Dynamical Power Density Spectrum

• Dynamical power density spectrum is a spectral measure that quantifies how fluctuation power is distributed over frequencies or wavenumbers, directly linking the data to underlying dynamics and instabilities.
• It is constructed using methods like finite-time Fourier transforms, Wiener–Khinchin relations, and statistical estimators, ensuring accurate extraction of key spectral features.
• Applications span fluid dynamics, stochastic processes, and nonlinear oscillators, where spectral exponents diagnose phenomena such as chaos, diffusion, resonance, and transient ballistic motion.

Searching arXiv for papers on dynamical power density spectra and closely related PSD formulations across fluids, stochastic processes, and dynamical systems. A dynamical power density spectrum is a spectral characterization of fluctuations generated by an evolving system, but its precise meaning depends on the underlying dynamical context. In the frequency-domain formulation used for stochastic time series, it is the power spectral density of a time-dependent observable and quantifies the amount of fluctuations at a given frequency (Dechant, 2023). In the setting of semiperiodic pulse trains, it captures dynamical behavior such as oscillations, chaos, and intermittent bursts through the frequency power spectral density of the signal (Theodorsen et al., 2021). In the fluid-mechanical setting of two-dimensional Poiseuille and Couette flow with the van der Waals effect, the corresponding object is not a temporal spectrum but a spatial Fourier power spectrum in streamwise wavenumber, averaged over time, of small fluctuations about laminar stationary states (Abramov, 19 Mar 2026). Across these formulations, the unifying theme is that a dynamical power density spectrum resolves how fluctuation power is distributed over spectral variables and links that distribution to the governing dynamics, their instabilities, and their characteristic scales.

1. Definitions and mathematical formulations

In the standard frequency-domain formulation for a real scalar signal $\Phi(t)$, the finite-time Fourier transform is

$$\mathcal{F}_\Phi(\omega) = \int_{-T/2}^{T/2} \Phi(t)\,e^{-i\omega t}\,dt ,$$

and the frequency power spectral density is

$$S_\Phi(\omega) = \lim_{T\to\infty} \frac{1}{T}\,\big\langle \big|\mathcal{F}_\Phi(\omega)\big|^2\big\rangle .$$

For a stationary process, this is equivalent, via the Wiener–Khinchin theorem, to the Fourier transform of the autocorrelation function $R_\Phi(\tau)$ (Theodorsen et al., 2021). In the Markov-process formulation, for an observable $z(\mathbf{x})$ measured along a trajectory, the PSD is defined by

$$S^z(\omega) = \lim_{\tau \to \infty} \frac{1}{\tau} \left( \big\langle |\hat{z}_\tau(\omega)|^2 \big\rangle - \big|\langle \hat{z}_\tau(\omega)\rangle \big|^2 \right),$$

with

$$\hat{z}_\tau(\omega) = \int_0^\tau dt \, e^{i \omega t}\, z(\mathbf{x}(t)),$$

and again

$$S^z(\omega) = \int_{-\infty}^{\infty} dt \, e^{i \omega t} \, \mathrm{Cov}\big(z(t),z(0)\big)$$

by Wiener–Khinchin (Dechant, 2023).

A distinct but related formulation appears for stochastic fields governed by linear autonomous stochastic differential equations. For a field $\phi$ satisfying

$$(\mathcal{L}\phi)(\mathbf{x},t) = \xi(\mathbf{x},t),$$

with Gaussian driving $$\mathcal{F}_\Phi(\omega) = \int_{-T/2}^{T/2} \Phi(t)\,e^{-i\omega t}\,dt ,$$0, the field covariance in Fourier space is diagonal under homogeneity, and the spectral density becomes

$$\mathcal{F}_\Phi(\omega) = \int_{-T/2}^{T/2} \Phi(t)\,e^{-i\omega t}\,dt ,$$1

where $$\mathcal{F}_\Phi(\omega) = \int_{-T/2}^{T/2} \Phi(t)\,e^{-i\omega t}\,dt ,$$2 is the Fourier representation of the linear operator $$\mathcal{F}_\Phi(\omega) = \int_{-T/2}^{T/2} \Phi(t)\,e^{-i\omega t}\,dt ,$$3 (Frank et al., 2017). In that framework, the spectral density is treated as the central object that encodes the dynamics of linear autonomous SDEs.

A further variation arises in finite-time, single-trajectory analysis. For Brownian motion, the single-trajectory PSD for one component is

$$\mathcal{F}_\Phi(\omega) = \int_{-T/2}^{T/2} \Phi(t)\,e^{-i\omega t}\,dt ,$$4

which is a random functional of a single path rather than an ensemble average (Krapf et al., 2018). For active Ornstein–Uhlenbeck particles, the finite-time PSD is defined by

$$\mathcal{F}_\Phi(\omega) = \int_{-T/2}^{T/2} \Phi(t)\,e^{-i\omega t}\,dt ,$$5

which remains meaningful for both stationary and non-stationary dynamics (Kim et al., 10 May 2026).

In the fluid case, the term is used in a spatial rather than temporal sense. For a scalar field $$\mathcal{F}_\Phi(\omega) = \int_{-T/2}^{T/2} \Phi(t)\,e^{-i\omega t}\,dt ,$$6, the procedure is: restrict to a transverse band, average over $$\mathcal{F}_\Phi(\omega) = \int_{-T/2}^{T/2} \Phi(t)\,e^{-i\omega t}\,dt ,$$7, compute a one-dimensional discrete Fourier transform in $$\mathcal{F}_\Phi(\omega) = \int_{-T/2}^{T/2} \Phi(t)\,e^{-i\omega t}\,dt ,$$8, take the modulus, then perform a time average over a late-time window. The resulting spectra $$\mathcal{F}_\Phi(\omega) = \int_{-T/2}^{T/2} \Phi(t)\,e^{-i\omega t}\,dt ,$$9 are spatial power spectra in streamwise wavenumber, averaged over time, and the reported power laws have the form

$$S_\Phi(\omega) = \lim_{T\to\infty} \frac{1}{T}\,\big\langle \big|\mathcal{F}_\Phi(\omega)\big|^2\big\rangle .$$0

(Abramov, 19 Mar 2026). This usage makes explicit that “dynamical” need not imply temporal Fourier analysis; it may refer instead to the spectrum of time-averaged fluctuations generated by the dynamics.

2. Relation to dynamics, covariance, and governing equations

The defining feature of a dynamical power density spectrum is its direct relation to the system’s equations of motion and correlation structure. For stationary stochastic dynamics, the PSD is the Fourier transform of the covariance function, so peaks, plateaus, and power-law tails correspond to characteristic timescales and correlation structures of the dynamics (Kim et al., 10 May 2026). In overdamped diffusions and continuous-time Markov jump processes, the low-frequency limit satisfies

$$S_\Phi(\omega) = \lim_{T\to\infty} \frac{1}{T}\,\big\langle \big|\mathcal{F}_\Phi(\omega)\big|^2\big\rangle .$$1

while the high-frequency limit obeys

$$S_\Phi(\omega) = \lim_{T\to\infty} \frac{1}{T}\,\big\langle \big|\mathcal{F}_\Phi(\omega)\big|^2\big\rangle .$$2

(Dechant, 2023). The spectrum therefore interpolates between long-time fluctuation statistics and short-time increments.

For linear stochastic differential equations, the same connection is expressed through the transfer structure of the dynamics. In the homogeneous, stationary, Gaussian setting, the observable PSD is $$S_\Phi(\omega) = \lim_{T\to\infty} \frac{1}{T}\,\big\langle \big|\mathcal{F}_\Phi(\omega)\big|^2\big\rangle .$$3, so the deterministic operator contributes through $$S_\Phi(\omega) = \lim_{T\to\infty} \frac{1}{T}\,\big\langle \big|\mathcal{F}_\Phi(\omega)\big|^2\big\rangle .$$4, while the driving appears through $$S_\Phi(\omega) = \lim_{T\to\infty} \frac{1}{T}\,\big\langle \big|\mathcal{F}_\Phi(\omega)\big|^2\big\rangle .$$5 (Frank et al., 2017). For the damped oscillator

$$S_\Phi(\omega) = \lim_{T\to\infty} \frac{1}{T}\,\big\langle \big|\mathcal{F}_\Phi(\omega)\big|^2\big\rangle .$$6

with white noise, the PSD becomes

$$S_\Phi(\omega) = \lim_{T\to\infty} \frac{1}{T}\,\big\langle \big|\mathcal{F}_\Phi(\omega)\big|^2\big\rangle .$$7

making resonance, damping, and high-frequency decay explicit in spectral form (Frank et al., 2017).

For mechanical Langevin oscillators, the power spectral density likewise emerges from the dynamical susceptibility. For the simple harmonic Langevin oscillator,

$$S_\Phi(\omega) = \lim_{T\to\infty} \frac{1}{T}\,\big\langle \big|\mathcal{F}_\Phi(\omega)\big|^2\big\rangle .$$8

the PSD is

$$S_\Phi(\omega) = \lim_{T\to\infty} \frac{1}{T}\,\big\langle \big|\mathcal{F}_\Phi(\omega)\big|^2\big\rangle .$$9

which is the baseline for the paper’s time-dependent and anharmonic generalizations (Amer et al., 2023).

In inertial channel flow with the van der Waals effect, the spectral behavior is tied to a specific instability mechanism. The governing variables are density $R_\Phi(\tau)$0, divergence $R_\Phi(\tau)$1, and vorticity $R_\Phi(\tau)$2, and the divergence equation contains the nonlinear term $R_\Phi(\tau)$3 together with the van der Waals coupling

$R_\Phi(\tau)$4

These two together generate an “inertial-range” direct cascade in $R_\Phi(\tau)$5 and $R_\Phi(\tau)$6, and the resulting power-law spectra are therefore linked to the dynamics of compressive modes rather than to rotational turbulence in the incompressible sense (Abramov, 19 Mar 2026).

3. Canonical spectral structures

Several recurrent spectral structures appear across the literature. A strictly periodic signal with period $R_\Phi(\tau)$7 has a Dirac-comb spectrum,

$R_\Phi(\tau)$8

with weights determined by the harmonic content of the waveform (Theodorsen et al., 2021). In the semiperiodic pulse framework, the full PSD factorizes into a pulse-spectrum envelope and a forcing spectrum determined by amplitude and arrival statistics: $R_\Phi(\tau)$9 Strict periodicity plus nonzero mean amplitude yields a Dirac comb modulated by the pulse spectrum; jitter or renewal waiting times suppress or broaden the comb, leaving mainly the pulse spectrum (Theodorsen et al., 2021).

For Brownian motion, the ensemble PSD is

$z(\mathbf{x})$0

so the canonical diffusive signature is a $z(\mathbf{x})$1 spectrum (Krapf et al., 2018). For active Ornstein–Uhlenbeck particles in free space, activity does not alter the Brownian $z(\mathbf{x})$2 exponent, but modifies its amplitude and introduces a crossover at the persistence frequency. The exact infinite-time PSD is

$z(\mathbf{x})$3

so both low and high frequencies remain $z(\mathbf{x})$4, with different amplitudes separated by $z(\mathbf{x})$5 (Kim et al., 10 May 2026).

Under harmonic confinement, richer structures arise. The stationary AOUP PSD is

$z(\mathbf{x})$6

which produces two characteristic signatures absent in both thermal systems and free AOUPs: a two-plateau structure from a double-trapping mechanism due to two noise sources, and a new $z(\mathbf{x})$7 spectral scaling associated with transient ballistic motion (Kim et al., 10 May 2026). This suggests that spectral exponents can diagnose not only diffusive versus ballistic motion but also the coexistence of multiple relaxation mechanisms.

In trapped nano-oscillator models, oscillatory frequency modulation leads to sideband spectra. For

$z(\mathbf{x})$8

the PSD becomes a sum of Lorentzians centered at $z(\mathbf{x})$9, with weights $$S^z(\omega) = \lim_{\tau \to \infty} \frac{1}{\tau} \left( \big\langle |\hat{z}_\tau(\omega)|^2 \big\rangle - \big|\langle \hat{z}_\tau(\omega)\rangle \big|^2 \right),$$0, so the power spectrum develops a Floquet-like sideband structure (Amer et al., 2023). By contrast, slow frequency drift broadens the PSD into an almost flat band, while the quadrature PSD remains approximately Lorentzian (Amer et al., 2023).

In laminar channel flow with the van der Waals effect, the spectra are power laws in wavenumber. For Poiseuille flow, the paper reports

$$S^z(\omega) = \lim_{\tau \to \infty} \frac{1}{\tau} \left( \big\langle |\hat{z}_\tau(\omega)|^2 \big\rangle - \big|\langle \hat{z}_\tau(\omega)\rangle \big|^2 \right),$$1

while for Couette flow

$$S^z(\omega) = \lim_{\tau \to \infty} \frac{1}{\tau} \left( \big\langle |\hat{z}_\tau(\omega)|^2 \big\rangle - \big|\langle \hat{z}_\tau(\omega)\rangle \big|^2 \right),$$2

The exponents depend both on the variable and on the background shear profile (Abramov, 19 Mar 2026).

4. Measurement, estimation, and inference

The practical construction of a dynamical power density spectrum depends strongly on the data model. In the semiperiodic pulse framework, the PSD is computed from the finite-time Fourier transform and interpreted through a filtered point-process model, in which the pulse shape determines the envelope and the arrival statistics determine line structure, broadening, or its disappearance (Theodorsen et al., 2021). This yields a direct statistical interpretation of departures from a spectral Dirac comb.

For steady-state Markov processes, the PSD can be constrained even without a complete dynamical model. The spectrum is bounded by rational functions involving two constants $$S^z(\omega) = \lim_{\tau \to \infty} \frac{1}{\tau} \left( \big\langle |\hat{z}_\tau(\omega)|^2 \big\rangle - \big|\langle \hat{z}_\tau(\omega)\rangle \big|^2 \right),$$3 and $$S^z(\omega) = \lim_{\tau \to \infty} \frac{1}{\tau} \left( \big\langle |\hat{z}_\tau(\omega)|^2 \big\rangle - \big|\langle \hat{z}_\tau(\omega)\rangle \big|^2 \right),$$4: $$S^z(\omega) = \lim_{\tau \to \infty} \frac{1}{\tau} \left( \big\langle |\hat{z}_\tau(\omega)|^2 \big\rangle - \big|\langle \hat{z}_\tau(\omega)\rangle \big|^2 \right),$$5 and in equilibrium these constants are the low- and high-frequency limits of the normalized PSD (Dechant, 2023). Out of equilibrium, the same bounds hold but $$S^z(\omega) = \lim_{\tau \to \infty} \frac{1}{\tau} \left( \big\langle |\hat{z}_\tau(\omega)|^2 \big\rangle - \big|\langle \hat{z}_\tau(\omega)\rangle \big|^2 \right),$$6 and $$S^z(\omega) = \lim_{\tau \to \infty} \frac{1}{\tau} \left( \big\langle |\hat{z}_\tau(\omega)|^2 \big\rangle - \big|\langle \hat{z}_\tau(\omega)\rangle \big|^2 \right),$$7 no longer coincide with the limiting values, which permits intermediate-frequency peaks associated with oscillatory modes (Dechant, 2023).

For linear stochastic fields, spectral density inference can be posed as an inverse problem. In the Information Field Theory formulation, the field covariance is diagonal in Fourier space and the spectral density is parameterized as

$$S^z(\omega) = \lim_{\tau \to \infty} \frac{1}{\tau} \left( \big\langle |\hat{z}_\tau(\omega)|^2 \big\rangle - \big|\langle \hat{z}_\tau(\omega)\rangle \big|^2 \right),$$8

The smooth component $$S^z(\omega) = \lim_{\tau \to \infty} \frac{1}{\tau} \left( \big\langle |\hat{z}_\tau(\omega)|^2 \big\rangle - \big|\langle \hat{z}_\tau(\omega)\rangle \big|^2 \right),$$9 captures the background spectral shape, while $$\hat{z}_\tau(\omega) = \int_0^\tau dt \, e^{i \omega t}\, z(\mathbf{x}(t)),$$0 captures divergent or peaked features. Noisy or incomplete observations are modeled by

$$\hat{z}_\tau(\omega) = \int_0^\tau dt \, e^{i \omega t}\, z(\mathbf{x}(t)),$$1

and the spectrum is inferred by minimizing a marginal Hamiltonian after analytically marginalizing over the field (Frank et al., 2017). This provides a non-parametric route to reconstructing dynamical spectra from noisy and masked data.

Mechanical-vibration analysis offers a different estimation paradigm. High Order Dynamic Mode Decomposition represents data as

$$\hat{z}_\tau(\omega) = \int_0^\tau dt \, e^{i \omega t}\, z(\mathbf{x}(t)),$$2

or equivalently

$$\hat{z}_\tau(\omega) = \int_0^\tau dt \, e^{i \omega t}\, z(\mathbf{x}(t)),$$3

with $$\hat{z}_\tau(\omega) = \int_0^\tau dt \, e^{i \omega t}\, z(\mathbf{x}(t)),$$4 encoding damping and oscillation frequency (Tuor et al., 2023). A Kernel Density Spectrum is then formed by replacing each modal line with a kernel centered at the HODMD frequency, using Gaussian or Lorentzian kernels and weights such as $$\hat{z}_\tau(\omega) = \int_0^\tau dt \, e^{i \omega t}\, z(\mathbf{x}(t)),$$5 or $$\hat{z}_\tau(\omega) = \int_0^\tau dt \, e^{i \omega t}\, z(\mathbf{x}(t)),$$6. This produces a high-resolution, tunable spectral density from identified damped modes rather than FFT bins (Tuor et al., 2023).

In observational cosmology, the relevant objects are power spectra of galaxy density and momentum fields in redshift space. Weighted density and momentum fields are defined from galaxy catalogues, and the density–momentum cross power spectrum multipoles are estimated alongside auto-density and auto-momentum multipoles. The combined density monopole, momentum monopole, and cross dipole power spectrum are then fitted to infer the growth rate $$\hat{z}_\tau(\omega) = \int_0^\tau dt \, e^{i \omega t}\, z(\mathbf{x}(t)),$$7 (Qin et al., 2024). This suggests that “dynamical” power spectra can refer to spectra of fields that directly encode dynamical growth, not only to explicit time-series observables.

5. Variable dependence, mechanisms, and physical interpretation

A recurrent theme is that the spectral form is variable-dependent. In thermodynamic nonequilibrium settings, the PSD of an observable $$\hat{z}_\tau(\omega) = \int_0^\tau dt \, e^{i \omega t}\, z(\mathbf{x}(t)),$$8 is constrained by both the observable and the properties of the system, so different observables can have different limiting behaviors and peak structures (Dechant, 2023). In biological stochastic dynamical systems linearized at a fixed point, the matrix-valued PSD

$$\hat{z}_\tau(\omega) = \int_0^\tau dt \, e^{i \omega t}\, z(\mathbf{x}(t)),$$9

shows explicitly how each auto- and cross-spectrum depends on the Jacobian, dispersion, and diffusion matrices (Rawat et al., 2023). Each entry is a complex rational function of frequency whose denominator encodes the eigenvalues of $$S^z(\omega) = \int_{-\infty}^{\infty} dt \, e^{i \omega t} \, \mathrm{Cov}\big(z(t),z(0)\big)$$0, hence damping rates and oscillatory modes (Rawat et al., 2023).

In the van der Waals channel-flow problem, variable dependence is especially pronounced. The direct cascade and the power-law spectra are primarily a phenomenon of $$S^z(\omega) = \int_{-\infty}^{\infty} dt \, e^{i \omega t} \, \mathrm{Cov}\big(z(t),z(0)\big)$$1–$$S^z(\omega) = \int_{-\infty}^{\infty} dt \, e^{i \omega t} \, \mathrm{Cov}\big(z(t),z(0)\big)$$2 dynamics driven by the van der Waals term and the nonlinear strain term in the divergence equation, while vorticity acts mainly as a stationary background shear (Abramov, 19 Mar 2026). When vorticity is pinned to its background state and only density and divergence are evolved, the spectra of $$S^z(\omega) = \int_{-\infty}^{\infty} dt \, e^{i \omega t} \, \mathrm{Cov}\big(z(t),z(0)\big)$$3, $$S^z(\omega) = \int_{-\infty}^{\infty} dt \, e^{i \omega t} \, \mathrm{Cov}\big(z(t),z(0)\big)$$4, and the potential velocity components retain the same exponents as in the full system (Abramov, 19 Mar 2026). This strongly indicates that the underlying physics of the power spectra reside primarily in the density and velocity divergence variables, and are not directly related to the vorticity of the flow (Abramov, 19 Mar 2026).

In semiperiodic pulse trains, the relevant variable split is between pulse shape and forcing statistics. The final PSD always factorizes into a pulse-spectrum envelope and a forcing spectrum. The pulse shape controls the high-frequency envelope, while amplitude and timing statistics control whether the spectrum exhibits a Dirac comb, broadened peaks, or a smooth background (Theodorsen et al., 2021). For Lorentzian pulses, the envelope decays exponentially, so a smooth exponential PSD need not imply a special “chaos-only” mechanism; it may arise from pulse structure together with broad waiting-time statistics and nearly zero-mean amplitudes (Theodorsen et al., 2021).

A similar decomposition underlies active-particle spectra. In free AOUPs, the persistence time $$S^z(\omega) = \int_{-\infty}^{\infty} dt \, e^{i \omega t} \, \mathrm{Cov}\big(z(t),z(0)\big)$$5 sets a crossover frequency but does not change the $$S^z(\omega) = \int_{-\infty}^{\infty} dt \, e^{i \omega t} \, \mathrm{Cov}\big(z(t),z(0)\big)$$6 exponent. Under confinement, the interplay between persistence $$S^z(\omega) = \int_{-\infty}^{\infty} dt \, e^{i \omega t} \, \mathrm{Cov}\big(z(t),z(0)\big)$$7, trap relaxation $$S^z(\omega) = \int_{-\infty}^{\infty} dt \, e^{i \omega t} \, \mathrm{Cov}\big(z(t),z(0)\big)$$8, and activity strength creates multi-regime spectra, including the $$S^z(\omega) = \int_{-\infty}^{\infty} dt \, e^{i \omega t} \, \mathrm{Cov}\big(z(t),z(0)\big)$$9 range associated with transient ballistic motion (Kim et al., 10 May 2026). This suggests that spectral exponents can separate mechanisms even when mean-square displacement alone does not.

6. Nonequilibrium, finite-time effects, and interpretation limits

Dynamical power density spectra are particularly sensitive to nonequilibrium structure. For continuous-time Markov processes, the spectrum at arbitrary frequency is bounded by short- and long-time behavior in equilibrium, but out of equilibrium the constants entering the bounds can no longer be identified with the limiting behavior of the spectrum, allowing for peaks that correspond to oscillations in the dynamics (Dechant, 2023). A key finite-frequency entropy production bound is

$\phi$0

so the height of peaks in $\phi$1 above the high-frequency plateau is related to dissipation (Dechant, 2023).

Finite observation time introduces additional structure. For free AOUPs, the finite-time PSD develops a low-frequency plateau whose leading behavior is

$\phi$2

for large $\phi$3, together with high-frequency oscillations whose amplitude decays as $\phi$4 (Kim et al., 10 May 2026). Under confinement, the short-window plateau instead scales as

$\phi$5

and the oscillatory corrections decay as $\phi$6 with exponential dependence on $\phi$7 and $\phi$8 (Kim et al., 10 May 2026). This difference shows that finite-time effects can distinguish non-stationary free motion from stationary confined motion.

Single-trajectory analysis imposes further limitations. For Brownian motion, the single-trajectory PSD has the same $\phi$9 scaling as the ensemble PSD in the regime of large $$(\mathcal{L}\phi)(\mathbf{x},t) = \xi(\mathbf{x},t),$$0, but the amplitude is a fluctuating random factor: $$(\mathcal{L}\phi)(\mathbf{x},t) = \xi(\mathbf{x},t),$$1 Thus the scaling exponent can be inferred from a single trajectory, but the numerical amplitude, and hence the diffusion coefficient, cannot be reliably extracted from a single PSD because the amplitude remains broadly distributed even as $$(\mathcal{L}\phi)(\mathbf{x},t) = \xi(\mathbf{x},t),$$2 at fixed nonzero frequency (Krapf et al., 2018). This shows that a dynamical power density spectrum may encode robust dynamical class information while leaving absolute transport coefficients poorly determined in finite or single-realization data.

A related interpretive caution appears in interferometric astronomy. When the power spectrum is estimated from reconstructed images rather than from visibilities, the image-based estimator acquires a scale-dependent bias due to incompleteness in baseline coverage, whereas the visibility-based estimator reproduces the true power spectrum (Nandakumar, 2018). This suggests that the reliability of a dynamical spectrum depends not only on the definition of the spectral object but also on the estimator and observation operator.

7. Broader significance and cross-domain usage

The phrase “dynamical power density spectrum” spans several scientific idioms, but the underlying role is consistent: it is a frequency- or wavenumber-resolved measure of fluctuations generated by a dynamical process, used to identify timescales, modes, transport regimes, and underlying mechanisms. In stochastic thermodynamics it serves as a frequency-resolved variance constrained by relaxation and dissipation (Dechant, 2023). In semiperiodic nonlinear dynamics it distinguishes periodic, quasi-periodic, and chaotic regimes through Dirac combs, broadened harmonics, or pulse envelopes (Theodorsen et al., 2021). In active matter it diagnoses persistence, confinement, and transient ballistic motion through plateaus and power-law crossovers (Kim et al., 10 May 2026). In fluid dynamics it reveals inertial-like cascades in nominally laminar flows and can isolate the variables carrying the instability (Abramov, 19 Mar 2026).

The term also generalizes beyond time series. In 21-cm studies of interstellar turbulence, the relevant “dynamical power density spectrum” is realized as spatial power spectra of density and velocity fields in wavenumber space, from which the energy associated with structures and the type of forcing of the turbulence are inferred (Nandakumar, 2018). In cosmology, joint power spectra of density and momentum fields in redshift space form a dynamical spectrum of large-scale structure growth, constraining $$(\mathcal{L}\phi)(\mathbf{x},t) = \xi(\mathbf{x},t),$$3 through the density monopole, momentum monopole, and density–momentum cross dipole (Qin et al., 2024). In biological stochastic dynamical systems, PSD matrices at fixed points provide an exact rational representation of the second-order statistics of fluctuations, with coefficients determined by the Jacobian, dispersion, and diffusion matrices (Rawat et al., 2023).

Taken together, these usages suggest a broad definition: a dynamical power density spectrum is a spectral representation of fluctuation power that is constructed from, and interpreted through, the governing dynamics. The spectral variable may be temporal frequency, spatial wavenumber, or a mixed representation. The observable may be a scalar time series, a vector field, or a set of coupled stochastic degrees of freedom. What makes the spectrum dynamical is not the notation $$(\mathcal{L}\phi)(\mathbf{x},t) = \xi(\mathbf{x},t),$$4 or $$(\mathcal{L}\phi)(\mathbf{x},t) = \xi(\mathbf{x},t),$$5 alone, but the explicit connection between spectral structure and the mechanisms that generate, transport, damp, or organize fluctuations.