Intermediate Scattering Function in Anisotropic ABPs

• Intermediate Scattering Function (ISF) is defined as the spatial Fourier transform of the n-resolved probability distribution, capturing all positional moments in active systems.
• Moment expansion techniques systematically truncate the infinite moment hierarchy, enabling analytical approximations that reveal oscillatory and non-Gaussian features.
• Experimental comparisons and simulations show that higher closure orders (Order 1 and 2) accurately predict ISF behavior in anisotropic active Brownian particles under moderate to high Péclet numbers.

The intermediate scattering function (ISF) for anisotropic active Brownian particles (ABPs) provides a compact, experimentally accessible statistical description of particle displacements arising from the interplay of self-propulsion, rotational diffusion, and anisotropic translational diffusion. As the spatial Fourier transform of the n-resolved probability distribution, the ISF encodes all positional moments and serves as a central observable for both theory and experiments, particularly in regimes dominated by non-Gaussian, persistent, and correlated motion. Addressing the complexity of the full Fokker–Planck equation, moment expansion and systematic closure schemes offer explicit, approximate analytical access to the ISF and related orientational fields—a key advantage in regimes where exact formulations are only implicit.

1. Formal Definition and Role of the ISF

For a single ABP, the ISF is defined as

$$F(\mathbf{q}, t) = \int d\mathbf{r} \, e^{-i\mathbf{q}\cdot\mathbf{r}} \, \rho(\mathbf{r}, t)$$

where $\rho(\mathbf{r}, t)$ is the probability density of the particle's position at time $t$. In the full Fokker–Planck framework, the joint distribution $P(\mathbf{r}, \mathbf{n}, t)$ evolves under coupled translational and orientational stochastic dynamics, with $\mathbf{n}$ the orientation vector. For anisotropic ABPs, the translational diffusion tensor and the self-propulsion vector are not collinear, and their coupling to orientation generates nontrivial cross-correlations. The ISF thus not only determines the mean-square displacement (MSD) and higher moments of the trajectory, but also reflects persistent motion via oscillatory and non-Gaussian features, making it central for interpreting scattering experiments such as differential dynamic microscopy.

2. Moment Expansion of the Fokker–Planck Equation

To obtain tractable expressions, $P(\mathbf{r}, \mathbf{n}, t)$ is systematically expanded as a series of orientational moments:

• 0th order: scalar density $\rho(\mathbf{r}, t)$,
• 1st order: polarization field $\mathbf{T}(\mathbf{r}, t)$,
• 2nd order: nematic tensor $Q_{ij}(\mathbf{r}, t)$,
• Higher orders corresponding to higher multipole moments.

The hierarchy arises because the Fokker–Planck operator couples each moment to those of higher (and, via anisotropy, up to two) order(s). For example:

• The evolution of $\rho$ couples to $\nabla\cdot \mathbf{T}$;
• $\mathbf{T}$ is coupled to $\rho$, gradients thereof, and $\nabla\cdot Q$ terms;
• $Q$ receives source terms from $\mathbf{T}$, etc.

This results in an infinite coupled hierarchy, which in practice requires suitable closure strategies for analytical solutions.

3. Closure Strategies: Order 0, 1, and 2 Schemes

Several closure levels are compared and analyzed:

Closure at Order 0:

• Truncate higher-order fields ($\mathbf{T}=0$, $Q=0$), giving

$F_0(\mathbf{q}, t) = \exp(-D_0 q^2 t)$

where $D_0$ is the mean translational diffusion. Captures purely diffusive, Gaussian behavior; not valid for persistent-active regimes.

Refined Order 0 (Slaved T):

• Assume polarization is “enslaved” to density gradients:

$$\mathbf{T}(\mathbf{q}, t) \simeq -\frac{v}{d(d-1) D_r} i\mathbf{q} \, \rho(\mathbf{q}, t)$$

The ISF remains Gaussian but with activity-enhanced diffusion $D_{0,\text{eff}} = D_0 + \frac{v^2}{d(d-1) D_r}$.

Order 1:

• Retain polarization, solve coupled equations for $\rho$ and $\mathbf{T}$.
• The ISF acquires non-Gaussian structure, explicitly displaying intermediate-time oscillations due to self-propulsion:

$$F_1(q, t) \text{ is given by the inverse Laplace transform of a rational function in frequency, with poles reflecting persistent and relaxational modes.}$$

• Oscillations in $F_1(q, t)$ are tied to persistent swimming and only become prominent at intermediate $q$ (length scales comparable to the persistence length).

Order 2:

• Retain both polarization and nematic tensors, yielding a closed system for $\rho$, $\mathbf{T}$, and $Q$.
• ISF is given in terms of a sum of three exponentials (from residue analysis of the Fourier-transformed coupled equations).
• Captures both the period and the amplitude of ISF oscillations at larger $q$ or higher Péclet numbers, improving agreement with the exact solution and simulations.

Closure Level	Moments Retained	ISF Behavior	Regime Captured
0	$\rho$	Gaussian, single exp	Passive-like, large scales
0 (refined)	$\rho$, slaved $\mathbf{T}$	Gaussian, enhanced diffusion	Small $Pe$, short times
1	$\rho$, $\mathbf{T}$	Non-Gaussian, oscillatory	Active, persistent regime
2	$\rho$, $\mathbf{T}$, $Q$	Quantitative match to exact	Full range up to $qL_p\sim 10$

4. Comparison to Exact Solutions and Simulations

The moment expansion closures are benchmarked against:

• The formally exact ISF obtained via spectral decompositions in spheroidal wave functions.
• Brownian dynamics simulations for 3D anisotropic ABPs.

Key conclusions:

• Order 0 and its refined counterpart are accurate only at large length scales (small $q$), or short times.
• Order 1 captures the onset, frequency, and qualitative nature of ISF oscillations, but can overestimate their amplitude at higher $q$.
• Order 2 closure achieves quantitative agreement with exact numerics and simulations up to $qL_p \sim 10$, and across moderate Péclet numbers.
• The ISF error metric $E(q, Pe)$ demonstrates that higher closure order consistently improves accuracy at intermediate $q$ and higher persistence.

Increasing anisotropy (via $\Delta D / D_0$ or aspect ratio) for fixed $Pe$ enhances the amplitude of non-Gaussian features in the ISF but does not change their period.

5. Practical Applications and Extensions

The explicit, closed-form ISFs from the moment expansion framework are directly applicable to both analysis and design of scattering experiments:

• Differential dynamic microscopy (DDM): ISF data analysis can benefit from models at closure order 1 or 2, which yield not only $F(q, t)$ but also polarization and nematic field correlations—enabling deeper experimental probes of orientational correlations.
• Theoretical modeling: Moment expansion and systematic closure methods offer an efficient alternative to numerically intensive full eigenfunction decompositions, especially valuable for parameter inference or simulation studies requiring rapid evaluations over a range of $q$, $Pe$, and anisotropy.

The modular construction of the moment hierarchy allows extensions to more complex scenarios:

• Propulsion switching and non-Markovian orientation dynamics (via additional moment couplings);
• Confinement or external fields (via modifications to the Fokker–Planck operator and its moments);
• Cases lacking available functional eigenbases (e.g., for external potentials or complex propulsion statistics).

6. Regimes of Validity and Main Conclusions

The regime diagram for closure selection is dictated by length scale ($qL_p$), observation time, and Péclet number ($Pe$):

• Order 0 (and refined): Large scale/short time, negligible persistence.
• Order 1: Intermediate $q$, moderate $Pe$—captures first non-Gaussian corrections (oscillations).
• Order 2: Required for quantifying amplitude and period of oscillations at moderate/high $Pe$ and for higher $q$ (up to $qL_p \sim 10$).

For increasing anisotropy, the structure of the coupling in the moment hierarchy remains valid, and the closures are robust, although strong increases in $Pe$ may require retaining even higher-order moments for precise ISF quantification.

In summary, the moment expansion and systematic closure approach to the ISF in anisotropic ABPs offers explicit, efficient, and physically transparent analytical tools that bridge the gap between exact implicit solutions and experiment-relevant, multi-modal, non-Gaussian dynamics. This method supports accurate modeling and interpretation for a range of active-matter experiments and can be generalized to broader classes of active, confined, or interacting systems (Gautry et al., 30 Sep 2025).