1/F Noise from Nonlinear Stochastic Differential Equations

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Here x is the signal, η is the exponent of the multiplicative noise, ν defines the behavior of stationary probability distribution, and W is a standard Wiener process. SDE (1) has the simplest form of the multiplicative noise term, σxηdW . Multiplicative equations with the drift coef- ficient proportional to the Stratonovich drift correction for transformation from the Stratonovich to the Itôstochastic equation [40] generate signals with the power-law distributions [5]. Eq. (1) is of such type and has probability distribution of the power-law form P (x) x−ν . Because of the divergence of the power-law distribution and the requirement of the stationarity of the process,∼ the SDE (1) should be analyzed together with the appropriate restrictions of the diffusion in some finite interval. For simplicity, in this article we will adopt reflective boundary conditions at x = xmin and x = xmax. However, other forms of restrictions are possible. For example, exponential restriction of the diffusion can be obtained by introducing additional terms in Eq. (1), ν m x m m x m dx = σ2 η + min x2η−1dt + σxη dW . (2) − 2 2 x − 2 xmax Here m is some parameter. Equation (1) with the reflective boundary condition at xmin and xmax can be rewritten in a form that does not 2 2η−2 contain parameters σ and xmin. Introducing the scaled stochastic variable x x/xmin and scaled time t σ xmin t one transforms Eq. (1) to → → ν dx = η x2η−1dt + xηdW . (3) − 2 The scaled equation (3) has a boundary at x =1 and at x ξ = max . (4) xmin Further we will consider Eq. (3) only. In order to obtain 1/f β noise we require that the region of diffusion of the stochastic variable x should be large. Therefore, we assume that ξ 1. ≫

III. POWER SPECTRAL DENSITY FROM THE FOKKER-PLANCK EQUATION

According to Wiener-Khintchine relations, the power spectral density is ∞ ∞ S(f)=2 C(t)eiωtdt =4 C(t)cos(ωt)dt , (5) Z−∞ Z0 where ω =2πf and C(t) is the autocorrelation function. For the stationary process the autocorrelation function can be expressed as an average over realizations of the stochastic process, C(t)= x(t′)x(t′ + t) . (6) h i This average can be written as

C(t)= dx dx′ xx′P (x)P (x′,t x, 0) , (7) 0 x | Z Z ′ where P0(x) is the steady-state probability distribution function and Px(x ,t x, 0) is the transition probability (the conditional probability that at time t the signal has value x′ with the condition| that at time t = 0 the signal had the value x). The transition probability can be obtained from the solution of the Fokker-Planck equation with the initial ′ ′ condition Px(x , 0 x, 0) = δ(x x). Therefore, for the| calculation− of the power spectral density of the signal x we will use the Fokker-Planck equation instead of stochastic diﬀerential equation (3). The Fokker-Planck equation corresponding to the Itˆosolution of Eq. (3) is [41, 42] ∂ ν ∂ 1 ∂2 P = η x2η−1P + x2ηP. (8) ∂t − − 2 ∂x 2 ∂x2 The steady-state solution of Eq. (8) has the form

ν−1 −ν 1−ξ1−ν x , ν =1 , P0(x)= 1 −1 6 (9) ( ln ξ x , ν =1 . 3

The boundary conditions for Eq. (8) can be expressed using the probability current [42]

ν 1 ∂ S(x, t)= η x2η−1P x2ηP. (10) − 2 − 2 ∂x At the reﬂective boundaries xmin = 1 and xmax = ξ the probability current S(x, t) should vanish, and, therefore, the boundary conditions for Eq. (8) are

S(1,t)=0 , S(ξ,t)=0 . (11)

A. Eigenfunction expansion

We solve Eq. (8) using the method of eigenfunctions. An ansatz of the form

−λt P (x, t)= Pλ(x)e (12) leads to the equation

ν ∂ 1 ∂2 η x2η−1P + x2ηP = λP (x) , (13) − − 2 ∂x λ 2 ∂x2 λ − λ where Pλ(x) are the eigenfunctions and λ 0 are the corresponding eigenvalues. The eigenfunctions Pλ(x) obey the orthonormality relation [42] ≥

ξ Φ(x) e Pλ(x)Pλ′ (x)dx = δλ,λ′ , (14) Z1 where Φ(x) is the potential, associated with Eq. (8),

Φ(x)= ln P (x) . (15) − 0

It should be noted that the restriction of diﬀusion of the variable x by xmin and xmax ensures that the eigenvalue spectrum is discrete. Expansion of the transition probability density in a series of the eigenfunctions has the form [42]

P (x′,t x, 0) = P (x′)eΦ(x)P (x)e−λt . (16) x | λ λ Xλ Substituting Eq. (16) into Eq. (7) we get the autocorrelation function

−λt 2 C(t)= e Xλ . (17) Xλ Here

ξ Xλ = xPλ(x)dx (18) Z1 is the first moment of the stochastic variable x evaluated with the λ-th eigenfunction Pλ(x). Such an expression for the autocorrelation function has been obtained in Ref. [43]. Using Eqs. (5) and (17) we obtain the power spectral density λ S(f)=4 X2 . (19) λ2 + ω2 λ Xλ This expression for the power spectral density resembles the models of 1/f noise using the sum of the Lorentzian spectra [15–18, 27, 28, 44, 45]. Here we see that the Lorentzians can arise from the single nonlinear stochastic differential equation. Similar expression for the spectrum has been obtained in Ref. [14] where reversible Markov chains on finite state spaces were considered (Eq. (34) in Ref. [14] with γk,m playing the role of λ ). A pure 1/f β power spectrum is physically impossible because the total− power would be infinity. It should be noted that the spectrum of signal x, obeying SDE (3), has 1/f β behavior only in some intermediate region of 4 frequencies, f f f , whereas for small frequencies f f the spectrum is bounded. The behavior of min ≪ ≪ max ≪ min spectrum at frequencies fmin f fmax is connected with the behavior of the autocorrelation function at times ≪ ≪β β 1/fmax t 1/fmin. Often 1/f noise is described by a long-memory process, characterized by S(f) 1/f as f ≪0. An≪ Abelian-Tauberian theorem relating regularly varying tails shows that this long-range dependence∼ property→ is equivalent to similar behavior of autocorrelation function C(t) as t [46]. However, this behavior of the autocorrelation function is not necessary for obtaining required form of the→ power ∞ spectrum in a finite interval of the frequencies which does not include zero [47–49]. From Eq. (19) it follows that if the terms with small λ dominate the sum, then one obtains 1/f 2 behavior of the spectrum for large frequencies f. If the terms with 1/f β (with β < 2) are present in Eq. (19), then at sufficiently large frequencies those terms will dominate over the terms with 1/f 2 behavior. Since the terms with small λ lead to 1/f 2 behavior of the spectrum, we can expect to obtain 1/f β spectum in a frequency region where the main contribution to the sum in Eqs. (17) and (19) is from the large values of λ. Thus we need to determine the behavior of the eigenfunctions Pλ(x) for large λ. The conditions when eigenvalue λ can be considered as large will be investigated below.

B. Eigenfunctions of the Fokker-Planck equation

For η = 1, it is convenient to solve Eq. (13) by writing the eigenfunctions P (x) in the form 6 λ −ν 1−η Pλ(x)= x uλ(x ) . (20)

1−η The functions uλ(z) with z = x obey the equation

d2 1 d u (z) (2α 1) u (z)= ρ2u (z) , (21) dz2 λ − − z dz λ − λ where the coeﬃcients α and ρ are

ν 1 √2λ α =1+ − , ρ = . (22) 2(1 η) η 1 − | − | 1−η The area of diﬀusion of the variable z = x is restricted by the minimum and maximum values zmin and zmax,

ξ1−η , η> 1 , 1 , η> 1 , zmin = zmax = 1−η (23) (1 , η< 1 , (ξ , η< 1 .

The probability current Sλ(x), Eq. (10), rewritten in terms of functions uλ, is

1 ν−η ∂ S (z)= (η 1)z η−1 u (z) . (24) λ 2 − ∂z λ ′ ′ 1−η ′ Therefore, the boundary conditions for Eq. (21), according to Eq. (11) are uλ(1) = 0 and uλ(ξ ) = 0. Here uλ(z) is the derivative of the function uλ(z). The orthonormality relation (14) yields the orthonormality relation for functions uλ(z),

− 1−ν ξ1 η 1 ξ η−ν − z 1−η u (z)u ′ (z)dz = δ ′ . (25) (ν 1)(1 η) λ λ λ,λ − − Z1 The expression (18) for the ﬁrst moment Xλ of the stochastic variable x evaluated with the λ-th eigenfunction becomes

− ξ1 η 1 1+η−ν X = z 1−η u (z)dz . (26) λ 1 η λ − Z1

C. Solution of the equation for eigenfunctions

The solutions of equation (21) are [50]

α uλ(z)= z [c1Jα(ρz)+ c2Yα(ρz)] , (27) 5 where Jα(z) and Yα(z) are the Bessel functions of the ﬁrst and second kind, respectively. The coeﬃcients c1 and c2 needs to be determined from the boundary and normalization conditions for function uλ(z). The asymptotic expression for the function uλ(z) is

α− 1 − 1 u (z) c z 2 ρ 2 cos(ρz + a) , ρz 1 . (28) λ ≈ λ ≫

Here a is a constant to be determined from the boundary conditions and cλ is the constant to be determined from the normalization (25). The behavior of the power spectral density in Eq. (19) as 1/f β can be only due to terms with large λ. Therefore, we will consider the values of λ for which at least the product ρzmax is large, ρzmax 1. The first moment of the variable x in the expression for the autocorrelation function (17) is expressed via integral≫ (26). If the condition ρz 1 is ≫ satisfied for all z then the function uλ(z) has frequent oscillations in all the region of the integration, and the integral is almost zero. Consequently, the biggest contribution to the sum in Eq. (17) makes the terms corresponding to those values of λ, for which the condition ρz 1 is not satisfied for all values of z between zmin and zmax. Therefore, we will restrict the values of λ by the condition≫ ρz 1. Thus we will consider eigenvalues λ satisfying the conditions min ≪ 1/z ρ 1/z . (29) max ≪ ≪ min Explicitly, we have conditions 1 ρ ξη−1 if η > 1 and 1/ξ1−η ρ 1 if η< 1. ≪ ≪ ≪ ≪ The derivative of the function uλ(z), Eq. (27), is

′ α uλ(z)= ρz [c1Jα−1(ρz)+ c2Yα−1(ρz)] . (30)

′ Since we consider the case ρzmin 1, then, using Eq. (30), instead of the boundary condition uλ(zmin) =0 we can approximately take the condition≪

lim (c1Jα−1(y)+ c2Yα−1(y))=0 . y→0

If α > 1 then we get c2 = 0; if α < 1 then c2 = c1 tan(πα). Using those values of the coeﬃcient c2 we obtain the solutions of Eq. (21) −

α cλz J−α(ρz) , α< 1 , uλ(z) α (31) ≈ (cλz Jα(ρz) , α> 1 .

From approximate solution (31), using asymptotic expression for the Bessel functions, we can determine the parameter a in the asymptotic expression (28). We obtain that the parameter a depnends on α and does not depend on ρ.

D. Normalization

Taking λ = λ′ from Eq. (25) we get the normalization condition. Using Eq. (31) we have

1−η 1 ξ1−ν ξ 1 ξ1−ν c2 ρzmax c2 − zJ 2 (ρz)dz = − λ yJ 2 (y)dy 1 . λ (ν 1)(1 η) ±α (ν 1) 1 η ρ2 ±α ≈ − − Z1 − | − | Zρzmin

Taking into account the condition ρzmin 1 and replacing the lower limit of integration by 0, we obtain that the integral is approximately equal to ≪

ρzmax ρz yJ 2 (y)dy max . ±α ≈ π Z0 Here we assumed that ρz 1. Therefore, the normalization constant c is max ≫ λ 1 η ν 1 cλ | − | −1−ν πρ . (32) ≈ s zmax 1 ξ − 6

IV. CALCULATION OF THE POWER SPECTRAL DENSITY

A. Estimation of the ﬁrst moment Xλ of the stochastic variable x

The expression (17) for the autocorrelation function contains the ﬁrst moment Xλ of the variable x, expressed as an integral of the function uλ, Eq. (26). Using Eq. (31) we get

1−η c ξ c ρzmax X λ zβ−1J (ρz)dz = λ yβ−1J (y)dy . λ ≈ 1 η ±α 1 η ρβ ±α − Z1 | − | Zρzmin Here ν 3 β =1+ − (33) 2(η 1) − and “+” sign is for α> 1, while “ ” is for α< 1. − If α + β > 0, taking into account that ρzmin 1, we can replace the lower limit of the integration by 0, ρz X ± cλ 1 max yβ−1J (y)dy . We get that≪ the integral in the expression for X approximately does not λ ≈ |1−η| ρβ 0 ±α λ depend on the lower limit of integration ρz . We can integrate the integral by parts and use the properties of the R min Bessel functions to obtain

ρzmax cλ β−1 ρzmax β−2 Xλ y J±(α−1)(y) (β + α 2) y J±(α−1)(y)dy . (34) ≈ 1 η ρβ ∓ ρzmin ± − Zρzmin | − | ′ ′ Using expression (31) for the function uλ(z), the boundary conditions uλ(zmin) = 0 and uλ(zmax) = 0 leads to

J±(α−1)(ρzmin)=0 , J±(α−1)(ρzmax)=0 . (35)

5 Therefore, the ﬁrst term in the expression (34) for Xλ is zero. If β < 2 , taking into account that ρzmax 1, we can extend the upper limit of integration to + . We get that the integral for X approximately does not depend≫ on the ∞ λ upper limit of integration ρzmax. Therefore, the ﬁrst moment Xλ of the variable x is proportional to the expression c 1 λ . (36) 1 η ρβ | − | Now we are ready to estimate the power spectral density.

B. Power spectral density

′ Since ρzmax 1 , from the boundary condition uλ(zmax) = 0 using the asymptotic expression (28) for the function u (z) we obtain≫ the condition sin(ρz + a) = 0 and ρz = πn a. Then λ max max − 2 (1 η) 2 λn −2 (πn a) . (37) ≈ 2zmax − Equation (37) shows that the density of eigvalues D(λ) is proportional to 1/√λ. Since the parameter a does not depend on λ, it follows that the density of eigenvalues and, consequently, the autocorrelation function do not depend on th parameter a. In order to estimate the sum in expression (17) for the autocorrelation function, we replace summation by the integration,

C(t) e−λtX2D(λ)dλ (38) ≈ λ Z Such a replacement is valid when ρzmax 1. Using the approximate expressions (32) and (36) we get the expression for the autocorrelation function ≫

−2 zmin −β −λt β−1 −2 −2 C(t) λ e dλ = t Γ(1 β,zmaxt) Γ(1 β,zmint) . (39) ∼ −2 − − − Zzmax 7

∞ a−1 −t 2 2 Here Γ(a,z)= z t e dt is the incomplete Gamma function. When zmin t zmax we have the following lowest powers in the expansion of the approximate expression (39) for the autocorrelation≪ ≪ function in the power series of t: R 2(β−1) 2(β−2) zmax tzmax β−1 β−2 , β> 2 2 − −2 zmax + (γ 1)t + t ln(zmaxt) , β =2 2(β−1) −  z C(t)  max + tβ−1Γ(1 β) , 1 <β< 2 (40) ∼  β−1  −2 −  γ ln(zmaxt) , β =1 −1 −  t1−β Γ(1 β) , β< 1  −  Here γ 0.577216 is the Euler’s constant. Similar ﬁrst terms in the expansion of the autocorrelation function in the power series≈ of time t has been obtained in Ref. [5]. Similarly, when ρzmax 1, replacing in Eq. (19) the summation by the integration we obtain the power spectral density ≫ λ S(f) 4 X2D(λ)dλ . (41) ≈ λ2 + ω2 λ Z Equation, similar to Eq. (41) has been obtained in Ref. [8] by considering a relaxing linear system driven by white noise (Eq. (27) in Ref. [8]). Similar equation also has been obtained in Ref. [14] where reversible Markov chains on ﬁnite state spaces were considered. In both Refs. [8], [14] the power spectral density is expressed as a sum or an integral over the eigenvalues of a matrix describing transitions in the system. Using the approximate expressions (32) and (36) we get the equation

− z 2 min 1 1 S(f) dλ . (42) ∼ −2 λβ−1 λ2 + ω2 Zzmax −2 −2 When zmax ω zmin then the leading term in the expansion of the approximate expression (42) for the power spectral density≪ in≪ the power series of ω is

ω−β , β< 2 , S(f) (43) ∼ ω−2 , β 2 . ( ≥ The second term in the expansion is proportional to ω−2. In the case of β < 2, the term with ω−β becomes larger than −2 −2 β 2(η−1) the term with ω when zmax ω. Therefore, we obtain 1/f spectrum in the frequency interval 1 ω ξ if η > 1 and the frequency interval≪ 1/ξ2(1−η) ω 1 if η < 1. It should be noted that time t and≪ frequency≪ ω in our analysis are dimensionless. ≪ ≪ Eq. (41) shows that the shape of the power spectrum depends on the behavior of the eigenfunctions and the 2 2 −β eigenvalues in terms of the function XλD(λ). This function XλD(λ) should be proportional to λ in order to obtain 1/f β behavior. Similar condition has been obtained in Ref. [14]. Eq. (13) for discrete time process and the unnumbered 2 −β equation after Eq. (34) for a continuous time process in Ref. [14] are analogous to the condition XλD(λ) λ since the density of eigenvalues in Ref. [14] is proportional to 1/ γ′(x) . Equations (39) and (42) for the power∼ spectral density and autocorrelation function resembles those obtained| from| the sum of Lorentzian signals with appropriate weights in Ref. [27].

V. NUMERICAL EXAMPLES

If ν = 3 we get that β = 1 and stochastic differential equation (1) should give signal exhibiting 1/f noise. We 5 1 will solve numerically two cases: η = 2 > 1 and η = 2 < 1. For the numerical solution we use Euler-Marujama approximation, transforming differential equations to difference− equations. Equation (44) with η = 5/2 was solved using variable step of integration, solution Eq. (45) with η = 1/2 was performed using a fixed step of integration. − 5 5 2 4 2 When η = 2 and ν = 3 then equation (1) is dx = σ x dt + σx dW . Using exponential restriction of the diffusion region we have the equation

2 1 xmin 1 x 4 5 dx = σ 1+ x dt + σx 2 dW . (44) 2 x − 2 x max 8

102 100 101 10-4 100 10-1 P(x) 10-8 S(f) 10-2 -3 10-12 10 10-4 -16 10 10-5 10-1 100 101 102 103 104 10-3 10-2 10-1 100 101 102 103 104 x f

FIG. 1: (Color online) Probability distribution function P (x) (left) and power spectral density S(f) (right) for the stochastic process deﬁned by the stochastic diﬀerential equation (44). Dashed green lines are analytical expression (9) for the steady-state 2 distribution function P0(x) on the left and the slope 1/f on the right. Parameters used are xmin = 1, xmax = 10 , and σ = 1 .

107 6 100 10 105 -2 10 104 P(x) S(f) 103 10-4 102 1 10-6 10 100 1 10 100 10-6 10-5 10-4 10-3 10-2 10-1 x f

FIG. 2: (Color online) Probability distribution function P (x) (left) and power spectral density S(f) (right) for the stochastic process deﬁned by the stochastic diﬀerential equation (45). Dashed green lines are analytical expression (9) for the steady-state 2 distribution function P0(x) on the left and the slope 1/f on the right. Parameters used are xmin = 1, xmax = 10 , and σ = 1 .

The equation was solved using the variable step of integration, ∆t = κ2/x3 , with κ 1 being a small parameter. k k ≪ The steady-state probability distribution function P0(x) and the power spectral density S(f) are presented in Fig. 1. We see a good agreement of the numerical results with the analytical expressions. The 1/f interval in the power −1 2 spectral density in Fig. 1 is approximately between fmin 2 10 and fmax 2 10 . The width of this region is much narrower than the width of the region 1 2πf ≈106 ×(ξ = 102) predicted≈ in× the previous section. When η = 1/2 and ν = 3 then equation (1)≪ is ≪ − σ2 σ dx = 2 dt + dW (45) − x2 √x

We used reflective boundary conditions at xmin = 1 and xmax = 100. The equation was solved with a constant step of integration. The steady-state probability distribution function P0(x) and the power spectral density S(f) are −6 presented in Fig. 2. The 1/f interval in the power spectral density in Fig. 2 is approximately between fmin 10 −4 −6 ≈ and fmax 2 10 . The width of this region is much narrower than the width of the region 10 2πf 1 (ξ = 102) predicted≈ × in the previous section. ≪ ≪ Numerical solution of the equations confirms the presence of the frequency region for which the power spectral density has 1/f β dependence. The width of this region can be increased by increasing the ratio between minimum and maximum values of the stochastic variable x. In addition, the region in the power spectral density with the power-law behavior depends on the exponent η: if η = 1 then this width is zero; the width increases with increasing the difference η 1 . However, the estimation of the width of the region, obtained in the previous section, is too | − | 9 broad, the width obtained in numerical solutions is narrower. Such a discrepancy can be explained as the result of various approximations, made in the derivation.

VI. DISCUSSION

In summary, we derived the behavior of the power spectral density from the nonlinear stochastic differential equation. In Refs. [29, 30] only the values of the exponent of the multiplicative noise η greater than 1 has been used. Here we showed, that it is possible to obtain 1/f β noise from the same nonlinear SDE for η< 1, as well. The analysis reveals that the power spectrum may be represented as a sum of the Lorentzian spectra with the coefficients proportional to the squared first moments of the stochastic variable evaluated with the appropriate eigenfunctions of the corresponding Fokker-Planck equation. Nonlinear SDE, corresponding to a particular case of Eq. (1) with η = 0, i.e., with linear noise and non-linear drift, was considered in Ref. [9]. It was found that if the damping is decreasing with increase of x , then the solution of such a nonlinear SDE has long correlation time. | |As Eq. (41) shows, the shape of the power spectrum depends on the behavior of the eigenfunctions and the 2 eigenvalues in terms of the function XλD(λ), where D(λ) is the density of eigenvalues. The SDE (3) considered in this article gives the density of eigenvalues D(λ) proportional to 1/√λ. One obtains 1/f β behavior of the power 2 −β spectrum when this function XλD(λ) is proportional to λ for a wide range of eigenvalues λ, as is the case for SDE (3). Similar condition has been obtained in Ref. [14]. One of the reasons for the appearance of the 1/f β spectrum is the scaling property of the stochastic differential equation (1): changing the stochastic variable from x to x′ = ax changes the time-scale of the equation to t′ = a2(1−η)t , leaving the form of the equation invariant. From this property it follows that it is possible to eliminate the eigenvalue λ in Eq. (13) by changing the variable from x to z = λ1/2(1−η)x. The dependence of the eigenfunction on eigenvalue λ then enters only via the boundary conditions. Such scaling properties were used estimating the norm of the eigenfunction and the first moment Xλ of the stochastic variable x evaluated with the λ-th eigenfunction. Other factor in obtaining the power-law spectrum is wide range of the region of diffusion of the stochastic variable x.

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