Neural-Network-Assisted Boltzmann Approach for Dilute Microswimmer Suspensions

TL;DR

A neural-network-assisted Boltzmann framework that learns the binary-collision map of microswimmers directly from data and uses it to evaluate collision integrals efficiently and enables a linear-stability analysis of isotropy against polar ordering in dilute suspensions.

Abstract

We introduce a neural-network-assisted Boltzmann framework that learns the binary-collision map of microswimmers directly from data and uses it to evaluate collision integrals efficiently. Using a representative model swimmer, the learned map quantitatively predicts translational and rotational diffusivities and enables a linear-stability analysis of isotropy against polar ordering in dilute suspensions. The resulting predictions closely match direct simulations. The present framework is agnostic to active matter models and broadly applicable: once two-body collision data are obtained -- either from simulations or experiments -- the same surrogate can be used to evaluate kinetic transport across dilute conditions where binary collisions dominate. Because the workflow relies only on pre- and post-collision statistics, the present approach provides a general data-driven route linking particle-scale interactions to macroscopic transport and collective behavior in active suspensions.

Figures (4)

• Figure 1: (color online) (a) Schematic of our model microswimmer resembling E. coli. The body is treated as rigid, whereas the flagellum is modeled as a massless "phantom" particle. Both the body and the flagellum are represented by three overlapping spheres of radius $R$. Further details of the swimmer model and its construction are provided in Ref.Hayano2022. (b) Pre-collisional state of two microswimmers specified by the relative angle $\theta$, impact parameter $b$, and azimuthal angle $\epsilon$. The vector $\hbox{\boldmath $b$}$ connecting the centers of the two swimmers at the closest approach in the absence of any interactions is perpendicular to the relative velocity: $\hbox{\boldmath $b$} \perp \hbox{\boldmath $v$}_{\rm rel}$. See Supplementary Material for the details of the initial configuration in our simulations. (c) Schematic of a binary collision between two microswimmers with relative angle $\theta$. The pre-collisional orientations $\hat{\hbox{\boldmath$n$}}_1$ and $\hat{\hbox{\boldmath$n$}}_2$ change to the post-collisional orientations $\hat{\hbox{\boldmath$n$}}'_1$ and $\hat{\hbox{\boldmath$n$}}'_2$ after the collision. We denote the orientation changes as $\Delta\hat{\hbox{\boldmath$n$}}_1 = \hat{\hbox{\boldmath$n$}}'_1 - \hat{\hbox{\boldmath$n$}}_1$ and $\Delta\hat{\hbox{\boldmath$n$}}_2 = \hat{\hbox{\boldmath$n$}}'_2 - \hat{\hbox{\boldmath$n$}}_2$.
• Figure 2: (color online) Predicted orientational change of the swimmer 1, $|\Delta \hat{\hbox{\boldmath$n$}}_1| = |\hat{\hbox{\boldmath$n$}}'_1 - \hat{\hbox{\boldmath$n$}}_1|$, as a function of the relative angle $\theta$, azimuthal angle $\epsilon$ and impact parameter $b$. The color indicates the magnitude of the orientational change.
• Figure 3: (color online) (a) Rotational diffusion coefficient $D_{\rm r}$ from many-body simulations (solid line) and the binary-collision prediction $D_{\rm r}^{\rm (bc)}$ from Eq. \ref{['eq:DrBC2']} (dashed line) versus swimmer volume fraction $\phi$. (b) Translational diffusion coefficient $D_{\rm t}$ from simulations (solid line) and $D_{\rm t}^{\rm (bc)}$ from Eq. \ref{['eq:DtBC']} (dashed line) versus $\phi$.
• Figure 4: (color online) Alignment metric $\Delta A \equiv (\Delta\hat{\hbox{\boldmath $n$}}_{1}+\Delta\hat{\hbox{\boldmath $n$}}_{2}) \!\cdot\! (\hat{\hbox{\boldmath $n$}}_{1}+\hat{\hbox{\boldmath $n$}}_{2})$ as a function of relative angle $\theta$, azimuth $\epsilon$, and impact parameter $b$. Colors indicate the magnitude of the alignment (negative values: disalignment).