Simple mathematical model for a pairing-induced motion of active and passive particles

Abstract

We propose a simple mathematical model that describes a pairing-induced motion of active and passive particles in a two-dimensional system, which is motivated by our previous paper [Ishikawa et al., Phys. Rev. E \textbf{106} (2022) 024604]. We assume the following features; the active and passive particles are connected with a linear spring, the active particle is driven in the direction of the current velocity, and the passive particle is repelled from the active particle. A straight motion, a circular motion, and a slalom motion were observed by numerical simulation. Theoretical analysis reproduces the bifurcation between the straight and circular motions depending on the magnitude of self-propulsion.

Figures (6)

• Figure 1: (Color online) Schematic illustration of our model. An active particle and a passive particle are connected with a linear spring. Additionally, the passive particle receives the force $f_1$ in the direction of the vector $\bm{e}_{ap}$ directing from the active particle to the passive particle. The active particle receives the force $f_2$ in its moving direction. The angles $\xi$ and $\phi$ show the direction of the center-of-mass (COM) velocity $\bm{v}$ and the angle between the two vectors $\bm{\ell}$ and $\bm{v}$.
• Figure 2: (Color online) Trajectories and the instant locations of active and passive particles after sufficient time for four characteristic modes of motion. (a) Passive-particle preceding straight (PPS) motion. (b) Passive-particle preceding circular (PPC) motion. (c) Active-particle preceding circular (APC) motion. (d) Slalom (SL) motion. Red and cyan curves show the trajectories of active and passive particles, respectively. The parameters were set to be (a) $f_2 = 0.1$, (b) $f_2 = 0.3$, (c) $f_2 = 1.0$, and (d) $f_2 = 2.5$. The parameters $f_1$ and $\eta$ were fixed at $f_1 = 0.5$ and $\eta = 0.5$, respectively. The dashed lines show the correspondence of active and passive particle positions at each instance taken every (a) 10, (b) 1, (c) 1, and (d) 3 time unit. The bar in each panel shows the spatial unit.
• Figure 3: (Color online) Time series of $\phi$ and $\xi$ at (a) $f_2 = 0.1$ for the PPS motion, (b) $f_2 = 0.3$ for the PPC motion, (c) $f_2 = 1.0$ for the APC motion, and (d) $f_2 = 2.5$ for the SL motion. Each panel corresponds to Fig. \ref{['fig2']}. The parameters $f_1$ and $\eta$ were fixed at $f_1 = 0.5$ and $\eta = 0.5$, respectively. The dark green thick curve shows $\phi$, which is the angle between the vector directing from the active particle to the passive particle $\bm{\ell} = \bm{r}_p - \bm{r}_a$ and the COM velocity vector $\bm{v}$. The magenta thin curve shows $\xi$, which is the direction of COM velocity $\bm{v}$.
• Figure 4: (Color online) Phase diagrams in the $f_1$-$f_2$ plane when $\eta$ was fixed at $\eta = 0.5$. Green, purple, red, blue, and cyan regions show the PPS motion, the PPC motion, the APC motion, bistability between the SL and the APC motions, and the ambiguous motion, respectively. The white dotted line corresponds to the situation shown in Fig. \ref{['fig5']}. The yellow dashed line shows the bifurcation line obtained by the theoretical analysis in Eq. \ref{['line_th']}.
• Figure 5: (Color online) Dependence of (a) averaged COM speed $\bar{v}$, (b) absolute values of maximum and minimum angle differences $\phi_\mathrm{max}$ and $\phi_\mathrm{min}$ between $\bm{v}$ and $\bm{\ell}$, (c) period $T$ of the circular and slalom motions, and (d) radii of the trajectories of the active and passive particles, $R_a$ and $R_p$, for the circular motion depending on $f_2$. In panel (b), $\phi_\mathrm{max}$ and $\phi_\mathrm{min}$ coincide for the circular motion. The inset of panel (c) shows the expanded plot. In panel (d), the upper and lower plots correspond to $R_a$ and $R_p$, respectively. The color of the plot points correspond to the one in Fig. \ref{['fig4']}. The parameters $f_1$ and $\eta$ were fixed at $f_1 = 0.5$ and $\eta = 0.5$, respectively.