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
Outline Introduction: 1 / f noise 1 Particular model of 1 / f noise: point process 2 Time-subordinated Langevin equations 3 Summary 4 Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 2 / 29
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 occasionally called “flicker noise” or “pink noise” Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 3 / 29
1 / f noise First observed (in 1925) by Johnson in vacuum tubes. Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 4 / 29
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
1 / f noise Many mathematical models: Superposition of relaxation processes � γ 2 γ 2 + ω 2 ❞ γ ≈ π N N S ( f ) = 2 ω , γ 1 ≪ ω ≪ γ 2 γ 1 Dynamical systems at the edge of chaos x n + 1 = x n + x z mod 1 n Alternating fractal renewal process Self-Organized Criticallity Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 6 / 29
Particular model of 1 / f noise: point process The signal of the model consists of pulses or events � I ( t ) = a δ ( t − t k ) k 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
Correlated inter-pulse durations Inter-pulse durations perform a random walk: τ k + 1 = τ k ± σ 10 4 10 3 10 2 10 1 S(f) 10 0 10 -1 10 -2 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 f Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 8 / 29
Correlated inter-pulse durations The spectrum is S ( f ) = ν f P τ ( τ ♠✐♥ ) in the frequency range � � σ 2 σ 2 1 ≪ f ≪ min , τ 3 τ ♠✐♥ τ 2 τ ♠❛① ♠❛① ♠❛① where P τ ( τ ) is the PDF of inter-pulse durations and � σ 2 = P ( τ k | τ k − 1 )( τ k − τ k − 1 ) 2 ❞ τ k Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 9 / 29
Point processes More general equation τ k + 1 = τ k + γτ 2 µ − 1 + στ µ k ε k k 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
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
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
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
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
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 or 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
Time-subordinated Langevin equations We consider the situation when the increments of the physical time are deterministic ❞ x τ = F ( x τ ) ❞ τ + ❞ W τ ❞ t τ = g ( x τ ) ❞ τ One can reduce the system of equations to a single equation in physical time with a multiplicative noise ❞ x t = F ( x t ) 1 g ( x t ) ❞ t + ❞ W t � g ( x t ) Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 15 / 29
Only positive values of x We choose the function g ( x ) as a power-law function of x : ❞ t τ = x − 2 η ❞ τ A simple Brownian motion ❞ x τ = ❞ W τ restricted to a interval between x ♠✐♥ and x ♠❛① leads to the equation in the physical time ❞ x t = x η t ❞ W t Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 16 / 29
Bessel process A Bessel process � 1 � η − λ ❞ x τ = ❞ τ + ❞ W τ 2 x τ leads to the equation in the physical time � η − λ � x 2 η − 1 ❞ t + x η ❞ x t = t ❞ W t t 2 Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 17 / 29
Geometric Brownian motion Geometric Brownian motion � � η − λ ❞ x τ = x τ ❞ τ + x τ ❞ W τ 2 together with the relation between the internal time and the physical time ❞ t τ = x − 2 ( η − 1 ) ❞ τ leads to the same equation in the physical time � η − λ � x 2 η − 1 ❞ t + x η ❞ x t = t ❞ W t t 2 Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 18 / 29
Numerical example 10 1 60 6 10 0 10 -1 40 4 τ /10 3 10 -2 x S(f) 10 -3 2 20 10 -4 10 -5 0 0 10 -1 10 0 10 1 10 2 10 3 10 4 0 0.5 1 1.5 2 2.5 3 t f Generated signal (red line) Spectrum of the signal (red curve). together with the corresponding Blue line shows the slope 1 / f internal time (blue line). The parameters are η = 5 / 2 and λ = 3 Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 19 / 29
Nonlinear SDEs � η − λ � x 2 η − 1 ❞ t + x η ❞ x t = t ❞ W t t 2 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
Estimation of spectrum from scaling properties � � η − λ x 2 η − 1 ❞ t + x η ❞ x t = t ❞ W t t 2 Steady state PDF has power-law form P 0 ( x ) ∼ x − λ The change of the magnitude of the stochastic variable x → ax is equivalent to the change of time scale t → a 2 ( η − 1 ) t . Trasnsition probability has a scaling property P ( ax ′ , t | ax , 0 ) = a − 1 P ( x ′ , a 2 ( η − 1 ) t | x , 0 ) Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 21 / 29
Estimation of spectrum from scaling properties Autocorrelation function can be written as � � ❞ x ′ xx ′ P 0 ( x ) P x ( x ′ , t | x , 0 ) C ( t ) = ❞ x The autocorrelation function C ( t ) has scaling property C ( at ) ∼ a β − 1 C ( t ) with λ − 3 β = 1 + 2 ( η − 1 ) Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 22 / 29
Both positive and negative values of x The Ornstein-Uhlenbeck process ❞ x τ = − γ x τ ❞ t + ❞ W τ The relation between the internal time and the physical time 1 ❞ t = 0 ) η ❞ τ τ + x 2 ( x 2 Resulting nonlinear SDE in physical time ❞ x t = − γ ( x 2 t + x 2 0 ) η x t ❞ t + ( x 2 t + x 2 η 2 ❞ W t 0 ) Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 23 / 29
Both positive and negative values of x Equation � η − λ � x τ ❞ x τ = ❞ τ + ❞ W τ τ + x 2 2 x 2 0 leads to SDE in the physical time � η − λ � ( x 2 t + x 2 0 ) η − 1 x t ❞ t + ( x 2 t + x 2 η ❞ x t = 0 ) 2 ❞ W t 2 Julius Ruseckas (Lithuania) Time-subordinated Langevin equations July 15, 2015 24 / 29
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