilibrium and non-equilibrium states in dynamicalsystems

• Equilibrium state: time average does not depend on apart i of a system

1

n

n−1∑k=0

fi ◦ T k → 〈f〉 as n → ∞

• Non-equilibrium steady state: time average exists butdepend on a part i of a system

1

n

n−1∑k=0

fi ◦ T k → 〈fi〉 as n → ∞

• Non-equilibrium non-steady (non-stationary) state:time average is random (converge in distribution)

1

an

n−1∑k=0

f ◦ T k L→〈f〉 as n → ∞

. . . . . .


Ergodicity in non-equilibrium non-stationary state

• Deterministic diffusion, T : R → RAlthough an invariant measure of T (x) can not be normalized(infinite measure), the reduced map of T (x), R(x), has aprobability measure.

0 1 2

1

2

3

x

T(x) 1

0 1

R(x

)

x

. . . . . .


Distributional limit theorem in diffusion process

Distribution of occupation times [D. A. Darling and M.Kac (1957)]Let x(t) be the two dimensional Brownian motion, x(0) = 0, and letV (x) be the characteristic function of a bounded plane set B ofnonzero Lebesgue measure.

limt→∞

Pr

{C

ln t

∫ t

0

V (x(τ ))dτ < x

}= 1 − e−x.

In a general Markoff process,

limt→∞

Pr

{C

tα

∫ t

0

V (x(τ ))dτ < x

}= Gα(x),

where Gα(x) is the Mittag-Leffler distribution.

. . . . . .


Random diffusion coefficients in experiments

- I. Golding and E. Cox, PRL, 96, 098102 (2006)- A. Graneli et al., PNAS, 103, 1221 (2006)

Time-averaged mean square displacement (TAMSD)

δx(t)2 ≡1

T − t

∫ T−t

0

[x(t′ + t) − x(t′)]2dt′

Diffusion coefficients are intrinsically random.

. . . . . .


Purpose

Foundation of ergodicity in non-equilibriumnon-stationary phenomena

• Distributional limit theorems in deterministic anomalousdiffusion (Infinite Ergodic theory)

• Relation between dynamical instabilities and transportcoefficients

. . . . . .


Infinite Ergodic Theory

. . . . . .


Infinite measure

ergodicity

A dynamical system (T,X,B, µ) is ergodic if

T−1A = A,A ∈ B ⇒ µ(A) = 0 or µ(Ac) = 0

Infinite measure: µ(X) = ∞

• Intermittent map (z ≥ 2)

T (x) =

x + 2z−1xz x ∈ (0, 1/2)

2x − 1 x ∈ (1/2, 1)

ρ(x) ∝ x1−z (x → 0)

. . . . . .


Distributional limit theorems

Time average of an observation function f(x) converges indistribution:

1

an

n−1∑k=0

f ◦ T k L→Yα as n → ∞.

Distribution of a random varianble Yα

• f ∈ L1+(µ)1: Mittag-Leffler distribution (an ∝ nα)

[J. Aaronson, J. D’Analyse Math. (1981)]

• f /∈ L1(µ): Generalized arc-sine or stable distribution[M. Thaler (1998), M. Thaler and R. Zweimuller (2006), T.Akimoto (2008)]

1f ∈ L1(µ) means∫X

|f |dµ < ∞.

. . . . . .


Dynamical instability

Lyapunov exponent

λ ≡ limn→

1

n

n−1∑k=0

ln |T ′(xk)| ⇒ ∆x(t) ∼ ∆x(0) exp(λt)

Subexponential instability

∆x(t) ∼ ∆x(0) exp(λαtα)

ln |T ′(x)| ∈ L1+(µ) ⇒ λα is a random variable.

λα(n) ≡1

an

n−1∑k=0

ln |T ′(xk)|L→Mα

[N. Korabel and E. Barkai, PRL (2009), T. Akimoto and Y.Aizawa, Chaos (2010)]

. . . . . .


Deterministic Subdiffusion

. . . . . .


Dynamical systems

-2

-1

0

1

2

-1 0 1

T(x)

x

0

0R(x)

x

1/2 -1/2

1/2

-1/2 c -1/4 1/4

Reduced map, R(x) = x + 4z−1xz, x ∈ (0, 1/4), has aninfinite invariant measure µR (z ≥ 2).

c: asymmetric parameter (1/4 ≤ c ≤ 1/2)

c = 0.375 corresponds to an unbiased random walk

. . . . . .


Consctruction of Continuous Time Random Walk

0

R(x)

x +1 -1 +1 -1

f(x) =

−1 x ∈ (−1/2,−c]+1 x ∈ (−c,−1/4)0 x ∈ [−1/4, 1/4]−1 x ∈ (1/4, c]+1 x ∈ (c, 1/2]

. . . . . .


Time-averaged mean square displacement

Time-averaged mean square displacement (TAMSD)

δx2m ≡

1

an

n−1∑k=0

(xk+m − xk)2

Because fm(x) ≡ (Tm(x) − x)2 ∈ L1+(µR), we have

1

an

n−1∑k=0

fm ◦ T k L→〈fm〉Mα as n → ∞,

where Mα is a random variable with Mittag-Leffler distribution.In an unbiased case (c = 0.375), 〈fm〉 ∝ m (m � 1), i.e.,

δx2m = Dm for m � 1 (N → ∞)

. . . . . .


Time-averaged drift

Time-averaged drift

δxm =1

an

n−1∑k=0

(xk+m − xk)

Although vm(x) ≡ Tm(x) − x/∈L1+(µR), we have

1

an

n−1∑k=0

vm ◦ T k L→〈vm〉Mα as n → ∞.

In a biased case (c 6= 0.375), 〈vm〉 ∝ m (m � 1), i.e.,

δxm = V m for m � 1 (N → ∞)

. . . . . .


Generalized Lyapunov exponent and Hopf’s ergodictheorem

By ln |T ′(x)| ∈ L1+(µR)

λ ≡1

an

n−1∑k=0


D,V and λ converge in distribution to a Mittag-Lefflerdistribution:

D,V , λL→Mα

However, by Hopf’s ergodic theorem,

D(x)

λ(x)→ χD and

V (x)

λ(x)→ χV

do not depend on an initial point x and converge to a constant.

. . . . . .


Generalized Lyapunov exponent and Hopf’s ergodictheorem

By ln |T ′(x)| ∈ L1+(µR)

λ ≡1

an

n−1∑k=0


D,V and λ converge in distribution to a Mittag-Lefflerdistribution:

D,V , λL→Mα

However, by Hopf’s ergodic theorem,

D(x)

λ(x)→ χD and

V (x)

λ(x)→ χV

do not depend on an initial point x and converge to a constant.

. . . . . .


Susceptibility

0.2

0.22

0.24

0.26

0.28

0.3

0.32

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4

χ D

z

0.25

0.3

0.35

0.4

0.45

0.5

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4

χ V

z

Figure: Left: susceptibility χD (c = 0.375 and n = 106). Right:susceptibility χV (c = 0.35 and n = 106).

. . . . . .


Einstein relation

Linear response to a bias

〈V 〉F = ε〈D〉,

where 〈·〉F is an ensemble average under a bias ε = 3 − 8c.

Two assumptions:

• an injection to the set Jc is uniform2

• 〈∆n(k)〉F ∼= 〈∆n(k)〉, where ∆n(k) is the total numberof jumps in [k, k + m]

Einstein relationp = W0e

−F/2kBT ∼= 2 − 4c, q = W0e+F/2kBT ∼= 4c − 1

〈V 〉F =〈D〉F2kBT

2J = [−1/4, 1/4], x ∈ Jc is mapped to the neighboring cell.

. . . . . .


Einstein relation


〈V 〉F = ε〈D〉,


Two assumptions:




−F/2kBT ∼= 2 − 4c, q = W0e+F/2kBT ∼= 4c − 1



. . . . . .


Einstein relation


〈V 〉F = ε〈D〉,


Two assumptions:




−F/2kBT ∼= 2 − 4c, q = W0e+F/2kBT ∼= 4c − 1



. . . . . .


Einstein relation

-1

-0.5

0

0.5

1

1 0.5 0 -0.5 -1

V

ε

. . . . . .


Relation between dynamical instability and transportcoefficient

Difference of the generalized Lyapunov exponent

∆λ ≡ 〈λ〉 − 〈λ〉F


∆λ =µR(Jc)

2

{(1 +

V

Vmax

)ln

(1 +

V

Vmax

)+

(1 −

V

Vmax

)ln

(1 −

V

Vmax

)},

where V ≡ 〈xn〉/n1/(z−1), Vmax is V for c = 0.25, and µR

is an invariant measure of R(x).

. . . . . .



Difference of the generalized Lyapunov exponent

∆λ ≡ 〈λ〉 − 〈λ〉F


∆λ =µR(Jc)

2

{(1 +

V

Vmax

)ln

(1 +

V

Vmax

)+

(1 −

V

Vmax

)ln

(1 −

V

Vmax

)},

where V ≡ 〈xn〉/n1/(z−1), Vmax is V for c = 0.25, and µR

is an invariant measure of R(x).

. . . . . .



0

0.2

0.4

0.6

0.8

1

1.2

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

∆λ

V

. . . . . .


Conclusion

Intrinsic random behavior innon-equilibrium non-stationary phenomena

is established by infinite ergodic theory.

• Random diffusion coefficient and drift are characterized bya Mittag-Leffler distribution.

• Relation between the difference of the generalizedLyapunov exponent and drift is obtained.

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Relationship between dynamical instability and transport ...wchaos11/abstracts/akimoto_wchaos11.pdf · Introduction In nite Ergodic Theory Deterministic Subdi usion Conclusion Equilibrium