Moment analysis of two-dimensional active Brownian run-and-tumble particles

TL;DR

This work introduces the active Brownian run-and-tumble particle (ABRTP) model, a 2D two-state active particle in a harmonic trap that combines ABP-like angular diffusion in the run state with a tumble state of zero velocity, where run/tumble times are exponentially distributed. Using Kac's direct-integration method, the authors derive exact Laplace-transformed moments $\mathscr{L}(\langle x^l\rangle)(\xi)$ and develop programmable recursion through Volterra equations and diagrammatic rules, unifying ABP and RTP limits. They validate the approach against BV2-cell experiments by comparing Chebyshev-moment expectations, and show that ABRTP captures both free-space diffusion (ballistic short-time and diffusive long-time MSD) and steady-state confinement behavior, with high-order moments revealing geometry-driven scaling in a trap. The findings indicate that high-order statistics and distribution inference via Chebyshev polynomials provide robust, experiment-relevant insights beyond MSD alone, and the Kac framework offers a powerful alternative to traditional Fokker–Planck analyses. Potential extensions include 3D diffusion on spheres and broader generalizations of run-and-tumble dynamics in active matter.

Abstract

We study an active Brownian run-and-tumble particle (ABRTP) model, that consists of an active Brownian run state during which the active velocity of the particle diffuses on the unit circle, and a tumble state during which the active velocity is zero, both with exponentially distributed time. Additionally we add a harmonic trap as an external potential. In the appropriate limits the ABRTP model reduces either to the active Brownian particle model, or the run-and-tumble particle model. Using the method of direct integration the equation of motion, pioneered by Kac, we obtain exact moments for the Laplace transform of the time dependent ABRTP, in the presence or absence of a harmonic trap. In addition we estimate the distribution moments with the help of the Chebyshev polynomials. Our results are in excellent agreement with the experiments.

Figures (4)

• Figure 1: Schematic drawing of the 2D ABRTP model. The velocity of the particle is a consequence of two components, the pull, $-br$, of harmonic potential $b\left|r\right|^{2}$, depending only on the position, and the active velocity, which switches randomly between zero (the tumble state), and a vector of magnitude $v$ (the run state). In the run state, the active component will diffuse along a circle from a randomly chosen starting point at the beginning of the run state. The time between these switches is exponentially distributed, with rate $\left(1-p\right)\gamma$ and $p\gamma$, respectively for the run and the tumble state, and each choice of the active velocity at the beginning of each run state is independent.
• Figure 2: The time depending expectation of Chebyshev polynomials of the active particles in free space. Dot: experimental data from the BV2 cells, Exp $x$ means results from the $x$ components of the experimental data, and Exp $y$ means the y components. See 2023_RunandTumbleDynamicsandMechanotaxisDiscoveredinMicroglialMigration for the details of the experiments; line: analytic calculation from Eq. \ref{['eq:Vdeq=00003D00003D00003D00003D00003D00003D000020ABRTP']}, with parameters $\gamma_{R}=0.58\min^{-1}$,$\gamma_{T}=0.091\min^{-1}$,$D=0.67\min^{-1}$ as reported in the same reference. As for the active velocity, $v_{e}=6.7\ \mu m/\min$ represents the averaged velocity of the run state in the interval of $0.5\min$ as reported, $v_{m}=7.9\ \mu m/\min$ is the inferred instantaneous velocity, or a model parameter that results in the experimentally averaged velocity $v_{e}$ according to the ABP model. In the calculation of the ABRTP model $v_{m}$ is used. It is obvious that the ABRTP model captures the experimental results better than the RTP model.
• Figure 3: Log (left column) and log-log (right column) plots of the (rescaled) moments as a function of their order, as calculated using Eq. \ref{['eq:Vdeq=00003D00003D00003D00003D00003D00003D000020ABRTP']}. $\gamma_{T}=1$ in all cases, but its exact value has little effect on the asymptotic behavior. Unlike the RTP that features an algebraic scaling law, the ABRTP seems to exhibit an geometric scaling law, and asymptotic to a line in the log plot.
• Figure 4: Steady state distributions inferred from the moments using methods outlined in Section \ref{["sec:Kac's-method-revisited"]}. The most obvious feature of such distribution are the possibly singular peaks near the center and/ or the boundary. At the boundary, the ABRTP behaves mostly like the ABP that clutter near, but not at, the boundary, thus a continuous peak near the boundary for small $\gamma_{R}$ and $D$. At the center, the ABRTP behaves mostly like the RTP, with a possible singular peak for small $\gamma_{T}$, and a continuous peak for large $\gamma_{R}$ or $D$. Note that when $\gamma_{R}=0$, the particle will be an ABP and never enters a tumble state, thus $\gamma_{T}$ has no effect on the distribution.