Anisotropic Active Brownian Particles

• Anisotropic active Brownian particles are self-propelled entities whose dynamics couple position and orientation, resulting in direction-dependent diffusion and stress.
• Systematic moment expansion and closure schemes capture both Gaussian and oscillatory non-Gaussian decay in the intermediate scattering function.
• Explicit formulations from the Fokker–Planck framework connect microscopic parameters to macroscopic observables in experimental setups such as synthetic colloids and biological microswimmers.

Anisotropic active Brownian particles (ABPs) are self-propelled microscopic or mesoscopic particles whose stochastic dynamics couple position and orientation, and whose translational and/or rotational transport coefficients, self-propulsion rules, or external environments explicitly break isotropy. Unlike classic isotropic ABPs—whose motility and fluctuations are identical in all directions—anisotropic ABPs exhibit direction-dependent diffusion, propulsion, stress, and response functions, yielding rich nonequilibrium behavior relevant to biological microswimmers, synthetic active colloids, glassy matter, and complex soft materials.

1. Moment Expansion Hierarchy and Systematic Closures

The dynamical evolution of anisotropic ABPs is governed by a Fokker–Planck equation for the joint probability density $P(\mathbf{r},\mathbf{n};t)$, where $\mathbf{r}$ is the position and $\mathbf{n}$ the orientation. Anisotropic motility and diffusion, often encoded in the translational diffusion tensor

$$D_{ij} = D_\parallel n_i n_j + D_\perp (\delta_{ij} - n_i n_j),$$

render the system fundamentally nontrivial: moments of the distribution over orientation, such as the density $\rho(\mathbf{r},t) = \int d\Omega\,P$, polarization $$\mathbf{T}(\mathbf{r},t) = \int d\Omega\,\mathbf{n}\,P$$, and traceless nematic tensor $$Q_{ij}(\mathbf{r},t) = \int d\Omega\,(n_i n_j-\delta_{ij}/d)P$$, obey a coupled infinite hierarchy reminiscent of the BBGKY or moment hierarchies in kinetic theory.

To proceed, a closure scheme truncates the expansion at a given order:

• Order 0: Neglecting both polarization and higher moments, the density obeys a simple isotropic or anisotropic diffusion equation, yielding a Gaussian intermediate scattering function (ISF), $F_0(q,t) = \exp(-D_0 q^2 t)$.
• Refined Order 0: By assuming steady-state and spatially uniform polarization, but neglecting nematic and higher moments, an effective enhanced diffusion emerges:

$D_{0,\text{eff}} = D_0 + \frac{v^2}{d(d-1)D_r}$

and $$F_{0,\text{Act}}(q,t) = \exp\left(-D_{0,\text{eff}} q^2 t\right)$$.

• Order 1: Retaining the polarization as a dynamic field while neglecting the nematic, the ISF becomes

$$F_1(q,t) = \textrm{prefactor} \times \exp(-b t/2) \left[\cos(\sqrt{\delta}\, t/2) + \cdots\right]$$

for coefficients $b$, $\delta$ specified by propulsion speed $v$, orientational relaxation $D_r$, and translational anisotropy.

• Order 2: Including the nematic tensor $Q$ generates a closed $3\times3$ linear system in Fourier space, giving the ISF as a sum over three exponentials with explicit residues.

The explicit coupled equations in real or Fourier space demonstrate that, at each order, the closure relates the dynamics of the lower moment to the next higher, and the propagation of anisotropy through the hierarchy is essential for capturing activity-induced correlations and ISF structure.

2. Mathematically Explicit Formulations

The closures are constructed via systematic expansions:

• Moments:

$$\begin{aligned} &\rho(r,t) = \int d\Omega\, P(r, n; t), \ &T_i(r,t) = \int d\Omega\, n_i\, P(r,n;t), \ &Q_{ij}(r,t) = \int d\Omega\, \left[n_i n_j-\frac{1}{d}\delta_{ij}\right] P(r,n;t). \end{aligned}$$

• Evolution equations (schematic, in the spatial Fourier domain):

$$\begin{aligned} \partial_t \rho(q, t) &= -i v q T_\parallel(q,t) - D_0 q^2 \rho(q,t) + \Delta D\,q^2 Q(q,t),\ \partial_t T_\parallel(q, t) &= -i \frac{v}{d} q \rho(q,t) - D_1 q^2 T_\parallel(q,t) - (d-1) D_r T_\parallel(q,t),\ \partial_t Q(q, t) &= \cdots \,. \end{aligned}$$

Here, $\Delta D = D_\parallel - D_\perp$ encodes the anisotropy in translational diffusion.

The ISF, $F(q,t)$, which is the primary observable in light scattering experiments, is directly related to these moments:

• At order 0 and refined order 0, $F$ is purely Gaussian (with diffusion coefficient depending on self-propulsion and rotational dynamics).
• At order 1 and higher, $F$ exhibits oscillatory and non-Gaussian features: signatures of persistence and anisotropy in ABP motion become manifest.

Key mathematical steps:

• Fourier/Laplace transforms reduce the hierarchy to tractable algebraic structures.
• At order 1: a quadratic equation in frequency [as in the denominator of

$$\widetilde{\rho}(q,\omega) = \frac{N(\omega)}{D(\omega)}$$

].

• At order 2: determinant of a $3\times3$ system.

3. Non-Gaussian Effects and Comparison to Exact Results

Higher-order moment closure schemes capture oscillatory decay and non-monotonicity in the ISF observed in simulations and exact theory. The exact ISF is expressible in terms of generalized spheroidal wave functions: $$F(q, t) = \frac{1}{2} e^{-D_\perp q^2 t} \sum_{\ell} e^{-D_r A_\ell^0 t} \left[\int_{-1}^1 d\eta\, \text{Ps}_\ell^0(c, R, \eta)\right]^2,$$ where $A_\ell^0$ and Ps are eigenvalues and eigenfunctions from a spectral problem set by $v$, $D_r$, and the anisotropy. This exact result is numerically tractable but analytically implicit.

Comparisons show:

• For small and large wavenumbers (or long and short observation scales), the simplest closures suffice: ISF is well-described by Gaussian decay.
• At intermediate wavenumbers (e.g., $qL_p\sim 1$ to $10$ for persistence length $L_p = v/D_r$), order-1 and, crucially, order-2 closures are necessary to reproduce oscillatory ISF and polarization/nematic response.
• Explicit formulae and agreement with Brownian dynamics simulations confirm the quantitative accuracy of the closure scheme for moderate Peclet number (Pe) and across observation times.

4. Extension to Complex and Experimental Regimes

A major advantage of the moment-based expansion is practical applicability:

• The closure scheme is generic and basis-independent, allowing the inclusion of more complex phenomena:
  • Propulsion-mode switching (e.g., run–tumble dynamics).
  • Confinement and boundary effects.
  • External fields (e.g., gravity, alignment torques).
• The explicit expressions for ISF, polarization, and nematic tensor can be directly compared with experimental measurements from scattering or imaging, enabling extraction of parameters such as $v$, $D_r$, $D_\parallel$, and $D_\perp$ or the influence of complex environments.
• The formalism is adaptable to systems beyond simple ABPs, e.g., mixtures, particles with switching internal states, or externally driven colloids, provided the underlying Fokker–Planck dynamics can be formulated.

Explicit transition criteria are given in the paper (Gautry et al., 30 Sep 2025) for when each closure level is appropriate, determined by the Pe number, wavenumber $q$, and accessible timescales.

5. Explicit LaTeX Formulas and Representations

Key formulas that emerge from the closure analysis include: $$\begin{align*} D_{ij} &= D_\parallel n_i n_j + D_\perp (\delta_{ij} - n_i n_j), \ T_\parallel(q,t) &=- \frac{v}{d(d-1)D_r} i q \rho(q,t), \ D_{0,\text{eff}} &= D_0 + \frac{v^2}{d(d-1)D_r}, \ F_{0,\text{Act}}(q,t) &= \exp\left[-D_{0,\text{eff}}q^2 t\right], \ \widetilde{\rho}(q,\omega) &= \frac{i\omega + \left(D_1 + \frac{2}{d+2}\Delta D\right)q^2 + (d-1)D_r}{\left[i\omega + \left(D_1+\frac{2}{d+2}\Delta D\right)q^2+(d-1)D_r\right]\left(i\omega + D_0q^2\right) + \frac{v^2q^2}{d}}. \end{align*}$$ At the highest considered closure,

$$F_2(q, t) = \sum_{i=1}^3 \prod_{j \neq i} \frac{\mathcal{P}_1(\omega_i) e^{i \omega_i t}}{(\omega_i - \omega_j)},$$

with $\omega_i$ the roots of a closure-derived cubic polynomial in frequency.

6. Broader Implications and Experimental Relevance

The moment expansion for anisotropic ABPs provides:

• A precise link between microscopic motility parameters and experimentally accessible observables such as ISF, polarization, and nematic order.
• Explicit predictions for the emergence of non-Gaussianity and memory effects in dynamic correlators at moderate Pe—characteristic of active materials with persistent propulsion and orientational relaxation.
• A flexible theoretical tool for rationalizing and guiding new types of experiments (e.g., with Janus colloids, bacteria, or synthetic microswimmers) and for the interpretation of scattering data or local probe measurements.

The method's extensibility makes it indispensable for complex scenarios where exact eigenfunction expansions are either unavailable or intractable—for example, under complex boundary conditions, with internal state dynamics, or in the presence of nontrivial interactions.

7. Summary Table: Closure Schemes and ISF Features

Closure Order	Retained Fields	ISF Functional Form	Non-Gaussian Effects
0	Density	Gaussian	No
Refined 0	Density, Polarization (steady)	Gaussian, activity-renormalized	No
1	Density, Polarization	Oscillatory/exponential	Yes
2	Density, Polarization, Nematic	Sum of exponentials (explicit)	Yes (enhanced)

Selection of closure order is practically determined by the values of Pe, $qL_p$, and the time window of the observational protocol.

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In conclusion, the moment expansion with systematic closure for anisotropic active Brownian particles provides a hierarchy of explicit, analytically tractable models that interpolate between simple diffusive (Gaussian) behavior and full non-Gaussian, persistence-dominated dynamics. It enables direct, quantitative comparison with experiments, and serves as a blueprint for theoretical analysis and extension to more complex active systems (Gautry et al., 30 Sep 2025).