Intermediate scattering function of a gravitactic circle swimmer

TL;DR

This work addresses gravitaxis in a Brownian circle swimmer by deriving the intermediate scattering function $F(\mathbf{k},t)$ through a spectral-theory solution of the Fourier-space Fokker-Planck equation, with a Dyson expansion used to extract cumulants up to fourth order. The authors present an explicit ISF expression in terms of eigenvalues and eigenfunctions of the operators $\mathcal{L}$ and $\mathcal{L}+\delta\mathcal{L}_{\mathbf{k}}$, and analyze mean motion, variance, skewness, and excess kurtosis for four observation directions, including a harmonic-approximation regime near the gravitactic bifurcation. They validate the analytical results with Langevin-dynamics simulations and discuss transforming to a comoving frame to remove drift-induced oscillations, revealing diffusion at small wavenumbers, circular motion at intermediate wavenumbers, and directed motion at large wavenumbers. The findings highlight pronounced non-Gaussian features near the bifurcation and anisotropic diffusion in the locked phase, providing a rigorous, experimentally accessible framework for interpreting dynamic light scattering and differential dynamic microscopy data on gravitactic microswimmers.

Abstract

We analyze gravitaxis of a Brownian circle swimmer by deriving and characterizing analytically the experimentally measurable intermediate scattering function (ISF). To solve the associated Fokker-Planck equation we use a spectral-theory approach and find formal expressions in terms of eigenfunctions and eigenvalues of the overdamped-noisy-driven-pendulum problem. We further perform a Taylor series of the ISF in the wavevector to read off the cumulants up to the fourth order. We focus on the skewness and kurtosis analyzed for four observation directions in the 2D-plane. Validating our findings involves conducting Langevin-dynamics simulations and interpreting the results using a harmonic approximation. The skewness and kurtosis are amplified as the orienting torque approaches the intrinsic angular drift of the circle swimmer from above, highlighting deviations from Gaussian behavior. Transforming the ISF to the comoving frame, again a measurable quantity, reveals gravitactic effects and diverse behaviors spanning from diffusive motion at low wavenumbers to circular motion at intermediate and directed motion at higher wavenumbers.

Figures (8)

• Figure 1: Tilted washboard potential for the angle $\vartheta$ and different torques $\gamma$ above, exactly at, and below the classical bifurcation. The harmonic approximation (HA) for the case of $\gamma /\omega = 2$ is shown in dashed lines. The inset is a zoom close to the inflection point.
• Figure 2: Real part of the lowest eigenvalues $\lambda_n$ for $n \geq 0$ of the unperturbed operator $\mathcal{L}$ for $D_\mathrm{rot}/\omega =0.005$ for increasing values of $\gamma$ above the bifurcation. Points correspond to numerical values, and lines display the harmonic approximation $\lambda^\text{HA}_n$.
• Figure 3: (a) The angle $\arctan (v_y/v_x)$ of the direction of motion and (b) the absolute value of the mean velocity $|\mathbf{v}|$ for various rotational diffusion constants $D_\text{rot}$. The black line corresponds to the deterministic motion $D_\text{rot}/\omega=0$.
• Figure 4: Real part of the ISF for different values of the wave vectors in $x$ direction $\mathbf{k} = k \mathbf{n}_x$ for a diffusion coefficient $D_\mathrm{rot}/\omega=0.025$ and $\gamma / \omega =1.5$. Full lines correspond to the spectral theory and symbols to Langevin-simulation results. The dashed lines correspond to the harmonic approximation (HA). (a) Real part of the ISF. (b) Same quantity in the comoving frame. (c) Same upon rescaling time by $k^2v^2/\omega$.
• Figure 5: The real and imaginary parts of the ISF in the comoving frame for different orienting torques $\gamma$ and for various values and two directions of the wave vector $\mathbf{k}$. The rotational diffusion coefficient $D_\text{rot}=0.025 \omega$ is chosen. For three different lengths $k=0.25, 1.0,5.0$ there is a plot for two different directions, parallel $\mathbf{n}_\parallel$ ((a)-(c) and (g)-(i)) and perpendicular $\mathbf{n}_\perp$ ((d)-(f) and (j)-(l)) to the direction of the mean velocity. Full lines correspond to the spectral theory and symbols to the Langevin-simulation results. The red dotted line in (a) and (d) correspond to the effective diffusion of a free circle swimmer $\exp(-D_0k^2t)$. Each legend in the first row is valid for the whole column.