1fnoise and anomalous scaling in L evy noise-driven on-o intermittency Adrian van Kan1and Fran cois P etr elis2

2025-04-30 0 0 2.23MB 23 页 10玖币
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1/f noise and anomalous scaling in L´evy
noise-driven on-off intermittency
Adrian van Kan1and Fran¸cois P´etr´elis2
1Department of Physics, University of California at Berkeley, Berkeley, California
94720, USA
2Laboratoire de Physique de l’Ecole normale sup´erieure, ENS, Universit´e PSL,
CNRS, Sorbonne Universit´e, Universit´e de Paris, F-75005 Paris, France
October 2022
Abstract. On-off intermittency occurs in nonequilibrium physical systems close to
bifurcation points, and is characterised by an aperiodic switching between a large-
amplitude “on” state and a small-amplitude “off” state. L´evy on-off intermittency
is a recently introduced generalisation of on-off intermittency to multiplicative L´evy
noise, which depends on a stability parameter αand a skewness parameter β. Here,
we derive two novel results on L´evy on-off intermittency by leveraging known exact
results on the first-passage time statistics of L´evy flights. First, we compute anomalous
critical exponents explicitly as a function of arbitrary L´evy noise parameters (α, β) for
the first time, by a heuristic method, extending previous results. The predictions are
verified using numerical solutions of the fractional Fokker-Planck equation. Second,
we derive the power spectrum S(f) of L´evy on-off intermittency, and show that it
displays a power law S(f)∝fκat low frequencies f, where κ∈(−1,0) depends
on the noise parameters α, β. An explicit expression for κis obtained in terms of
(α, β). The predictions are verified using long time series realisations of L´evy on-off
intermittency. Our findings help shed light on instabilities subject to non-equilibrium,
power-law-distributed fluctuations, emphasizing that their properties can differ starkly
from the case of Gaussian fluctuations.
arXiv:2210.10197v2 [physics.flu-dyn] 5 Dec 2022
1/f noise and anomalous scaling in L´evy on-off intermittency 2
1. Introduction
Instabilities arise at parameter thresholds in many systems. Real physical systems
are typically embedded in an uncontrolled noisy environment, with the noise deriving
from high-dimensional chaotic dynamics. The fluctuating properties of the environment
affect the control parameter(s) of an instability, which leads to multiplicative noise.
If this multiplicative noise is dominant over additive noise close to an instability
threshold, the resulting behaviour is on-off intermittency, which is characterised by an
aperiodic switching between a large-amplitude “on” state and a small-amplitude “off”
(or ”laminar”) state, separated by some small threshold. It was extensively studied
in the context of low-dimensional deterministic chaos and nonlinear maps [1, 2, 3, 4],
and has also been observed in numerous experimental setups ranging from electronic
devices [5], spin-wave instabilities [6], liquid crystals [7, 8] and plasmas [9] to multistable
laser fibers [10], sediment transport [11], human balancing motion [12], oscillator
synchronisation [13], as well as blinking quantum dots in semiconductor nanocrystals
[14, 15], and measurements of earthquake occurence [16]. On-off intermittency has
also been observed in studies of in quasi-two-dimensional turbulence [17, 18, 19], and
magneto-hydroydnamic flows [20, 21, 22]. In addition, similar bursting behaviour
is found in other contexts, including hydrodynamic [23, 24, 25] and neural systems
[26]. On-off intermittency has been investigated theoretically in the framework of
nonlinear stochastic differential equations [27, 28, 29] such as a pitchfork bifurcation
with fluctuating growth rate,
dX
dt = (f(t) + µ)X−γX3,(1)
where µ∈Rand f(t) is typically Gaussian white noise, with hf(t)i= 0, hf(t)f(t0)i=
2δ(t−t0), in terms of the ensemble average h·i. Early studies of closely related models can
be found in [30, 31, 32]. We can take Xto be positive without loss of generality, since (1)
does not allow sign changes. In the following, we adopt the Stratonovich interpretation of
equation (1). A practical implication of this choice is that the rules of standard calculus
apply to equation (1). For Gaussian white noise, the stationary probability distribution
function (PDF) of the system is known to be of the form pst(x) = Nx−1+µe−γx2/2with
normalisation N, cf. [30]. At small µ≥0 the moments of Xscale as hXni ∝ µcn
with cn= 1 for all n > 0, which is different from the deterministic “mean-field” scaling
cn=n/2. This defines anomalous scaling, a well-known phenomenon in the context of
continuous phase transitions (where noise is of thermal origin) and critical phenomena
[33, 34], as well as in turbulence [35, 36]. In addition to anomalous scaling, the result
cn= 1 for all nalso implies multiscaling, which is defined by cnnot being proportional to
n. Multiscaling occurs in a variety of contexts including turbulence [37], finance [38] and
rainfall statistics [39]. In addition to its non-trivial scaling properties, the intermittent
dynamics resulting from the multiplicative noise in equation (1) are reflected in the
form of the power spectral density (PSD) of X. Denoting the two-time correlation
function by C(t) = hX(0)X(t)i, the Fourier transform of C(t) defines the PSD of X,
1/f noise and anomalous scaling in L´evy on-off intermittency 3
S(f) = ReiftC(t)dt, according to the Wiener-Khinchin theorem [40, 41]. This has been
exploited to explain the S(f)∝f−1/2range observed at low frequencies for small µ > 0,
cf. [42, 43]. Such behaviour, namely the existence of a wide range in log(f), at small f,
for which the PSD S(f) is of power-law form with exponent smaller than 0 and greater
than −2, is generically called 1/f noise, also known as Flicker noise, or pink noise.
It has been observed in a wide variety of systems, ranging from voltage and current
fluctuations in vacuum tubes and transistors, where this behaviour was first recognised
[44, 45, 46], to astrophysical magnetic fields [47] and biological systems [48], climate
[49], turbulent flows [50, 51, 52], reversing flows [53, 54, 55, 56], traffic [57], as well as
music and speech [58, 59], to name a few, and is also found in fractional renewal models
[60]. In addition, 1/f noise has also been observed for L´evy flights in inhomogeneous
environments [61, 62], but these studies did not consider any bifurcation points.
While the above-described case of Gaussian noise has been studied in depth, non-
Gaussian fluctuations arise in many systems. For example, out-of-equilibrium dynamics,
such as turbulent fluid flows, typically exhibit non-Gaussian statistics, see e.g. [63],
implying that instabilities developing on a turbulent background generally exhibit non-
Gaussian growth rates, cf. [19, 64]. Power-law-distributed fluctuations in particular
are found in a variety of systems, including the human brain [65], climate [66], finance
[67] and beyond. An important example of random motion resulting from additive
non-Gaussian noise is given by L´evy flights (a term coined by Mandelbrot [68]), which
are driven by L´evy noise. L´evy noise follows a heavy-tailed α-stable distribution that
depends on a stability parameter α∈(0,2] and a skewness parameter β∈[−1,1].
Stable distributions come in different forms: the case α= 2 corresponds to the Gaussian
distribution, while at α < 2 the distribution has power-law tails with exponent −1−α.
The main interest lies in the parameter regime 1 < α ≤2, where there is a finite mean,
but an infinite variance. While the parameter regime 0 < α ≤1 is formally admissible,
it is of little practical interest, since the noise distribution has a diverging mean in
this case. The reason why Gaussian random variables are common in physics is their
stability: by the central limit theorem [69], the Gaussian distribution constitutes an
attractor in the space of PDFs with finite variance. Similarly, by the generalised central
limit theorem [70, 71], non-Gaussian α-stable distributions constitute an attractor in
the space of PDFs whose variance does not exist. Stable distributions can be symmetric
(β= 0) or asymmetric (β6= 0), giving rise to symmetric and asymmetric L´evy flights.
L´evy flights have since found numerous applications in many areas both in physics
[72, 73, 74, 75, 76, 77] and beyond, including climatology [78], finance [79], ecology [80]
and human travel [81].
L´evy statistics and on-off intermittency can be present in the same system.
Examples include experiments of human balancing motion [12, 82, 83], blinking quantum
dots [84, 85] and the intermittent growth of three-dimensional instabilities in quasi-
two-dimensional turbulence [19]. In a recent study [86], the problem of L´evy on-off
intermittency was formally introduced as the case where f(t) in equation (1) is L´evy
noise with 1 < α < 2. In this case, if X(t) solves (1), then log(X(t)) performs a L´evy
1/f noise and anomalous scaling in L´evy on-off intermittency 4
flight in an anharmonic potential. The asymptotics of the stationary PDF of Xwere
derived from the fractional Fokker-Planck equation associated with (1). However, an
analytical solution for the full stationary PDF is only known in the Gaussian case
(α= 2). From the asymptotics of the stationary PDF, the moments hXniwere
computed heuristically in [86]. Anomalous scaling of the moments with the distance
µ > 0 from the instability threshold was observed, with critical exponents cnthat differ
in general from the Gaussian case and depend on the stability and skewness parameters
αand βof the L´evy noise. However, the explicit dependence of the critical exponents
on α, β could only be computed for certain special cases in [86]. Specifically, for all
−1< β < 1, the expression for the critical exponents obtained in [86] contained a
heuristic, numerically estimated constant. Therefore, it remains an open problem to
determine the explicit dependence of the critical exponents on α, β at a theoretical
level. In this paper we derive, for the first time, an explicit expression for the critical
exponents in L´evy noise parameters with arbitrary parameters α, β, using heuristic
arguments. Moreover, although the power spectral density in on-off intermittency with
L´evy statistics has been experimentally measured for human balancing motion, where a
low-frequency exponent close to −1/2 was found [12], no theoretical results exist for the
PSD of L´evy on-off intermittency, and the dependence of the noise parameters remains
unknown. Here, we present a heuristic derivation of the low-frequency PSD in L´evy
on-off intermittency. Both derivations will be explicated later on in the text.
In addition to critical scaling, another important characteristic of on-off
intermittency is given by the statistics of the duration Toff of laminar phases. These have
received much attention, in particular since they are rather easily accessible numerically
[2, 3, 4] and in experiments [5, 6, 7, 8, 9, 10, 12, 11, 14, 15, 16]. In many studies, Toff is
found to follow a PDF with a power-law tail p(Toff )∝(Tof f )m, with m=−3/2. The
value of the exponent has been explained in terms of first-passage time statistics: on
a logarithmic scale, the linear dynamics in the laminar phase can be mapped onto
a random walk on the negative half line, so that the duration of laminar periods
corresponds to the first-passage time through the origin of the random walk. For
symmetric random walks, this quantity is known to follow a PDF with a power-law tail
whose exponent is −3/2 [87]. According to the Sparre Andersen theorem [88, 89], this
result holds for any symmetric step size distribution, as long as steps are independent,
including symmetric L´evy flights, for which β= 0, and even in the presence of finite
spatio-temporal correlations [90]. Despite the large body of research corroborating the
scenario leading to the exponent m=−3/2, some studies on blinking quantum dots,
bubble dynamics and other systems [91, 92, 93, 94] find a different behaviour. There,
the duration of laminar phases also follows a power-law distribution, but with exponent
m6=−3/2 varying between −1 and −2. Similarly, Manneville [42] finds m=−2 for
a chaotic discrete map. For L´evy flights, there exist exact results for the asymptotics
of the first-passage time distribution. The distribution features a power-law tail with
exponent m=m(α, β)∈(−1,−2), whose dependence on (α, β) is known explicitly. A
summary and derivation of these results is given in [95]. The goal of the present paper
1/f noise and anomalous scaling in L´evy on-off intermittency 5
is to leverage these exact first-passage time results to better understand two aspects
of L´evy on-off intermittency: its anomalous critical exponents close to the threshold of
instability, and its power spectral density (PSD).
For the case of Gaussian white noise f(t) in equation (1), where the critical
exponents can be calculated directly from the known stationary PDF, an alternative
derivation was presented in the work of Aumaˆıtre et al. [28], where a heuristic argument
based on the knowledge of p(Toff ) and simple properties of the on-phases leads to the
same result. In the present study, we first generalise the argument given by Aumaˆıtre et
al. to L´evy on-off intermittency, where the stationary PDF is not fully known. We thus
derive, for the first time, explicit expressions for the critical exponents valid for arbitrary
noise parameters α, β. First-passage time statistics are also known to be linked to the
two-time correlation function C(t) = hx(t)x(0)iin on-off intermittency, as described
in [42, 43]. In the second part of this paper, we generalise these arguments to L´evy
on-off intermittency to show that it displays 1/f noise with a spectral power-law range
S(f)∝fκwhose exponent κ∈(−1,0) is computed explicitly for the first time, and
shown to depend on the noise parameters.
The remainder of this paper is structured as follows. In section 2, we describe the
theoretical background of this study. Next, in section 3 we present a derivation of the
critical exponents in L´evy on-off intermittency, comparing the results to the findings
of [86] and additional numerical solutions of the fractional Fokker-Planck equation
associated with equation (1). In section 4, we present a spectral analysis of L´evy on-off
intermittency. We describe the arguments relating first-passage time distributions to
1/f noise and again verify the predictions numerically. Finally, in section 5, we discuss
our results and conclude.
2. Theoretical background
In this section we define stable distributions, recall results on L´evy on-off intermittency,
and introduce relevant properties of L´evy flight first-passage time PDFs.
2.1. Definition of α-stable distributions
For parameters α∈(0,2], β ∈[−1,1], we denote the alpha-stable PDF for a random
variable Yby ℘α,β(y). It is defined by its characteristic function (i.e. Fourier transform),
ϕα,β(k) = exp − |k|α[1 −iβsgn(k)Φ(k)],(2)
with
Φ(k) = tan πα
2for α6= 1,Φ(k) = −2
πlog(|k|) for α= 1,(3)
see [71]. We note that (2) is not the most general form possible: there may be a scale
parameter in the exponential, which we set equal to one. One refers to αas the stability
parameter. For α= 2, one recovers the Gaussian distribution, independently of the
摘要：

1=fnoiseandanomalousscalinginLevynoise-drivenon-o intermittencyAdrianvanKan1andFrancoisPetrelis21DepartmentofPhysics,UniversityofCaliforniaatBerkeley,Berkeley,California94720,USA2LaboratoiredePhysiquedel'Ecolenormalesuperieure,ENS,UniversitePSL,CNRS,SorbonneUniversite,UniversitedeParis,F-750...

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