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Interacting Particle Systems Modeling Self-Propelled Motions

Saori Morimoto, Makoto Katori, Hiraku Nishimori

TL;DR

This work extends the deterministic camphor-disk model of Nishimori et al. by introducing stochastic dynamics that couple self-propelled disks to a random-walk representation of camphor molecules on the water surface. It formulates a stochastic Newtonian-motion model in which disk motion obeys $m \frac{d \mathbf{v}}{dt} = -\mu \mathbf{v} + \mathbf{F}(\mathbf{r}, \Xi_t)$ with $N$ random walkers modeling supply, diffusion, and sublimation, and also derives a one-dimensional viscous-limit dynamical system for mean behavior, $v(t+1) = \frac{\alpha \ell}{4 \sqrt{\pi} D^{3/2}} v(t) e^{-v(t)^2/(4D)}$. The analysis identifies three dynamic regimes controlled by $C = \frac{4 \sqrt{\pi} D^{3/2}}{\alpha \ell}$: rest, steady propulsion, and oscillatory motion, and compares stochastic simulations with the deterministic system, showing good agreement for mean trajectories and revealing fluctuations around them. The study provides a tractable framework linking microscopic stochastic fluctuations to macroscopic self-propelled dynamics, with potential applications to multi-disk configurations and other active-matter settings.

Abstract

In non-equilibrium statistical physics, active matters in both living and non-living systems have been extensively studied. In particular, self-propelled particle systems provide challenging research subjects in experimental and theoretical physics, since individual and collective behaviors of units performing persistent motions can not be described by usual fluctuation theory for equilibrium systems. A typical example of man-made self-propelled systems which can be easily handled in small-sized experiments is a system of camphor floats put on the surface of water. Based on the experimental and theoretical studied by Nishimori et al. (J. Phys. Soc. Jpn. 86 (2017) 101012), we propose a new type of mathematical models for complex motions of camphor disks on the surface of water. In the previous mathematical models introduced by Nishimori et al. are coupled systems of the equations of motion for camphor disks described by ordinary differential equations and the partial differential equation for the concentration field of camphor molecules in water. Here we consider coupled systems of equation of motions of camphor disks and random walks representing individual camphor molecules in water. In other words, we take into account non-equilibrium fluctuations by introducing stochastic processes into the deterministic models. Numerical simulation shows that our models can represent self-propelled motions of individual camphor disk as well as repulsive interactions among them. We focus on the one-dimensional models in which viscosity is dominant, and derive a dynamical system of a camphor disk by taking the average of random variables of our stochastic system. By studying both of stochastic models and dynamical systems, we clarify the transitions between three phases of motions for a camphor disk depending on parameters.

Interacting Particle Systems Modeling Self-Propelled Motions

TL;DR

This work extends the deterministic camphor-disk model of Nishimori et al. by introducing stochastic dynamics that couple self-propelled disks to a random-walk representation of camphor molecules on the water surface. It formulates a stochastic Newtonian-motion model in which disk motion obeys with random walkers modeling supply, diffusion, and sublimation, and also derives a one-dimensional viscous-limit dynamical system for mean behavior, . The analysis identifies three dynamic regimes controlled by : rest, steady propulsion, and oscillatory motion, and compares stochastic simulations with the deterministic system, showing good agreement for mean trajectories and revealing fluctuations around them. The study provides a tractable framework linking microscopic stochastic fluctuations to macroscopic self-propelled dynamics, with potential applications to multi-disk configurations and other active-matter settings.

Abstract

In non-equilibrium statistical physics, active matters in both living and non-living systems have been extensively studied. In particular, self-propelled particle systems provide challenging research subjects in experimental and theoretical physics, since individual and collective behaviors of units performing persistent motions can not be described by usual fluctuation theory for equilibrium systems. A typical example of man-made self-propelled systems which can be easily handled in small-sized experiments is a system of camphor floats put on the surface of water. Based on the experimental and theoretical studied by Nishimori et al. (J. Phys. Soc. Jpn. 86 (2017) 101012), we propose a new type of mathematical models for complex motions of camphor disks on the surface of water. In the previous mathematical models introduced by Nishimori et al. are coupled systems of the equations of motion for camphor disks described by ordinary differential equations and the partial differential equation for the concentration field of camphor molecules in water. Here we consider coupled systems of equation of motions of camphor disks and random walks representing individual camphor molecules in water. In other words, we take into account non-equilibrium fluctuations by introducing stochastic processes into the deterministic models. Numerical simulation shows that our models can represent self-propelled motions of individual camphor disk as well as repulsive interactions among them. We focus on the one-dimensional models in which viscosity is dominant, and derive a dynamical system of a camphor disk by taking the average of random variables of our stochastic system. By studying both of stochastic models and dynamical systems, we clarify the transitions between three phases of motions for a camphor disk depending on parameters.

Paper Structure

This paper contains 15 sections, 1 theorem, 56 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Stochastic Newtonian-Motion model
  3. Newtonian equation coupled with random walks
  4. Simulation Results
  5. Single camphor disk
  6. Two camphor disks
  7. Three and four camphor disks
  8. Stochastic Viscous-Motion Models and Dynamical Model in One-Dimension
  9. Stochastic models
  10. Brownian motion limit
  11. Mean-value motion
  12. Dynamical system
  13. Analysis of dynamical system
  14. Comparison between stochastic models and dynamical systems
  15. Concluding Remarks and Future Problems

Key Result

Proposition 3.1

Figures (6)

  • Figure 1: Single camphor disk (red disk) and random walkers representing camphor molecules (blue dots) in the $(x, z)$-plane. The dotted rectangular around the red disk indicates $\Lambda(\bm{r})$.
  • Figure 3: Two camphor disks (red and green disks) and random walkers representing camphor molecules (blue dots) in the $(x, z)$-plane.
  • Figure 5: The functions $y=F(x)$ given by \ref{['eq:F1']} and $y=C x$ are drawn by a blue curve and a red line, respectively for three cases. (i) $C \geq 1$, (ii) $e^{-1} < C < 1$, and (iii) $0 < C < e^{-1}$. The stable (resp. unstable) fixed points are indicated by red (resp. black) dots.
  • Figure 6: Comparison between the the simple case of stochastic viscous-motion model \ref{['eq:simple_viscous']} and the dynamical system \ref{['eq:dynamics3']}. (i-a) for the stochastic model with $\widehat{\ell}=0.04$ and (ii-b) for the dynamical system with $\ell=0.01$.
  • Figure 7: Comparison between the the simple case of stochastic viscous-motion model \ref{['eq:simple_viscous']} and the dynamical system \ref{['eq:dynamics3']}. (ii-a) for the stochastic model with $\widehat{\ell}=0.2$ and (ii-b) for the dynamical system with $\ell=0.1$.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Proposition 3.1