Number fluctuations distinguish different self-propelling dynamics

Abstract

In nonequilibrium suspensions, static number fluctuations $N$ in virtual observation boxes reveal remarkable structural properties, but the dynamic potential of $N(t)$ signals remains unexplored. Here, we develop a theory to learn the dynamical parameters of self-propelled particle models from $N(t)$ statistics. Unlike traditional trajectory analysis, $N(t)$ statistics distinguish between models, by sensing subtle differences in reorientation dynamics that govern re-entrance events in boxes. This paves the way for quantifying advanced dynamic features in dense nonequilibrium suspensions.

Figures (4)

• Figure 1: Probing self-propelled particle models. (a) Simulated trajectories of 3 models -- Active Brownian (ABP), Run and tumble (RTP) or Active Ornstein-Uhlenbeck Particles (AOUP) -- overlayed with virtual observation boxes of size $L = 3~\unit{\mu m}$. Here $v = 3~\unit{\mu m.s^{-1}}$, $D_r = 1~\unit{s^{-1}}$, and for illustrative purposes we turn off translational diffusion. (b) The mean-squared displacements for many particles are indistinguishable between different models and captured by a single theory Eq. \ref{['eq:MSD']}. Here $v = 5~\unit{\mu m.s^{-1}}$, $D_t = 0.1~\unit{\mu m^2.s^{-1}}$ and $D_r = 1~\unit{s^{-1}}$. (c) "Countoscope" approach where we probe the number of particles in virtual observation boxes of a simulation.
• Figure 2: Number fluctuations $\langle \Delta N^2(t) \rangle$ exhibit 3 distinct regimes in time, for (a) Active Brownian particles, (b) Run and Tumble particles, and (c) Active Ornstein-Uhlenbeck particles. Box sizes are the same for all 3 plots, and go from small (in light color) to large boxes (in dark color). Parameters are the same as in Fig. \ref{['fig:fig1']}b. Symbols: simulation; lines: theory -- see text.
• Figure 3: Limit regimes of advection or diffusion captured by scaling laws. Number fluctuations for ABPs, same data as in Fig. \ref{['fig:Countoscope_fig']}a, with (a) advective rescaling in time and (b) diffusive rescaling. (c-d) Schematics illustrating the probability distribution clouds to find a particle in a box after some time $t$ (yellow) given it started in the box initially (blue) in a purely (c) advective or (d) diffusive case, with $D = D_t$ or $D_{\rm eff}$.
• Figure 4: Number correlation functions $C_N(t)$ distinguish different models. (a) Number correlations from simulations (symbols) and theory (lines) for different box sizes; (b) $P_{\rm stay}(t)$ (open) and $P_{\rm return}(t)$ (full symbols) from simulations, taking $L = 5~\unit{\mu m}$. (inset) Lin-log scale of the same plot keeping just ABP data. Color codes for particle models are the same as in (a) and the gray color represents a passive case with diffusion coefficient $D_{\rm eff}$. Numerical parameters are the same as in Fig. \ref{['fig:fig1']}b.