Cascade Amplification of Fluctuations Michael Wilkinson, Marc - - PowerPoint PPT Presentation

Cascade Amplification of Fluctuations

• Michael Wilkinson, Marc Pradas
• Department of Mathematics and Statistics,

The Open University, Walton Hall, Milton Keynes, MK7 6AA, England

• Robin Guichardaz, Alain Pumir
• Laboratoire de Physique, Ecole Normale Superieure de Lyon,

F-69007, Lyon, France

An unsolved problem?

Anomalous diffusion is widely observed. It is often ‘explained’ by postulating another power-law in the equation of motion (for example, a waiting-time distribution).

• However, fundamental physical laws do not contain non-

integer exponents. It is desirable to find more mechanisms where power-laws emerge naturally. This talk describes a new source of non-integer power-laws.

hx2i ⇠ tα , α 6= 1

Our investigation

˙ x = v(x, t) + √ 2Dη(t)

Separations of nearby particles in complex flows were investigated, with thermal noise: We consider a case with negative Lyapunov exponent. Without noise, nearby trajectories coalesce. We expect that, with noise, an Ornstein-Uhlenbeck model is applicable. This implies a Gaussian distribution of separations.

hη(t)i = 0

hη(t)η(t0)i = δ(t t0)

λ < 0

∆ ˙ x = λ ∆x + √ 2D η(t)

P∆x = C exp ✓ −|λ|∆x2 4D ◆

Small particles in turbulent flows

200 400 600 800 1000

• 0.5

0.5 1 1.5

200 400 600 800 1000

• 0.5

0.5 1 1.5

˙ x = v ˙ v = γ[u(x, t) − v]

Equations of motion for small particles (1-d model): For sufficiently large damping the paths of particles coalesce: we are interested in this case where the Lyapunov exponent is negative: Z(t) = δ ˙ x δx λ = lim

t!1

1 t Z t dt0 Z(t0)

Effects of noise

˙ x = v + √ 2Dη(t) ˙ v = γ[u(x, t) − v] hη(t)i = 0 hη(t)η(t0)i = δ(t t0)

Add Brownian diffusion to the model: The distribution of separations of particles was found to be non-Gaussian. It has well-defined power-law tails:

P(∆x) ∼ |∆x|−(1+α) , α > 0

Power-law distribution

P∆x ∼ |∆x|−(1+α)

The probability density of separations is a power-law: the exponent depends upon the damping coefficient:

0.1 x0.985

a

x P(x) 0.1 1 10 100 P( x) x1.34

b

104 103 102 101 x

Intermittency

5000 10000 15000 1 1 2 3 4 5 Time x 2500 5000 7500 10000 12500 15000 0.5 0.5 Time x

0.5 0.5 x 0.5 0.5 x

Numerical experiments also show that the particle separations are intermittent, with occasional large excursions.

Cascade amplification of noise

δ ˙ x = Z(t)δx + 2 √ Dη(t) Z(t) = ∂v ∂x(x(t), t) Y = ln(∆x) PY ∼ exp(−αY ) P∆x ∼ |∆x|−(1+α)

The instantaneous Lyapunov exponent is negative most of the time, but has occasional positive excursions, with frequency independent of the particle separation. Scale invariance indicates a logarithmic variable: Linearised equation of motion for particle separations:

Equation for the exponent

∂ ∂Z  (γZ + Z2) + Dγ2 ∂ ∂Z

• ρ(Z) + αZρ(Z) = 0

Seek a joint PDF in the form For short correlation time, instantaneous Lyapunov exponent has equation of motion:

˙ Z = −γZ − Z2 + √ 2D ζ(t)

P(Y, Z) = exp(αY )ρ(Z)

The exponent satisfies a Fokker-Planck equation and eigenvalue condition (see arXiv:1502:05855): Z ∞

−∞

dZ Z ρ(Z) = 0

g(∆x) ∼ |∆x|D2−1 α = −D2

Negative fractal dimensions

0.5 1 1.5 2 2.5 3 1 0.5 0.5 1

• When the Lyapunov exponent is

positive, the pair correlation function is a power-law, with exponent defining the correlation dimension: This corresponds to the expression for the particle separation due to Brownian motion if the exponent is a negative fractal dimension:

Summary

• The separation of particles due to Brownian

fluctuations shows intermittency, and a power-law distribution.

• This is a consequence of a cascade amplification

effect: there are episodes of instability which multiply the particle separation.

• The effect is very general, and it is observed in other

variables (e.g. distributions of angles describing shapes of constellations of particles).

• The effect provides a new route to explain some types
• f anomalous diffusion, intermittency, and an

interpretation of negative fractal dimensions.