Spectral Power-Law Regimes

• Spectral power-law regimes are specific frequency or wavenumber ranges where the power spectrum decays algebraically with clearly defined exponents.
• They are observed across turbulence, stochastic processes, and quantum phenomena, with breakpoint transitions marking shifts in underlying physical mechanisms.
• Analytical and experimental methods, including inertial cascade models and fixed-point analyses, provide actionable insights into complex system behavior.

Spectral power-law regimes refer to spatial or temporal frequency intervals in which the power spectrum of a physical, stochastic, or dynamical process exhibits scaling of the form $S(k)\propto k^{-\alpha}$ or $S(f)\propto f^{-\beta}$ with well-defined exponents, often separated by transition or break scales. Such regimes are ubiquitous across turbulence, condensed matter, random matrix theory, nonlinear dynamical systems, and stochastic processes, manifesting universal or nonuniversal scaling laws depending on constraints and physical mechanisms. The taxonomy, origin, mathematical structure, and phenomenology of these power-law intervals underpin the analysis and modeling of complex systems on arXiv and in the broader literature.

1. Mathematical Definition and Phenomenology

A spectral power-law regime is a frequency or wavenumber range in which the spectral density of some observable—correlation function, fluctuation field, or response—follows an asymptotic algebraic law

$$S(f) \sim f^{-\beta},\quad\text{or}\quad S(k)\sim k^{-\alpha},$$

for $f$ (frequency) or $k$ (wavenumber) within a specific range $[f_1, f_2]$ or $[k_1, k_2]$.

Commonly, but not universally, such scaling emerges:

• In the inertial ranges of fluid, plasma, or magnetohydrodynamic (MHD) turbulence (Meyrand et al., 2010, Sasmal et al., 25 Apr 2025, Cheng et al., 2018).
• In stochastic models, e.g., for time series generated by filtered Poisson or subordinator-driven processes (Korzeniowska et al., 8 Oct 2024, Garcia et al., 2016, Kazakevicius et al., 2015).
• In quantum many-body systems at quantum criticality, Mott transitions, or in Luttinger liquids (Jin et al., 2018, Eickhoff, 28 Sep 2025).
• In the spectral density (DoS) for random matrices with heavy-tailed variance structure (Safonova et al., 27 Aug 2024, Fleig, 15 Dec 2025).

Distinct intervals ("regimes") often display different exponents, separated by characteristic scales dictated by physical mechanisms (e.g., inertial vs. dissipative scales) or by external parameters (e.g., system size, correlation length, temperature).

2. Origin and Universality in Turbulence and MHD

In classical MHD turbulence, multiple spectral regimes arise:

• Inertial range: Above ion scales, the magnetic fluctuation spectrum obeys $E_B(k)\propto k^{-5/3}$ (Kolmogorov scaling), reflecting universal energy-cascade dynamics.
• Dispersive regime: Between ion and electron scales, $E_B(k)\propto k^{-\alpha}$ with $\alpha\simeq2.5\pm0.2$ (attributed to Hall-MHD or wave-mediated cascades).
• Sub-electron range: Below the electron inertial length $d_e$, a third regime appears, characterized by

$E_B(k) \propto k^{-11/3},$

derivable from an exact third-order law in electron-MHD (Meyrand et al., 2010). This regime is a genuine inertial cascade governed by electron inertia, not merely a dissipation range.

In non-axisymmetric MHD turbulence, universal algebraic constraints relate the parallel and perpendicular scaling exponents. For strong turbulence, the reduced parallel spectrum is always $E(k_\parallel)\sim k_\parallel^{-2}$, independent of alignment scenario, while the perpendicular exponent varies according to the nature and scale-dependence of velocity-magnetic field alignment:

• Scale-independent alignment: $E(k_\perp)\sim k_\perp^{-5/3}$
• Scale-dependent (Boldyrev) alignment: $E(k_\perp)\sim k_\perp^{-3/2}$ (Sasmal et al., 25 Apr 2025).

In stable atmospheric boundary layers, three spectral subranges are distinguished:

• Buoyancy subrange: $k<k_B$, $E(k)\sim k^{-5/3}$.
• Transition region: $k_B<k<k_O$, exhibiting non-universal, system-dependent exponents.
• Isotropic inertial subrange: $k>k_O$, $E(k)\sim k^{-5/3}$ (Cheng et al., 2018).

3. Stochastic Models and Nonuniversality

Beyond dynamically universal cascades, power-law spectral regimes emerge in stochastic processes:

• Superpositions of uncorrelated pulses (e.g., Lorentzian or exponential shapes) with random durations $\tau$ produce

$S(f) \propto f^{-\beta},\quad \beta=3-\alpha,$

for a pulse duration distribution $P_\tau(\tau)\sim \tau^{-\alpha}$, $1<\alpha<3$ (Korzeniowska et al., 8 Oct 2024).

• The range of scale-invariance is controlled by the cutoff spans of $\tau$; finite-size effects significantly narrow the observable power-law interval.
• In time-subordinated Langevin equations generating subdiffusive dynamics, spatial inhomogeneity can yield intermediate-frequency spectral exponents $\beta\geq1$, with explicit dependence on the trap inhomogeneity and drift/diffusion exponents, while homogeneous limits produce only $\beta<1$ (Kazakevicius et al., 2015).
• Poisson/Lorentzian superposition models explain the transition between exponential and power-law spectral tails, reproducing both exponential fall-off for narrow duration distributions and algebraic scaling for broad ones (Garcia et al., 2016).

4. Quantum and Statistical Many-Body Systems

In strongly correlated quantum systems, nontrivial power-law regimes arise:

• In the orbital-selective Mott (OSM) phase of multi-orbital Hubbard models, single-particle spectra display distinct power-law exponents for different orbitals:

$$\rho_f(\omega)\propto |\omega|^{1/3},\quad \rho_c(\omega)\propto |\omega|^{-1/3},$$

and $\omega/T$ scaling holds for both single-particle and two-particle (spin/charge) susceptibilities (Eickhoff, 28 Sep 2025).

• In interacting 1D quantum wires, the nonlinear-Luttinger-liquid theory predicts a power-law singularity at the spectral edge, $A_1(k,E)\propto |E-\epsilon(k)|^{-\mu(k)}$, with momentum-dependent exponent $\mu(k)$. This emergent universality manifests even in the presence of a finite length scale, observed experimentally in tunneling spectroscopy (Jin et al., 2018).

In "random-matrix universality" with heavy-tailed entries, the mean spectral density exhibits algebraic decay in the tails,

$\rho(E) \sim |E|^{-1-\mu},$

for a Lévy-stable off-diagonal law with tail index $0<\mu\leq2$. Phase transitions between ergodic, fractal, and localized regimes in Lévy–Rosenzweig–Porter ensembles are sharply manifest in the appearance or suppression of these power-law tails, with the DoS acting as an order parameter (Safonova et al., 27 Aug 2024). Similar transitions and the effect of system-size dependent normalization ("renormalization group for spectral collapse") emerge in random-matrix models with power-law variance profiles, with analytic control of scaling exponents and collapse fixed points across $N$ (Fleig, 15 Dec 2025).

5. Physical and Experimental Systems

Experimental realizations of spectral power-law regimes span astrophysical, atmospheric, and laboratory contexts:

• Solar and wind energy: The power spectrum of minute-resolved solar irradiance under clear sky conditions shows a breakpoint between intermediate ($\alpha\sim1.8$ to $2.2$) and high-frequency ($\alpha\sim0.1$ to $0.4$) power laws, with exponents sensitive to latitude—crucial for photovoltaic engineering and grid-stability analysis (Bel et al., 2018).
• Wind turbine arrays: Wind farm power output fluctuations exhibit two key scaling regimes: $\Phi_P(f)\sim f^{-2}$ at low frequencies (due to large-scale flow-turbine interactions) and $\Phi_P(f)\sim f^{-11/3}$ at inertial-range frequencies, inherited from Kolmogorov turbulence cascades coupled to inertial rotor response (Liu et al., 2017).
• Solar flare statistics: The histogram of GOES soft X-ray fluxes and their temperature distributions obey robust power-law statistics, $dN/dT\propto T^{-2}$ at large $T$ (Li et al., 2016). However, power-law indices of flux distributions are found to vary systematically with energy channel, placing constraints on self-organized criticality models.

6. Mechanisms of Spectral Power-Law Generation

Spectral power-law regimes can originate from diverse mechanisms:

• Nonlinear inertial cascades: Conservation laws and scale-local transfer dictate universal exponents (e.g., $-5/3$ in Kolmogorov turbulence).
• Sequential and recursive processes: Sequential acceleration (as in flare magnetic islands) yields power-law particle energy spectra

$$f(E)\propto E^{-\delta'},\quad \delta'=1-\frac{\ln t}{\ln r},$$

with tunable exponent set by the hop fraction $t$ and energy gain per stage $r$ (Guidoni et al., 2022).

• Scale-local versus nonlocal drive/damping: In driven-dissipative PDEs or generalized Kuramoto–Sivashinsky models, nonuniversal exponents arise when the ratio of linear to nonlinear characteristic timescales remains nearly constant over a spectral interval (1305.4111).
• Hidden structural heterogeneity: In random matrices and covariance estimation with power-law variance profiles, tail exponents and spectral collapse are controlled by the scaling properties of the variance index, as captured by an RG flow (Fleig, 15 Dec 2025).

The precise interval of power-law scaling is often limited by finite-size effects, crossover to other regimes (e.g., dissipation), or by parameter-dependent breaks tied to underlying distributions.

7. Analytical Structures, Universality, and Applications

The analytical foundation for spectral power-law regimes includes:

• Exact laws in turbulence (e.g., third-order structure functions in MHD), yielding explicit exponents and robust universality (Meyrand et al., 2010, Sasmal et al., 25 Apr 2025).
• Self-consistent fixed-point equations for random matrix spectra, providing finite-size scaling and phase classification (Fleig, 15 Dec 2025, Safonova et al., 27 Aug 2024).
• Characterization of pulse superpositions and subordinator-driven stochastic processes in terms of convolution integrals and asymptotic tail analysis (Korzeniowska et al., 8 Oct 2024, Kazakevicius et al., 2015).

The ability to diagnose, parametrize, and interpret power-law spectral regimes is central to extracting underlying physics—ranging from constraining turbulence models (MHD, solar wind, atmospheric), interpreting kinetic acceleration in flares and reconnection, to benchmarking convergence rates of optimization algorithms in machine learning under power-law spectral conditions (Velikanov et al., 2022).

Applications extend to practical design and diagnostics: grid management in renewable energy, separating signal/noise in large-scale data, and inference of microscopic parameters from spectral features in experimental measurements.

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References:

• (Meyrand et al., 2010): "A Universal Law for Solar-Wind Turbulence at Electron Scales"
• (Sasmal et al., 25 Apr 2025): "Universal relations between parallel and perpendicular spectral power law exponents in non-axisymmetric magnetohydrodynamic turbulence"
• (Korzeniowska et al., 8 Oct 2024): "Long-range correlations with finite-size effects from a superposition of uncorrelated pulses with power-law distributed durations"
• (Garcia et al., 2016): "Power law spectra and intermittent fluctuations due to uncorrelated Lorentzian pulses"
• (Kazakevicius et al., 2015): "Anomalous diffusion in nonhomogeneous media: Power spectral density of signals generated by time-subordinated nonlinear Langevin equations"
• (1305.4111): "Nonuniversal power-law spectra in turbulent systems"
• (Eickhoff, 28 Sep 2025): "Power-Law Spectra and Asymptotic $\omega/T$ Scaling in the Orbital-Selective Mott Phase of a Three-Orbital Hubbard Model"
• (Jin et al., 2018): "Momentum-dependent power law measured in an interacting quantum wire beyond the Luttinger limit"
• (Safonova et al., 27 Aug 2024): "Spectral properties of Levy Rosenzweig-Porter model via supersymmetric approach"
• (Fleig, 15 Dec 2025): "Renormalization group for spectral collapse in random matrices with power-law variance profiles"
• (Bel et al., 2018): "Geographic Dependence of the Solar Radiation Spectrum at Intermediate to High Frequencies"
• (Cheng et al., 2018): "Turbulence Spectra in the Stable Atmospheric Boundary Layer"
• (Liu et al., 2017): "Towards uncovering the structure of power fluctuations of wind farms"
• (Li et al., 2016): "On the Power-Law Distributions of X-ray Fluxes from Solar Flares Observed with GOES"
• (Guidoni et al., 2022): "Spectral Power-law Formation by Sequential Particle Acceleration in Multiple Flare Magnetic Islands"
• (Kilian et al., 2020): "Exploring the acceleration mechanisms for particle injection and power-law formation during trans-relativistic magnetic reconnection"
• (Velikanov et al., 2022): "Tight Convergence Rate Bounds for Optimization Under Power Law Spectral Conditions"