There exist natural and artificial systems with spontaneously arising large fluctuations relative to the mean values. The fluctuations are scale-invariant and demonstrate the power spectrum with a powerlike dependence on frequency: S(f) ~ 1/f  α, where exponent α lies in a range of 0.7 ≤ α ≤ 1.8. Distribution functions have powerlike “tails.” The energy of these fluctuations is concentrated at low frequencies and manifests itself in the form of the “flashes” or large-scale emissions [1–4]. The relaxation of disturbances decreases according to a power law, but not exponentially, as it is for small fluctuations. Scale-invariant processes with large fluctuations are stable without fine-tuning of the parameters of the process, although they arise in complex statistical systems far from thermodynamic equilibrium.

The appearance of the 1/f  α spectrum is sometimes associated with the concept of self-organized criticality when describing avalanche dynamics in computer models [3, 4]. Another means of mathematical description of random processes with a 1/f  α spectrum gives a fractional integration of white noise [5], but this approach is difficult to interpret physically. In turbulent fluid flows [6], the fluctuations in a wide range of scales can also develop.

Random processes with extreme fluctuations arise under the influence of white noise in interacting phase transitions [7]. When describing such random processes, the system of nonlinear stochastic equations is used:

$$\frac{{d\varphi }}{{dt}} = - \varphi {{\psi }^{2}} + \psi + {{\sigma }_{1}}{{\xi }_{1}}(t),$$
$$\frac{{d\psi }}{{dt}} = - {{\varphi }^{2}}\psi + \lambda \varphi + {{\sigma }_{2}}{{\xi }_{2}}(t),$$
((1))

where φ and ψ are dynamic variables, ξk is the Gaussian white noise with zero mean value 〈ξk〉 = 0, amplitude σk, and autocorrelation function 〈ξk(t)ξl(t')〉 = δ(t – t'), k, l = 1, 2. When the system is critical (1), the white noise level corresponds to the transition induced by the noise [7]. Coefficient λ > 1 in the second equation violates the potentiality of the field of forces and is associated with some uncompensated flow. In the critical state, random process φ(t), described by system (1), has a powerlike spectrum 1/f  α. In the case of isotropic noise (σ1 = σ2) at λ = 2, exponent α in the frequency dependence of the spectrum is equal to unit. The equations of system (1) are in the hierarchy of control and subordination. The second equation of system (1) is the governing one [8, 9]. As the argument increases, the probability density of variable ψ decreases exponentially, as for the Gaussian distribution. For large 1/f  α, the probability density distribution of variable φ decreases according to a power law ~φ−3. Figure 1 shows the probability density distributions for variables ψ and φ in the critical state.

Fig. 1.
figure 1

Probability density distribution of variables ψ (1) and φ (2) in logarithmic coordinates. The dashed line shows the dependence ~φ–3.

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The Gaussian distribution of variable ψ(t) makes it possible to use the usual formulas of statistical mechanics. In particular, the stability of the process will be described by the Gibbs–Shannon entropy maximum

$$H = - \sum\limits_n {{{P}_{n}}\log {{P}_{n}}} ,$$
((2))

where Pn is the normalized probability density distribution function of random process. Random process φ(t) with a 1/f  α power spectrum is a non-Gaussian process. The statistical Gibbs–Shannon entropy is thought to be inapplicable for such processes due to the nonintegrability of powerlike “tails” [10–12]. However, if there is complete information on a complex system, it is sufficient to examine the behavior of a “Gaussian” variable for analyzing the stability of the entire system.

In [8, 9, 13], using the principle of maximum entropy, it was shown that the critical state of system (1) is stable. By varying coefficient λ in (1), one can change exponent α within small limits while maintaining the dependence 1/f  α. In the case of isotropic noise, when the noise amplitudes are equal: σ1 = σ2, the fluctuations with a 1/f  α power spectrum, i.e., with exponent α = 1, are distributed according to the principle of maximum of the Gibbs–Shannon entropy [13].

In this paper, we consider a more general case of anisotropic noise, when σ1 ≠ σ2. It is shown that, by varying coefficient λ and the anisotropy of noise, one can change the values of α in the range of 0.7–1.7 while maintaining dependence ~1/f  α along with the implementation of the principle of maximum of the Gibbs–Shannon entropy.

Figure 2a shows the power fluctuation spectra of variables φ and ψ for the parameter σ1/σ2 = 0.75 of white noise anisotropy. The value of the exponent in the frequency dependence of the power spectrum of variable φ is α = 0.8. The power spectrum of variable ψ is close to the form 1/f  2. Figure 2b shows the corresponding spectra for anisotropy parameter σ1/σ2 = 4 corresponding to exponent α = 1.7.

Fig. 2.
figure 2

Power spectra of variables (1) ψ and (2) φ. (a) white noise anisotropy parameter σ1/σ2 = 0.75; dashed line: dependence 1/f 0.8; (b) white noise anisotropy parameter σ1/σ2 = 4, dashed line: 1/f 1.7.

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The principle of maximum of statistical Gibbs–Shannon entropy was used as a stability criterion. The corresponding value was found through the distribution function of squared variable ψ using the control stochastic equation of system (1). Figure 3a shows the maximum of Gibbs–Shannon entropy H obtained from numerical calculations of random process ψ(t), corresponding to different values of exponent α in the frequency dependence of the power spectrum of variable φ. We can see that dependence H has a broad maximum. This means that random processes with 1/f  α power spectrum are stable in a rather wide interval of variation of exponent α.

Fig. 3.
figure 3

Parameters of statistical distributions depending on the exponent α in the frequency dependence of the power spectrum of variable φ. (a) Gibbs–Shannon entropy; (b) exponent n characterizing the tails of the distribution of variable φ.

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For large φ, the probability density of random process φ decreases according to the power law P(φ) ~ φ–n. Figure 3b shows the dependence of exponent n of distribution φ on exponent α in the frequency dependence of the power spectrum.

In cases in which we are dealing with powerlike distributions, to which Gibbs–Shannon formula (2) is not applicable, expressions for the Tsallis [12] or Renyi [14] entropy are often used to analyze the stability of processes (see (3) and (4), respectively)

$${{H}^{T}} = \frac{1}{{1 - q}}\left( {\sum\limits_n {P_{n}^{q} - 1} } \right),$$
((3))
$${{H}^{R}} = \frac{1}{{1 - q}}\ln \sum\limits_n {P_{n}^{q}} .$$
((4))

These expressions depend on fitting parameter q determining the specific position of entropy maximum with an unknown value. In the case in which there is complete information on the system, parameter q in the expressions for Tsallis (3) and Renyi (4) entropy can be determined from the condition that the position of maximum of the Gibbs–Shannon entropy for the control equation coincides with the positions of the maxima of Tsallis and Renyi entropy of the distribution with powerlike tails for the subordinate equation. The Gibbs–Shannon entropy values were calculated numerically for the exponential distribution, while the Tsallis and Renyu entropy values were calculated for the distributions of variable φ with powerlike tails. All entropies correspond to the fluctuations with the 1/f  α power spectrum. The values of parameter q are determined based on the condition of coincidence of the positions of entropy maxima. In the range of variation of parameter α, in which the 1/f  α dependence of the spectral density takes place (0.8 ≤ α ≤ 1.7), the obtained values of parameter q remained approximately constant; namely, q ≈ 0.6 for both Tsallis and Renyi entropy. It should be noted that, in contrast to the case in which the entire density distribution function is used to calculate the entropy using formula (2), when calculating the Tsallis and Renyi entropies by formulas (3) and (4), the main contribution stems from the powerlike “tails” of the distributions that contain fewer points compared to a full distribution. Therefore, the accuracy of calculation of the Tsallis and Renyi entropy by formulas (3) and (4) and, accordingly, parameter q is lower than for calculating the Gibbs–Shannon entropy by formula (2) for Gaussian distributions.

Thus, it is shown that fluctuations with the 1/f  α spectrum corresponds to the entropy maximum. This indicates the stability of processes with 1/f  α power spectra at various values of exponent α.

Our results are useful in studying the stability of random processes with powerlike distribution functions in the case of limited information about the system, particularly in the analysis of experimental data.

FUNDING

This work was supported in part by Comprehensive Program of the Ural Branch of the Russian Academy of Sciences (project no. 18-2-2-3) and the Russian Foundation for Basic Research (project no. 19-08-00091-a).