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

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.

💡 Research Summary

In this paper the authors introduce a minimalistic two‑dimensional mathematical model to capture the “pairing‑induced motion” observed when an active particle and a passive particle are coupled. The model rests on three key assumptions: (i) the two particles are linked by a linear spring of natural length R and stiffness k, (ii) the active particle experiences a constant self‑propulsion force of magnitude f₂ that is always aligned with its instantaneous velocity, and (iii) the passive particle feels a constant non‑reciprocal repulsive force of magnitude f₁ directed from the active particle toward the passive one. Both particles share the same mass m and viscous drag coefficient η. After nondimensionalisation (mass, length and time scaled by m, R and √(m/k) respectively) the governing equations reduce to a set that depends only on the three dimensionless parameters η, f₁ and f₂.

By introducing centre‑of‑mass (COM) coordinates r = (r_a + r_p)/2, relative vector ℓ = r_p − r_a, COM velocity v = (v_a + v_p)/2 and relative velocity w = v_p − v_a, the dynamics become a six‑dimensional autonomous system (r evolves trivially as the integral of v). The equations for ℓ, v and w contain nonlinear terms such as |ℓ| v̂ and |v − w/2| (v − w/2) that generate rich behaviour while preserving the essential physics of the original system.

Numerical integration (fourth‑order Runge‑Kutta, Δt = 10⁻⁴, up to t = 10⁴) was performed for a fixed drag η = 0.5 and repulsion strength f₁ = 0.5, while the propulsion strength f₂ was varied from 0.1 to 2.5. Initial conditions were chosen with a small offset to avoid the singularities at ℓ = 0 and v = w/2. Four distinct motional regimes emerged as f₂/f₁ increased:

1. Passive‑Particle‑Preceding Straight (PPS) – for small f₂ the passive particle leads and both particles travel along a straight line; the angle φ between ℓ and v converges to zero.
2. Passive‑Particle‑Preceding Circular (PPC) – at intermediate f₂ the pair follows a circular trajectory with the passive particle ahead; φ settles to a constant value between 0 and π/2.
3. Active‑Particle‑Preceding Circular (APC) – for larger f₂ the active particle leads the circular motion; φ converges to a value between π/2 and π.
4. Slalom (SL) – at high f₂ the active particle leads while the passive particle trails in a wavy, zig‑zag path; φ oscillates around π.

The authors quantified these regimes using the time series of φ and the COM direction ξ, defining thresholds on the extrema of cos φ and on the sweep of ξ to classify the motion. Phase diagrams in the (f₁, f₂) plane reveal clear boundaries between the regimes, a narrow bistable region where SL and APC coexist, and a thin “ambiguous” zone near the SL boundary where the dynamics are sensitive to initial conditions.

Linear stability analysis of the straight‑motion fixed point (ℓ constant, v constant, φ = 0) shows that it loses stability via a supercritical pitchfork bifurcation when f₂ exceeds a critical value (f_{2c}= \sqrt{\eta(\eta+2f_1)}). Beyond this point two symmetric non‑zero‑φ branches appear, corresponding to the PPC and APC circular motions. The transition from PPS to PPC is continuous in both average COM speed and trajectory radius, confirming the pitchfork nature. The boundary between PPC and APC is not a standard bifurcation but rather a smooth crossover where the passive particle momentarily stops and the active particle orbits around it. The SL–APC bistability indicates the presence of coexisting attractors, a feature that could be explored further with basin‑of‑attraction analysis.

Overall, the study demonstrates that a very simple set of forces—linear attraction, non‑reciprocal repulsion, and velocity‑aligned self‑propulsion—is sufficient to reproduce a rich repertoire of pairwise motions observed experimentally in systems such as camphor‑driven disks paired with metal washers, or chemically active droplets interacting via concentration fields. The model abstracts away the detailed chemistry or hydrodynamics yet captures the essential bifurcation structure governing the transition from straight to circular motion. The authors suggest that extensions incorporating nonlinear springs, time‑dependent propulsion, or many‑particle interactions could elucidate more complex collective behaviours like clustering, swarming, or pattern formation in heterogeneous active matter.