Intermediate Scattering Function

• Intermediate scattering function is a time-dependent correlation function that encapsulates density fluctuations and particle displacements via Fourier transform relations.
• It reveals various dynamical regimes including ballistic motion, diffusive behavior, and anomalous transport in systems like liquids, glasses, and porous media.
• It acts as a generating function for moments and cumulants, enabling extraction of transport coefficients and in-depth analysis in scattering experiments.

The intermediate scattering function (ISF) is a time-dependent correlation function of density fluctuations and, for the self part, a characteristic function of particle displacement. In scattering theory it is related to the dynamic structure factor by temporal Fourier transform, and in probability-theoretic language it is the Fourier transform of the displacement distribution or of the self part of the van Hove function. Across surface diffusion, liquids, glasses, porous media, active matter, colloids in periodic fields, and quantum diffusion, the ISF serves as a compact observable that encodes ballistic motion, caging, diffusion, anomalous transport, oscillatory persistence, and quantum recoil (Torres-Miyares et al., 17 Sep 2025, Zhao et al., 2024, Townsend et al., 2018).

1. Formal definitions and notation

In quasielastic helium atom scattering, the measured differential reflection coefficient is proportional to the dynamic structure factor $S(\Delta K,\omega)$, and the ISF is obtained via the frequency Fourier transform

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$

In neutron scattering, the ISF can be written as the Fourier transform of the autocorrelation of the particle-density operator, or, in terms of the van Hove space-time correlation function. Within the Born approximation, $S$ and $I$ can be expressed in terms of the generalized pair-distribution function (van Hove function) (Torres-Miyares et al., 17 Sep 2025).

For the self part, one standard definition is

$$F_s(k,t)=\frac{1}{N}\sum_{i=1}^N \left\langle \exp\!\big[i\,k\cdot (r_i(t)-r_i(0))\big]\right\rangle,$$

and the corresponding van Hove relation is

$F_s(k,t)=\int dr\, e^{i k\cdot r} G_s(r,t),$

where $G_s(r,t)$ is the probability density for a particle to move by $r$ in time $t$. In collective form, the ISF is

$$F(q,\tau)=\frac{1}{N}\sum_{j,k=1}\left\langle \exp\!\left[i\vec{q}\,(\vec{r_j}(\tau)-\vec{r_k}(0))\right]\right\rangle,$$

with $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$0, the static structure factor (Zhao et al., 2024, Martinez et al., 2010).

Surface-diffusion work often writes

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$1

where $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$2 are the diagonal elements of the reduced density matrix, and for displacement $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$3,

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$4

This identifies the self ISF as the characteristic function of the displacement probability distribution (Torres-Miyares et al., 17 Sep 2025).

Notation varies by subfield. Surface diffusion commonly uses $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$5; liquid-state and glass literature uses $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$6, $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$7, or $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$8; active-matter papers often use $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$9 for a single particle; depolarized dynamic light scattering in vertical–horizontal geometry uses the field autocorrelation $S$0 as the relevant intermediate scattering function (Passow et al., 2015, Kurzthaler et al., 2019).

2. Characteristic-function structure, moments, and cumulants

Because the self ISF is a characteristic function, its derivatives at zero wavevector generate displacement moments and cumulants. In the one-dimensional form used for surface diffusion,

$S$1

In multiple dimensions, the derivatives are taken with respect to the Cartesian components of $S$2, and moments and cumulants are tensors (Torres-Miyares et al., 17 Sep 2025).

The formal cumulant expansion at small $S$3 can be written as

$S$4

and retaining only the second cumulant tensor $S$5 gives

$S$6

For continuous isotropic diffusion in two dimensions, the standard Gaussian result is

$S$7

In the one-dimensional projected jump model used for surface diffusion, the first moment vanishes and the second moment equals the second cumulant,

$S$8

with

$S$9

This relation follows by expanding the exact ISF for small $I$0 (Torres-Miyares et al., 17 Sep 2025).

The same low-$I$1 logic appears in other contexts. For an anisotropic active Brownian particle in three dimensions,

$I$2

and the exact mean-square displacement is

$I$3

For long times $I$4, this implies

$I$5

The quartic moment then yields the non-Gaussian parameter $I$6 (Kurzthaler et al., 2017).

In periodic potentials, time-dependent perturbation theory provides the low-order moments directly from the ISF. For a single overdamped Brownian colloid in a one-dimensional cosine potential, the time-dependent diffusivity is defined by

$I$7

and the long-time diffusivity is

$I$8

This construction makes explicit that the ISF is not merely a decay function: it is a generating object for transport coefficients and higher-order deviations from Gaussian dynamics (Rusch et al., 1 Jul 2025).

3. Dynamical regimes encoded by the ISF

The short-time ballistic limit is explicit in several formulations. In vibration–transit theory for a monatomic liquid,

$I$9

and this free-particle behavior holds only up to a short ballistic time $$F_s(k,t)=\frac{1}{N}\sum_{i=1}^N \left\langle \exp\!\big[i\,k\cdot (r_i(t)-r_i(0))\big]\right\rangle,$$0. At long times, when $$F_s(k,t)=\frac{1}{N}\sum_{i=1}^N \left\langle \exp\!\big[i\,k\cdot (r_i(t)-r_i(0))\big]\right\rangle,$$1, the SISF reduces to

$$F_s(k,t)=\frac{1}{N}\sum_{i=1}^N \left\langle \exp\!\big[i\,k\cdot (r_i(t)-r_i(0))\big]\right\rangle,$$2

V–T theory organizes the dynamics into a vibrational interval, a crossover interval, and a diffusive interval, with the full function factorized as

$$F_s(k,t)=\frac{1}{N}\sum_{i=1}^N \left\langle \exp\!\big[i\,k\cdot (r_i(t)-r_i(0))\big]\right\rangle,$$3

(Wallace et al., 2016).

In glassy and glass-like systems, the ISF resolves caging and structural relaxation. In a two-dimensional dodecagonal quasicrystal, the cage-relative self-ISF $$F_s(k,t)=\frac{1}{N}\sum_{i=1}^N \left\langle \exp\!\big[i\,k\cdot (r_i(t)-r_i(0))\big]\right\rangle,$$4 displays the canonical two-step relaxation across the quasicrystal temperatures studied $$F_s(k,t)=\frac{1}{N}\sum_{i=1}^N \left\langle \exp\!\big[i\,k\cdot (r_i(t)-r_i(0))\big]\right\rangle,$$5: a fast, weakly $$F_s(k,t)=\frac{1}{N}\sum_{i=1}^N \left\langle \exp\!\big[i\,k\cdot (r_i(t)-r_i(0))\big]\right\rangle,$$6-dependent $$F_s(k,t)=\frac{1}{N}\sum_{i=1}^N \left\langle \exp\!\big[i\,k\cdot (r_i(t)-r_i(0))\big]\right\rangle,$$7 process, an intermediate-time plateau (cage regime), and a slow, strongly $$F_s(k,t)=\frac{1}{N}\sum_{i=1}^N \left\langle \exp\!\big[i\,k\cdot (r_i(t)-r_i(0))\big]\right\rangle,$$8-dependent $$F_s(k,t)=\frac{1}{N}\sum_{i=1}^N \left\langle \exp\!\big[i\,k\cdot (r_i(t)-r_i(0))\big]\right\rangle,$$9 decay. The fitting form used there is

$F_s(k,t)=\int dr\, e^{i k\cdot r} G_s(r,t),$0

with $F_s(k,t)=\int dr\, e^{i k\cdot r} G_s(r,t),$1 defined by $F_s(k,t)=\int dr\, e^{i k\cdot r} G_s(r,t),$2. In the same work, $F_s(k,t)=\int dr\, e^{i k\cdot r} G_s(r,t),$3 versus $F_s(k,t)=\int dr\, e^{i k\cdot r} G_s(r,t),$4 bends upwards below $F_s(k,t)=\int dr\, e^{i k\cdot r} G_s(r,t),$5, signaling non-Arrhenius growth of relaxation times upon cooling (Zhao et al., 2024).

In disordered porous media near a localization transition, the ISF becomes a probe of anomalous transport and fractal geometry. For tracer dynamics on the infinite cluster of the Lorentz model at criticality,

$F_s(k,t)=\int dr\, e^{i k\cdot r} G_s(r,t),$6

with $F_s(k,t)=\int dr\, e^{i k\cdot r} G_s(r,t),$7, and the long-time decay at small $F_s(k,t)=\int dr\, e^{i k\cdot r} G_s(r,t),$8 approaches

$F_s(k,t)=\int dr\, e^{i k\cdot r} G_s(r,t),$9

where $G_s(r,t)$0. The relaxation time defined by $G_s(r,t)$1 obeys

$G_s(r,t)$2

These results establish that the terminal behavior need not be exponential and that the ISF can directly reveal dynamic scaling controlled by percolation geometry (Spanner et al., 2012).

Active systems produce another distinct class of signatures. For an anisotropic active Brownian particle in three dimensions, the most prominent feature of the ISF is an oscillatory behavior at intermediate wavenumbers reflecting the persistent swimming motion, whereas at small length scales bare translational and at large length scales an enhanced effective diffusion emerges. For an anisotropic Brownian circle swimmer in two dimensions, persistent circular motion produces oscillations at an elevated plateau, and in the deterministic circular limit

$G_s(r,t)$3

with plateau height $G_s(r,t)$4 and oscillation frequency $G_s(r,t)$5 (Kurzthaler et al., 2017, Kurzthaler et al., 2019).

Periodic landscapes can also produce two-step relaxation, but now the mechanism is intra-well equilibration followed by inter-well hopping. For colloids in a one-dimensional periodic laser field, the diagonal ISF exhibits single-step decay for moderate barriers and two-step relaxation for larger $G_s(r,t)$6: fast approach to an intra-well plateau, followed by slow decay at small $G_s(r,t)$7 governed by

$G_s(r,t)$8

This suggests that superficially similar plateau structures may originate from caging, confinement by a periodic substrate, or persistent active motion, and the ISF distinguishes these possibilities through its detailed $G_s(r,t)$9- and $r$0-dependence (Rusch et al., 1 Jul 2025).

4. Experimental realizations and analysis protocols

The ISF is directly accessible or reconstructible in a broad range of scattering and imaging experiments. Helium spin echo directly measures $r$1 in the time domain, with

$r$2

In surface diffusion, a common diffusive-regime fit is

$r$3

and for nearest-neighbor diffusion on Pt(111) the total jump rate is extracted through

$r$4

For the geometry considered there, $r$5, $r$6, and $r$7, with coverage $r$8 ML (Torres-Miyares et al., 17 Sep 2025).

Dynamic light scattering and X-ray photon correlation spectroscopy measure the normalized intensity autocorrelation function

$r$9

which is linked to the normalized field correlation through the Siegert relation

$t$0

For diffusive density fluctuations, the short-time form is

$t$1

and a useful diagnostic is the width function

$t$2

whose linear growth in $t$3 identifies diffusive regimes without requiring numerical differentiation. In concentrated hard-sphere suspensions, long-time collective diffusion is only observed at $t$4 for sufficiently high $t$5; away from $t$6, the ISF reaches the experimental noise floor before any clear long-time diffusive plateau appears (Martinez et al., 2010).

Differential dynamic microscopy provides a direct route to the ISF in active and driven systems. For chiral swimmers, the image structure function obeys

$t$7

while for gravitactic circle swimmers the measured signal with a known mean drift $t$8 is

$t$9

This shows that the comoving-frame ISF $$F(q,\tau)=\frac{1}{N}\sum_{j,k=1}\left\langle \exp\!\left[i\vec{q}\,(\vec{r_j}(\tau)-\vec{r_k}(0))\right]\right\rangle,$$0 can be accessed by shifting images at the known drift velocity (Kurzthaler et al., 2019, Rusch et al., 2024).

Depolarized dynamic light scattering in vertical–horizontal geometry uses the field autocorrelation $$F(q,\tau)=\frac{1}{N}\sum_{j,k=1}\left\langle \exp\!\left[i\vec{q}\,(\vec{r_j}(\tau)-\vec{r_k}(0))\right]\right\rangle,$$1. In dilute, ergodic suspensions of independent anisotropic particles in the single-scattering regime,

$$F(q,\tau)=\frac{1}{N}\sum_{j,k=1}\left\langle \exp\!\left[i\vec{q}\,(\vec{r_j}(\tau)-\vec{r_k}(0))\right]\right\rangle,$$2

and the measured first cumulant

$$F(q,\tau)=\frac{1}{N}\sum_{j,k=1}\left\langle \exp\!\left[i\vec{q}\,(\vec{r_j}(\tau)-\vec{r_k}(0))\right]\right\rangle,$$3

is generally not simply proportional to $$F(q,\tau)=\frac{1}{N}\sum_{j,k=1}\left\langle \exp\!\left[i\vec{q}\,(\vec{r_j}(\tau)-\vec{r_k}(0))\right]\right\rangle,$$4, because translational–rotational coupling and rotational multipole weights both contribute (Passow et al., 2015).

5. Exact and semi-exact theoretical constructions

A major class of exact results arises from jump processes. For Poissonian nearest-neighbor jumps on a periodic lattice with symmetry-equivalent jump vectors $$F(q,\tau)=\frac{1}{N}\sum_{j,k=1}\left\langle \exp\!\left[i\vec{q}\,(\vec{r_j}(\tau)-\vec{r_k}(0))\right]\right\rangle,$$5, the ISF is the characteristic function of a compound Poisson process,

$$F(q,\tau)=\frac{1}{N}\sum_{j,k=1}\left\langle \exp\!\left[i\vec{q}\,(\vec{r_j}(\tau)-\vec{r_k}(0))\right]\right\rangle,$$6

In the one-dimensional specialization used for surface diffusion,

$$F(q,\tau)=\frac{1}{N}\sum_{j,k=1}\left\langle \exp\!\left[i\vec{q}\,(\vec{r_j}(\tau)-\vec{r_k}(0))\right]\right\rangle,$$7

and for multiple jump shells,

$$F(q,\tau)=\frac{1}{N}\sum_{j,k=1}\left\langle \exp\!\left[i\vec{q}\,(\vec{r_j}(\tau)-\vec{r_k}(0))\right]\right\rangle,$$8

These expressions are the Chudley–Elliott form and its multi-shell generalization (Torres-Miyares et al., 17 Sep 2025).

For monatomic liquids, V–T theory derives the self-ISF from harmonic vibrations in a random valley plus local transits between valleys. The vibrational contribution is

$$F(q,\tau)=\frac{1}{N}\sum_{j,k=1}\left\langle \exp\!\left[i\vec{q}\,(\vec{r_j}(\tau)-\vec{r_k}(0))\right]\right\rangle,$$9

and the transit damping factor satisfies

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$00

The resulting factorization,

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$01

was shown to reproduce molecular-dynamics data with extremely small deviations at all $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$02 and $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$03 in liquid Na at $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$04 K (Wallace et al., 2016).

Active-particle formulations often proceed by spectral theory on orientation space. For an anisotropic active Brownian particle in three dimensions, the exact ISF is an expansion in generalized spheroidal wave functions,

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$05

with $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$06 and $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$07. For an anisotropic Brownian circle swimmer in two dimensions, the exact ISF is obtained from generalizations of the Mathieu functions,

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$08

In both cases, non-Hermitian spectra generate oscillatory contributions when eigenvalues form complex-conjugate pairs (Kurzthaler et al., 2017, Kurzthaler et al., 2019).

Periodic media motivate a generalized, matrix-valued ISF. For a single overdamped Brownian colloid in a one-dimensional periodic potential, the generalized ISF is

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$09

with $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$10 in the first Brillouin zone and $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$11. The diagonal elements recover the conventional ISF, $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$12, while off-diagonal elements are finite in periodic media and are required to reconstruct the propagator. The spectral representation follows from Bloch’s theorem and diagonalization of the Smoluchowski operator in a plane-wave basis (Rusch et al., 1 Jul 2025).

Mode-coupling theory enters most prominently in porous-media problems. For the Lorentz model, the exact equation of motion is written in Zwanzig–Mori form,

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$13

with $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$14. In the generalized-hydrodynamics approximation, the theory predicts $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$15, $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$16, and $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$17, capturing certain aspects of the anomalous critical ISF surprisingly well while missing pronounced long-time tails (Spanner et al., 2012).

6. Quantum, recoil, and memory-friction formulations

Quantum-mechanical treatments reinterpret the ISF as a correlation of thermally prepared states perturbed at different times. In the one-dimensional single-particle reduction,

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$18

Using thermal wave packets with random phases,

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$19

the ISF becomes an ensemble average of overlaps between “kick-then-evolve” and “evolve-then-kick” states,

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$20

In the ballistic proof of concept,

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$21

The real part is the classical ballistic mean-square displacement, while the imaginary part produces a quantum recoil phase (Bindech et al., 2024).

For a quantum particle diffusing in a harmonic potential and linearly coupled to a harmonic bath, the exact non-Markovian result is

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$22

with

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$23

and

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$24

The imaginary part of the exponent is proportional to an accumulated phase and is an antisymmetric function of the correlation time $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$25. Two universal properties follow directly: the accumulated phase has a universal gradient at the origin,

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$26

and for unconfined diffusion its large-$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$27 limit is

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$28

The plateau depends only on the classical diffusion coefficient $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$29 and is therefore independent of the detailed memory properties of the friction kernel (Townsend et al., 2018).

A closely related classical-memory treatment starts from the generalized Langevin equation with exponential memory friction,

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$30

for diffusion on a flat surface. In this case the exact ISF is

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$31

where $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$32 solve

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$33

The long-time decay rate remains

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$34

independent of $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$35, but the amplitude of the long-time exponential tail is suppressed relative to the Markovian case by

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$36

This separates the memory timescale from the asymptotic diffusion coefficient in a way that is directly relevant for quasi-elastic line-shape analysis (Townsend et al., 2018).

7. Assumptions, limitations, and interpretive issues

Interpretation of the ISF depends strongly on which part of the correlation function is being measured. A common source of confusion is the distinction between self and collective ISFs. The hard-sphere study measures the collective ISF, not the self part, whereas the quasicrystal, active-matter, porous-media, and most surface-diffusion analyses focus on the self contribution (Martinez et al., 2010, Zhao et al., 2024).

Many closed-form expressions rely on restrictive stochastic assumptions. In surface diffusion, the exponential Chudley–Elliott form assumes a Markovian jump or tunneling process with independent increments, instantaneous jumps, and stationarity. In the Pt(111) application, adsorbate–adsorbate interactions are not treated explicitly, coverage is low $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$37 ML), and the analysis uses a specific lattice geometry and measurement angle $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$38 (Torres-Miyares et al., 17 Sep 2025).

Gaussian or cumulant-truncated approximations are likewise regime-dependent. The small-$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$39 form

$$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$40

is controlled by the second cumulant and is therefore reliable only when higher cumulants are negligible. In active particles, porous media, and periodic substrates, higher moments and non-Gaussian parameters become central observables rather than small corrections (Kurzthaler et al., 2017, Spanner et al., 2012, Rusch et al., 1 Jul 2025).

Two-dimensional systems require additional care because long-wavelength fluctuations can contaminate the standard self-ISF. In the dodecagonal quasicrystal study, cage-relative definitions of MSD, $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$41, and $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$42 are used to mitigate 2D fluctuations and finite-size artifacts associated with Mermin–Wagner motions. This suggests that an observed plateau in a conventional two-dimensional ISF need not be a purely local caging signal unless long-wavelength contributions are removed (Zhao et al., 2024).

Another misconception is that a long-time diffusive regime is always observable if measurements extend far enough. In concentrated hard-sphere suspensions, long-time collective diffusion is only observed at $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$43 for sufficiently high $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$44; away from $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$45, no clear long-time diffusive plateau appears within the accessible dynamic range. The width function $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$46 is therefore emphasized as a more robust diagnostic than direct numerical differentiation of $$I(\Delta K,t)=\frac{1}{2\pi}\int d\omega\, e^{i\omega t} S(\Delta K,\omega).$$47 (Martinez et al., 2010).

Finally, the ISF need not be purely monotone or purely real. Off-diagonal generalized ISFs in periodic media can be non-monotone, with minima and maxima at intermediate times. Active and chiral swimmers exhibit oscillatory ISFs at intermediate wavenumbers. Quantum ISFs acquire a phase from recoil, and non-Markovian baths can make the accumulated phase non-monotonic. These cases show that the ISF is not merely a relaxation envelope; it is a full spatiotemporal correlator whose detailed structure depends on symmetry, conservation laws, geometry, and the stochastic or quantum dynamics that generate the underlying displacement process (Rusch et al., 1 Jul 2025, Kurzthaler et al., 2019, Townsend et al., 2018).