Closures of moment expansion of anisotropic active Brownian particles

TL;DR

This work tackles the non-Gaussian dynamics of anisotropic active Brownian particles by developing a systematic moment-closure framework for the Fokker–Planck equation. By deriving evolution equations for density, polarization, and nematic fields and exploring closures at orders 0, 1, and 2, the authors obtain explicit ISF expressions that capture activity-induced non-Gaussian features, including oscillations, and provide predictions for the polarization and nematic components. They validate the closures against exact spheroidal-wave-function results and 3D Brownian dynamics simulations, mapping the validity domain in terms of Péclet number, wavenumber, and timescales. The order-2 closure stands out as the most accurate across a broad regime, while the approach offers a practical, explicit alternative to fully implicit exact solutions and can be extended to more complex driving, confinement, or field scenarios. Overall, the moment-closure framework yields tangible, parameterizable tools for interpreting ISF experiments and guiding theoretical models of anisotropic active matter.

Abstract

We study analytically the dynamics of anisotropic active Brownian particles (ABPs), and more precisely their intermediate scattering function (ISF). To this end, we develop a systematic closure scheme for the moment expansion of their Fokker-Planck equation. Starting from the coupled evolution of translational and orientational degrees of freedom, we derive equations for the density, polarization, and nematic tensor fields, which naturally generate an infinite hierarchy of higher-order moments. To obtain explicit solutions, we investigate truncation strategies and analyze closures at different orders. While the closure at lowest order yields Gaussian dynamics with an effective translational diffusion, closures at higher orders incorporate orientational correlations and reproduce non-Gaussian features in the ISF. By confronting these approximations with exact solutions based on spheroidal wave functions and with Brownian dynamics simulations, we identify their range of validity in terms of Péclet number, wavenumber, and observation timescales. An advantage of this method is its ability to yield approximate yet explicit expressions not only for the ISF but also for polarization and nematic fields, which are often neglected but relevant in scattering experiments and theoretical modeling. Beyond providing a practical guide to select the appropriate closure according to the spatiotemporal regime, our framework highlights the efficiency of moment-based approaches compared to exact yet implicit formulations. This strategy can be systematically extended to more complex situations, such as propulsion switching, confinement, or external fields, where functional bases for exact solutions are generally unavailable.

Figures (9)

• Figure 1: Left: Sketch of the anisotropic active Brownian particle under study. Its position and orientation are denoted by $\boldsymbol{r}$ and $\hat{\boldsymbol{n}}$ respectively. The longitudinal and transverse translational diffusion coefficients are denoted by $D_\parallel$ and $D_\perp$, while $D_r$ denotes the orientational diffusion coefficient. In the particular case of an ellipsoidal particle, $a$ and $b$ are its major and minor axes, respectively. Right: intermediate scattering function, as a function of time, for different wavevectors, obtained from numerical simulations (see Appendix \ref{['app:numerical']} for details) and from the exact solution recalled in Appendix \ref{['app:Kurzthaler_result']}. Parameters: persistence length $L_p = 14$, aspect ratio $a/b = 3$, $v = 12$, $D_\parallel = 1.42$, $D_\perp = 1.16$, $D_r = 0.85$ ($\tau_{\text{rot}} = 1.17$).
• Figure 2: Left: Simulated intermediate scattering function $F(q,t)$ of an anisotropic ABP as a function of time and for different wavenumbers $q$. The dashed blue lines represents the effective diffusion regimes that are retrieved at large and small wavenumbers. Right: Spatial Fourier transform of the first and second orientational moments (top: polarization; bottom: nematic tensor) projected along the wavenumber $q$. Parameters: $\Delta D/D_0 = 0.21$, $a/b = 3$, $L_p = 14$, $\tau_{\text{rot}} = 1.17$.
• Figure 3: Comparison between the imaginary part of the spatial Fourier transform of the polarization $T_{\parallel}(q,t)$ and the imaginary part of the spatial Fourier transform of the gradient of the ISF for different wavenumbers $q$. Parameters: $\Delta D/D_0 = 0.21$, $a/b = 3$, $L_p = [7.0,28.1]$ (for the respective Péclet numbers), $\tau_{\text{rot}} = 1.17$.
• Figure 4: Comparison between the ISF measured in numerical simulations (colored symbols) and the analytical ISF computed with the closures at order 1 and 2 (blacked dashed line, Eqs. \ref{['eq:F1']} and \ref{['eq:F2']} respectively) for different wavenumbers $q$ and different Péclet number $\text{Pe}$. The different plots are shifted in the vertical direction for clarity -- all the ISF actually tend to $1$ in the limit $t\to 0$ and $0$ in the limit $t\to\infty$. Parameters: $\Delta D/D_0 = 0.21$, $a/b = 3$, $L_p = [7.0,14.0,28.1,56.2]$ (for the respective Péclet numbers), $\tau_{\text{rot}} = 1.17$.
• Figure 5: Comparison between the analytical expressions obtained with the different closures and the exact solution as a function of time. Parameters: $\Delta D/D_0 = 0.21$, $a/b = 3$, $L_p = 14$, $\tau_{\text{rot}} = 1.17$.