1 / f noise arising from time-subordinated Langevin equations - - PowerPoint PPT Presentation

▶
1 f noise arising from time subordinated langevin
SMART_READER_LITE
LIVE PREVIEW

1 / f noise arising from time-subordinated Langevin equations - - PowerPoint PPT Presentation

Nov 07, 2023 264 likes •595 views

1 / f noise arising from time-subordinated Langevin equations Julius Ruseckas and Bronsilovas Kaulakys Institute of Theoretical Physics and Astronomy, Vilnius University, Lithuania July 15, 2015 Julius Ruseckas (Lithuania) Time-subordinated

slide-1
SLIDE 1

1/f noise arising from time-subordinated Langevin equations

Julius Ruseckas and Bronsilovas Kaulakys

Institute of Theoretical Physics and Astronomy, Vilnius University, Lithuania

July 15, 2015

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 1 / 29

slide-2
SLIDE 2

Outline

1

Introduction: 1/f noise

2

Particular model of 1/f noise: point process

3

Time-subordinated Langevin equations

4

Summary

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 2 / 29

slide-3
SLIDE 3

What is 1/f noise?

1/f noise

a type of noise whose power spectral density S(f) behaves like S(f) ∼ 1/f β , β is close to 1

  • ccasionally called “flicker noise”
  • r “pink noise”

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 3 / 29

slide-4
SLIDE 4

1/f noise

First observed (in 1925) by Johnson in vacuum tubes.

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 4 / 29

slide-5
SLIDE 5

1/f noise

Fluctuations of signals exhibiting 1/f behavior of the power spectral density at low frequencies have been observed in a wide variety of physical, geophysical, biological, financial, traffic, Internet, astrophysical and other systems.

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 5 / 29

slide-6
SLIDE 6

1/f noise

Many mathematical models: Superposition of relaxation processes S(f) = γ2

γ1

N γ2 + ω2 ❞γ ≈ πN 2ω , γ1 ≪ ω ≪ γ2 Dynamical systems at the edge of chaos xn+1 = xn + xz

n

mod 1 Alternating fractal renewal process Self-Organized Criticallity

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 6 / 29

slide-7
SLIDE 7

Particular model of 1/f noise: point process

The signal of the model consists of pulses or events I(t) = a

  • k

δ(t − tk) Point processes arise in different fields such as physics, economics, ecology, neurology, seismology, traffic flow, financial systems and the Internet.

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 7 / 29

slide-8
SLIDE 8

Correlated inter-pulse durations

Inter-pulse durations perform a random walk: τk+1 = τk ± σ 10-2 10-1 100 101 102 103 104 10-5 10-4 10-3 10-2 10-1 100 S(f) f

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 8 / 29

slide-9
SLIDE 9

Correlated inter-pulse durations

The spectrum is S(f) = ν f Pτ(τ♠✐♥) in the frequency range σ2 τ 3

♠❛①

≪ f ≪ min

  • σ2

τ♠✐♥τ 2

♠❛①

, 1 τ♠❛①

  • where Pτ(τ) is the PDF of inter-pulse durations and

σ2 =

  • P(τk|τk−1)(τk − τk−1)2 ❞τk

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 9 / 29

slide-10
SLIDE 10

Point processes

More general equation τk+1 = τk + γτ 2µ−1

k

+ στ µ

k εk

Allows to obtain power-law exponent β in the spectrum different from 1. Used for modeling of the internote interval sequences of the musical rhythms

  • D. J. Levitin, P

. Chordia, and V . Menon, Proc. Natl. Acad. Sci. U.S.A. 109, 3716 (2012).

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 10 / 29

slide-11
SLIDE 11

Conclusion One of possible origins of 1/f noise Brownian motion in time axis leads to 1/f noise

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 11 / 29

slide-12
SLIDE 12

Question Can this way to 1/f noise be applied not only to a sequence of pulses?

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 12 / 29

slide-13
SLIDE 13

The main idea

In a sequence of pulses the pulse number can be interpreted as an internal time. Start from a stochastic differential equation Interpret the time as an internal parameter. Add an additional equation relating the physical time to the internal time. Increments of the physical time should be a power-law function of the magnitude of the signal.

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 13 / 29

slide-14
SLIDE 14

The main idea

In a sequence of pulses the pulse number can be interpreted as an internal time. Start from a stochastic differential equation Interpret the time as an internal parameter. Add an additional equation relating the physical time to the internal time. Increments of the physical time should be a power-law function of the magnitude of the signal.

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 13 / 29

slide-15
SLIDE 15

Why two times?

Impurities and regular structures in a medium results in a transport of variable speed, the particle may be trapped for some time or accelerated. The waiting time can depend on the particle position

  • r on the intensity of the signal.

Example: a diffusion on fractals and multifractals.

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 14 / 29

slide-16
SLIDE 16

Time-subordinated Langevin equations

We consider the situation when the increments of the physical time are deterministic ❞xτ =F(xτ)❞τ + ❞Wτ ❞tτ =g(xτ)❞τ One can reduce the system of equations to a single equation in physical time with a multiplicative noise ❞xt = F(xt) g(xt)❞t + 1

  • g(xt)

❞Wt

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 15 / 29

slide-17
SLIDE 17

Only positive values of x

We choose the function g(x) as a power-law function of x: ❞tτ = x−2η❞τ A simple Brownian motion ❞xτ = ❞Wτ restricted to a interval between x♠✐♥ and x♠❛① leads to the equation in the physical time ❞xt = xη

t ❞Wt

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 16 / 29

slide-18
SLIDE 18

Bessel process

A Bessel process ❞xτ =

  • η − λ

2 1 xτ ❞τ + ❞Wτ leads to the equation in the physical time ❞xt =

  • η − λ

2

  • x2η−1

t

❞t + xη

t ❞Wt

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 17 / 29

slide-19
SLIDE 19

Geometric Brownian motion

Geometric Brownian motion ❞xτ =

  • η − λ

2

  • xτ❞τ + xτ❞Wτ

together with the relation between the internal time and the physical time ❞tτ = x−2(η−1)❞τ leads to the same equation in the physical time ❞xt =

  • η − λ

2

  • x2η−1

t

❞t + xη

t ❞Wt

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 18 / 29

slide-20
SLIDE 20

Numerical example

20 40 60 0.5 1 1.5 2 2.5 3 2 4 6 x τ/103 t

Generated signal (red line) together with the corresponding internal time (blue line). The parameters are η = 5/2 and λ = 3

10-5 10-4 10-3 10-2 10-1 100 101 10-1 100 101 102 103 104 S(f) f

Spectrum of the signal (red curve). Blue line shows the slope 1/f

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 19 / 29

slide-21
SLIDE 21

Nonlinear SDEs

❞xt =

  • η − λ

2

  • x2η−1

t

❞t + xη

t ❞Wt

This nonlinear SDE has been proposed in

  • B. Kaulakys and J. Ruseckas, Phys. Rev. E 70, 020101(R) (2004).
  • B. Kaulakys and J. Ruseckas, V

. Gontis, and M. Alaburda, Physica A 365, 217 (2006).

Such nonlinear SDEs have been used to describe signals in socio-economical systems

V . Gontis, J. Ruseckas and A. Kononovicius, Physica A 389, 100 (2010).

  • J. Mathiesen, L. Angheluta, P

. T. H. Ahlgren and M. H. Jensen, Proc. Natl. Acad. Sci. 110, 17259 (2013).

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 20 / 29

slide-22
SLIDE 22

Estimation of spectrum from scaling properties

❞xt =

  • η − λ

2

  • x2η−1

t

❞t + xη

t ❞Wt

Steady state PDF has power-law form P0(x) ∼ x−λ The change of the magnitude of the stochastic variable x → ax is equivalent to the change of time scale t → a2(η−1)t. Trasnsition probability has a scaling property P(ax′, t|ax, 0) = a−1P(x′, a2(η−1)t|x, 0)

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 21 / 29

slide-23
SLIDE 23

Estimation of spectrum from scaling properties

Autocorrelation function can be written as C(t) =

  • ❞x
  • ❞x′ xx′P0(x)Px(x′, t|x, 0)

The autocorrelation function C(t) has scaling property C(at) ∼ aβ−1C(t) with β = 1 + λ − 3 2(η − 1)

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 22 / 29

slide-24
SLIDE 24

Both positive and negative values of x

The Ornstein-Uhlenbeck process ❞xτ = −γxτ❞t + ❞Wτ The relation between the internal time and the physical time ❞t = 1 (x2

τ + x2 0)η ❞τ

Resulting nonlinear SDE in physical time ❞xt = −γ(x2

t + x2 0)ηxt❞t + (x2 t + x2 0)

η 2 ❞Wt Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 23 / 29

slide-25
SLIDE 25

Both positive and negative values of x

Equation ❞xτ =

  • η − λ

2

  • xτ

x2

τ + x2

❞τ + ❞Wτ leads to SDE in the physical time ❞xt =

  • η − λ

2

  • (x2

t + x2 0)η−1xt❞t + (x2 t + x2 0)

η 2 ❞Wt Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 24 / 29

slide-26
SLIDE 26

Both positive and negative values of x

  • 60
  • 40
  • 20

20 40 60 0.5 1 1.5 2 2.5 3 2 4 6 8 x τ/103 t

Generated signal (red line) together with the corresponding internal time (blue line). The parameters are η = 5/2 and λ = 3

10-5 10-4 10-3 10-2 10-1 100 101 10-1 100 101 102 103 104 S(f) f

Spectrum of the signal (red curve). Blue line shows the slope 1/f

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 25 / 29

slide-27
SLIDE 27

Numerical approach

Using internal time we can obtain an effective way of solving non-linear SDEs. For example, let us consider the non-linear SDE ❞xt =

  • η − λ

2

  • x2η−1

t

❞t + xη

t ❞Wt

We introduce operational time τ by the equation ❞τt = x2η

t ❞t

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 26 / 29

slide-28
SLIDE 28

Numerical approach

Using internal time we can obtain an effective way of solving non-linear SDEs. For example, let us consider the non-linear SDE ❞xt =

  • η − λ

2

  • x2η−1

t

❞t + xη

t ❞Wt

We introduce operational time τ by the equation ❞τt = x2η

t ❞t

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 26 / 29

slide-29
SLIDE 29

Numerical approach

Discretizing the internal time τ with the step ∆τ and using the Euler-Marujama approximation for the SDE we get xk+1 =xk +

  • η − λ

2 1 xk ∆τ + √ ∆τεk , tk+1 =tk + ∆τ x2η

k

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 27 / 29

slide-30
SLIDE 30

Summary

1/f noise can be obtained by introducing the difference between the internal time and the physical time and also assuming that the increments of the physical time have power-law dependence on the intensity of the signal This difference between physical and internal times can arise due to presence of traps or other impurities in an inhomogeneous medium Introduction of internal time can be an effective way to solve highly non-linear SDEs.

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 28 / 29

slide-31
SLIDE 31

Thank you for your attention!

Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 29 / 29