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The Countoscope for self-propelled particles

Tristan Cerdin, Talia Calazans, Carine Douarche, Sophie Marbach

Abstract

Particle number fluctuations $N(t)$, measured in virtual observation boxes of an image or a simulation, offer a way to quantify particle dynamics when particle tracking is impractical, such as in high-density systems. While traditionally limited to equilibrium diffusive systems, we extend this approach -- named ``Countoscope'' -- to out-of-equilibrium self-propelled particles: Active Brownian (ABPs), Run and Tumble (RTPs), and Active Ornstein-Uhlenbeck Particles (AOUPs). For AOUPs, we leverage their Gaussian statistics to derive a general formula applicable to any Gaussian system. For ABPs and RTPs, we derive the intermediate scattering function (ISF) -- and thus the correlations of $N(t)$ -- using an exact perturbative expansion over the probability density fields, revealing key physical features of the ISF and of the number correlations. Our theoretical predictions for the mean-squared number difference $\langle ΔN^2(t) \rangle = \langle (N(t) - N(0))^2 \rangle$ match stochastic simulations and exhibit three time-dependent scaling regimes: diffusive, advective, and long-time enhanced diffusive, reflecting the regimes of the mean squared particle displacement. We further uncover limiting laws in each of these regimes that are useful to quantify self-propulsion properties.

The Countoscope for self-propelled particles

Abstract

Particle number fluctuations , measured in virtual observation boxes of an image or a simulation, offer a way to quantify particle dynamics when particle tracking is impractical, such as in high-density systems. While traditionally limited to equilibrium diffusive systems, we extend this approach -- named ``Countoscope'' -- to out-of-equilibrium self-propelled particles: Active Brownian (ABPs), Run and Tumble (RTPs), and Active Ornstein-Uhlenbeck Particles (AOUPs). For AOUPs, we leverage their Gaussian statistics to derive a general formula applicable to any Gaussian system. For ABPs and RTPs, we derive the intermediate scattering function (ISF) -- and thus the correlations of -- using an exact perturbative expansion over the probability density fields, revealing key physical features of the ISF and of the number correlations. Our theoretical predictions for the mean-squared number difference match stochastic simulations and exhibit three time-dependent scaling regimes: diffusive, advective, and long-time enhanced diffusive, reflecting the regimes of the mean squared particle displacement. We further uncover limiting laws in each of these regimes that are useful to quantify self-propulsion properties.

Paper Structure

This paper contains 27 sections, 70 equations, 10 figures.

Table of Contents

  1. Setup for Counting Self-Propelled Particles
  2. Counting Method
  3. Three models for self-propelled particles
  4. Calculating ISFs for self-propelled particles
  5. Non-gaussian models.
  6. Gaussian models.
  7. Hydrodynamic equations of motion for non gaussian systems
  8. Derivation of the probability densities
  9. Formulation in terms of an infinite hierarchy.
  10. Closure procedure and infinite continued fraction.
  11. Intermediate Scattering Functions (ISF)
  12. From frequency domain to real time.
  13. Convergence of the obtained expressions.
  14. Countoscope for self-propelled particles
  15. Number Fluctuations
  16. ...and 12 more sections

Figures (10)

  • Figure 1: Probing number fluctuations "Countoscope" approach where we probe the number of particles in virtual observation boxes of an image/simulation.
  • Figure 2: Intermediate Scattering Functions for Active Brownian Particles (top), and Run-and-Tumble particles (bottom) obtained at different orders in the truncation. k size are taken to be the same for all three plots and go from small length scales (in light colors) to large ones (in dark colors). Dynamic parameters are $v = 5~\unit{\mu m.s^{-1}}$, $D_t = 0.1~\unit{\mu m^2.s^{-1}}$, and $D_r = 1~\unit{s^{-1}}$. For order 5/Exact, in the top ABP graph the full line corresponds to an order $N = 5$ truncation, while in the bottom RTP plot, the full line is the exact result from the inverse FFT of Eq. \ref{['eq:a_0 RTPs']} and the dotted line from an order $N = 5$ truncation.
  • Figure 3: Intermediate Scattering Function of Active Ornstein-Uhlenbeck Particles. The dynamic parameters used are equivalent to the ones in Fig \ref{['fig:ISF_ABP_RT']}.
  • Figure 4: NMSD behavior for Run-and-Tumble particles. (a) NMSD behavior with respect to lag time for increasing box sizes going from yellow to brown; (b) same as (a) but where time is rescaled by a typical diffusion timescale $L^2/D_{\rm eff}$ and the NMSD by $\langle N \rangle$, and (c) same as (b) but where time is rescaled by a typical advection time $L/v$. In all of the plots, stars correspond to numerical simulations and lines to the theory obtained from integrating $F_2(k,t)$ given by Eq. \ref{['eq:f2kt']} inputted in Eq. \ref{['eq:Cnt']}. Physical parameters are the same as in Fig. \ref{['fig:ISF_ABP_RT']}. Here we have $t_{\rm adv} = \pi \frac{D_t}{v^2}$ and $t_{\rm diff} = \pi \frac{D_{\rm eff}}{v^2}$ the crossover times between regimes for the NMSD.
  • Figure 5: Probability distribution from geometric arguments. Schematics (a) illustrating the probability distribution clouds to find an advective particle in a box after some time $t$ (yellow) given it started in the box initially (blue). Probability distribution of ABPs or RTPs to remain in the box they start in (b) obtained from simulation against ones obtained from geometric arguments in the advective case. Simulation parameters are the same as in Fig.\ref{['fig:ISF_ABP_RT']} and we consider boxes of size $L = 5~\unit{\mu m}$.
  • ...and 5 more figures