1/f noise from the nonlinear transformations of the variables

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1 Modern Physics Letters B Vol. 9, No. 34 (015) (6 pages) c World Scientiic Publishing Company DOI: /S / noise rom the nonlinear transormations o the variables Bronislovas Kaulakys, Miglius Alaburda and Julius Ruseckas Institute o Theoretical Physics and Astronomy, Vilnius University, A. Goštauto 1, Vilnius, Lithuania bronislovas.kaulakys@tai.vu.lt miglius.alaburda@tai.vu.lt julius.ruseckas@tai.vu.lt Received 30 December 014 Accepted 8 June 015 Published 15 December 015 The origin o the low-requency noise with power spectrum 1/ β (also known as 1/ luctuations or licker noise) remains a challenge. Recently, the nonlinear stochastic dierential equations or modeling 1/ β noise have been proposed and analyzed. Here, we use the sel-similarity properties o this model with respect to the nonlinear transormations o the variables o these equations and show that 1/ β noise o the observable may yield rom the power-law transormations o well-known standard processes, like the Brownian motion, Bessel and similar stochastic processes. Analytical and numerical investigations o such techniques or modeling processes with 1/ β luctuations is presented. Keywords: 1/ noise; stochastic dierential equations; nonlinear transormations. 1. Introduction Dierent theories and models have been proposed or explanation o the ubiquitous 1/ β noise phenomena, observable or about 80 years in dierent systems rom physics to inancial markets (see, e.g. Res. 1 8 and reerences herein). Recently, the stochastic model o 1/ β noise, based on the nonlinear stochastic dierential equations, ( dx = η λ ) x η 1 dt + x η dw t, (1) where x is the signal with 1/ β spectrum, η 1 is the nonlinearity exponent, λ is the exponent o the steady-state distribution P ss x λ and W t is a Wiener Corresponding author

2 B. Kaulakys, M. Alaburda & J. Ruseckas process (Brownian motion), has been proposed and analyzed. 6,7 The relation o the exponent β in the spectrum to the parameters o Eq. (1) isgivenby β =1+ λ 3 (η 1). () Equation (1) may be derived rom the point process model, 6 rom scaling properties o the signal 7 or rom the agent-based herding model. 9 Here, we employ the sel-similarity property o Eq. (1) with respect to the powerlaw transormations o the variable. We show that processes with 1/ β spectrum may yield rom the nonlinear transormations o the variable o the widespread processes, e.g. rom the Brownian motion, Bessel or similar amiliar processes.. Transormations From the scaling o the power spectral density (PSD) β, S(a) a β, (3) according to the Wiener Khintchine theorem yields the scaling o the autocorrelation unction, C(at) a β 1 C(t) (4) in some time interval 1/ max t 1/ min. Recently, 7 it was shown that rom the scaling (4), it ollows the nonlinear stochastic dierential equation (SDE) (1). Due to this scaling, the nonlinear SDE dx = ( η x λ x generating signal x with PSD 1, β βx x =1+ λ x 3 (η x 1) ater the nonlinear transormation ) x ηx 1 dt + x ηx dw t, (5) (6) x = 1 y δ, (7) with δ being the transormation exponent, yields SDE or the variable y o the same orm, ( dy = η y λ ) y x ηy 1 dt + y ηy dw t (8) with η y =1 δ(η x 1), λ y =1 δ(λ x 1). (9) Equation (8) generates signal y with the PSD 1 βy, β y =1+ λ y 3 (η y 1) = β x + 1+δ δ(η x 1). (10)

3 1/ noise rom the nonlinear transormations o the variables Thereore, 1/ βx noise o the observable x may yield rom the nonlinear dependence (7) o this observable on another variable y resulted rom simple or more common Eq. (8), where β x = β y + 1+δ η y 1. (11) 3. Examples The simplest relation between the variable y and the observable x is the inverse transormation x =1/y. From the interrelations (10) and(11) between the parameters β x and β y, we obtain that simple equation without the drit term, dy = 1 dw t (1) y results in 1/ noise o the observable x =1/y, instead o very nonlinear equation, 10 dx = x 4 dt + x 5/ dw t, (13) or the observable x. It should be noted that Eq. (1) coincides with equation or the interevent time, dτ = 1 dw t, (14) τ obtained rom the Brownian motion, dτ k = dw k (15) o the interevent time τ k in the events space or k-space. 10 Figures 1 and demonstrate the appearance o 1/ noise o the observable x, as a result o the inverse transormation x =1/y o the variable y generated by the simple equation without the drit term (1) with approximate 1/ PSD o the variable y. P(y) y Fig. 1. (Color online) Steady-state distribution,, and approximately 1/ PSD,, o the variable y generated by the simple equation (1), red circles, as well as o y =1/x generated by Eq. (13), blue triangles, with the appropriate restrictions 7 o the diusion intervals

4 B. Kaulakys, M. Alaburda & J. Ruseckas P(x) x Fig.. (Color online) Steady-state distribution,, and 1/ PSD,, o the observable x = 1/y, where the variable y is generated by Eq. (1), red circles, as well as o x generated by Eq. (13), blue triangles. Note, that special cases o Eq. (8) are: (i) an equation with the additive noise, η y = 0, and nonlinear drit, i.e. the Bessel process o order N =1 λ y, (ii) the order N =(1 λ y ) squared Bessel process when η y =1/ andλ y =1 N/, (iii) the special exponential restrictions o the variable y in Eq. (8) yield constant elasticity o variance (CEV) process or (iv) Cox Ingersoll Ross (CIR) process when η =1/. 9,11 Here, we will demonstrate the possibility to obtain 1/ β noise rom the Bessel process and rom the Brownian motion. Transormation (7) with δ =1/(η x 1), i.e. y = x 1 ηx, yields the special orm o Eq. (8), i.e. the Bessel equation, dy = N 1 dt y + dw t (16) corresponding to N-dimensional Wiener process, or Bessel process with index ν = N/ 1. Here, On the other hand, N = λ x 1 η x 1 =1 λ y. (17) λ x =1+ N δ, β x =1+ N δ. (18) Thus, the Bessel process or N-dimensional Brownian motion can cause 1/ β luctuations o the observable x as a unction (7) o this process. So, the pure 1/ noise yields when N =δ, e.g. rom 1D Brownian motion o y or the observable x =1/ y and D Brownian motion o y or the observable x =1/y. Figure 3 demonstrates examples o the signals o 1D Brownian motion o the variable y, the inverse 1D Brownian motion o x =1/y yielding in 1/ noise and observable x =1/ y resulting in pure 1/ noise. In Fig. 4, the distribution density and PSD o the corresponding observables are presented

5 1/ noise rom the nonlinear transormations o the variables y(t) t x(t) t (c) x(t) t Fig. 3. Examples o the signals o 1D Brownian motion o the variable y,,otheinverse 1D Brownian motion o the observable x =1/y yielding in 1/ noise,, and o x =1/ y observable resulting in pure 1/ noise, (c). P(x) x Fig. 4. (Color online) Steady-state distribution,, and PSD,, o 1D Brownian motion o the variable y, yielding 1/ noise, red circles, the inverse 1D Brownian motion o x =1/y yielding in 1/ noise, blue triangles, and o the observable x =1/ y resulting in pure 1/ noise, green squares. P(x) x Fig. 5. (Color online) Steady-state distribution,, and PSD,, o the observable x as inverse o the distance rom the beginning, x =1/r, o the Brownian motion in D. In Fig. 5, the pure 1/ noise o the observable x as inverse o the distance rom the beginning, x =1/r, o the Brownian motion in D, r = B1 (t)+b (t), (19) i.e. o the Bessel process with the index ν = 0 is shown. Here, B 1 (t) andb (t) are independent Brownian motions. We see the pure 1/ noise

6 B. Kaulakys, M. Alaburda & J. Ruseckas 4. Conclusions Thus, the proper unction o the widespread process, e.g. the inverse transormation o the order two Bessel process and the inverse square root o the Brownian motion yield the pure 1/ noise, observable, e.g. in condense matter. Possible relevancies o these transormations to 1/ noise in condensed matter: (i) 1/ luctuations o the voltage, U = I/G, at constant current I as a result o Brownian luctuations o the conductivity G, (ii) 1/ luctuations o resistivity, ρ =1/σ, as Brownian luctuations o the conductivity σ. Reerences 1. M. B. Weissman, Rev. Mod. Phys. 60 (1988) T. Gisiger, Biol. Rev. 76 (001) M. Li and W. Zhao, Math. Probl. Eng. 01 (01) A. A. Balandin, Nat. Nanotechnol. 8 (013) E. Paladino, Y. M. Galperin, G. Falci and B. L. Altshuler, Rev. Mod. Phys. 86 (014) B. Kaulakys and M. Alaburda, J. Stat. Mech. 009 (009) P J. Ruseckas and B. Kaulakys, J. Stat. Mech. 014 (014) P M. A. Rodriguez, Phys. Rev. E 90 (014) J. Ruseckas, B. Kaulakys and V. Gontis, Europhys. Lett. 96 (011) B. Kaulakys and J. Ruseckas, Phys. Rev. E 70 (004) M. Jeanblanc, M. Yor and M. Chesney, Mathematical Methods or Financial Markets (Springer, London, 009)