Memory of elastic collisions drives high minority spin and oscillatory entropy in underdamped chiral spinners

Underdamped active matter consists of driven objects whose translational and rotational momenta persist between collisions, carrying memory forward in time and producing time-irreversible dynamics. While many active matter studies ignore inertia (low Reynolds number, reversible dynamics), inertial memory has been shown to induce phenomena absent in overdamped systems, such as enhanced motility-induced phase separation, enhanced diffusion, and stochastic dynamics. Rotational (chiral) inertial effects in elastic collisions have received less attention despite rich behavior in overdamped chiral systems. This work experimentally investigates inertial spin–orbital/angular momentum coupling in chiral spinners and how memory-preserving, nearly elastic collisions lead to unexpected collective phenomena: (i) violation of equipartition via minority spin pumping, and (ii) time-oscillatory spatial entropy linked to emergent vorticity. The central question is how elastic, underdamped collisions between chiral spinners redistribute spin and orbital angular momentum depending on relative handedness, and how this memory drives energy and entropy flows at the collective scale.

Prior studies of inertial active matter have highlighted linear momentum effects while often neglecting rotational degrees of freedom. Inertial memory has been implicated in phenomena such as enhancement of motility-induced phase separation, enhanced diffusion, and stochastic dynamics. Overdamped chiral active matter exhibits rich collective behavior, motivating exploration of rotational inertia in collisions. The present work extends this by focusing on underdamped, nearly elastic rotational interactions and spin–orbit coupling in chiral spinners, an area comparatively underexplored.

Experimental platform: Laser-cut acrylic toothed discs (24 teeth; spinner diameter ~7 cm) act as spinners driven by two oppositely directed blowers (SUNON UB3-500B) powered by a 3.7 V 400 mA-hr Li-ion battery (Adafruit 3898). Changing handedness is achieved by inverting the disc. Each spinner is lifted by a 60 mm plastic Petri dish to avoid gear-boundary interactions. Mass m ≈ 0.025 kg; moment of inertia about center I ≈ 1.01×10^-4 kg m^2 (measured). A barcode on each spinner allows tracking of position and rotation. Arena: Ealing Precision Air Table with honeycomb core; flatness ±0.025 cm; steel wire boundary for elastic rebounds; leveling screws for leveling. Imaging and tracking: iPhone camera at 30 fps records dynamics; custom tracking reads barcode orientation and center-of-mass trajectories to compute spin angular velocities and translational velocities over time. Batteries operated within 30 minutes post-charge to maintain consistent torque. Collision rule measurement: Binary collisions between spinners of same and opposite handedness were analyzed. The measured change in angular velocity follows Δω ≈ −β(ω1+ω2) with β ≈ 3/8 < 1/2, consistent with theory for nearly elastic, underdamped contacts. Same-sign collisions (++, −−) exhibit strong negative correlations (spin annihilation), while opposite-sign (+−) collisions show small or negligible spin change. Phenomenological model: Using conservation of angular momentum and measured collision rule, an inertia-dominated toy model is written for each spinner’s rotational dynamics, including average effects of same- and opposite-spin collisions and external drive/drag torques. After averaging, a steady-state expression for mean spin of each species is obtained: ⟨ω±⟩ = ±Ω × α(α + 3N∓ − 2N±)/[(α + 2N∓)(α + N∓)], with Ω the saturated spin rate and α a parameter set by spinner inertia and blower torque. Predictions for energy partitions are estimated: E_rot,all ∝ Ω^2 φ^2[1 − 3(1/2 − n_+)^2], and E_orb ∝ φ^2{[(3 − 2n_+)^2 + ε] n_+}, where n_+ = N_+/N, φ is area fraction, and ε the normalized variance of translational energy. Experiments: Three sets were performed.

• Experiment 1 (Minority spin pumping): Mixtures with fixed total N (e.g., N=18 at φ≈0.16) and varied spin ratio n_+ were studied. Time series of individual spins and ensemble-averaged rotational/orbital energies per spinner were recorded. Binary collision statistics validated Eq. (1).
• Experiment 2 (Entropy oscillations and mixing): For denser collectives (e.g., N=36, φ≈0.32), spinners were initially partitioned by position. Positional mixing entropy S(τ) was computed by tracking the fraction of spinners crossing regions over intervals τ, ensemble-averaging many intervals after initial transients. Dephasing times and vortex counts were extracted; a phase diagram in (n_+, φ) was mapped.
• Experiment 3 (Spin frustration under confinement): Four spinners were confined within a floating ring (inner diameter D ≈ 6 spinner radii) allowing collisions but preventing exchange of center-of-mass positions. Two topologies with zero net spin were examined: side-by-side parallel and diagonal opposite placements. Ring rotation and internal energy distributions were measured to assess frustration and energy transmission to the ring. Simulations: Geometry-accurate spinner models used line-segment representations (24 teeth, 96 line segments per spinner; 96^2 pairwise interactions per pair). Collision forces used a spring-dashpot model tuned to match experimental restitution and Δω = β(ω1+ω2). Forces and torques included collision F_coll, blower torque τ_drive with sign s_i ∈ {+1,−1}, rotational drag τ_drag = −Π_r ω, translational drag −η v, air-current force F_air (empirically fitted), and wall forces. Representative parameters: m=0.025 kg; R_o=0.035 m; R_i=0.030 m; I=1.01×10^-4 kg m^2; Ω≈32 rad s^-1; rotational driving acceleration γ≈3.4 rad s^-2 (τ_drive=Iγ≈3.4×10^-3 N m); translational drag η≈1.6×10^-5 kg s^-1; rotational drag Π_r≈1.06×10^-6 N m s; collision stiffness k_n=10 N/m; damping k_d=2×10^4 N s/m^2. Numerical integration used a symplectic velocity-Verlet scheme with Δt≈10^-3 s. Validation runs without drive/drag showed equipartition between rotational (1 dof) and translational (2 dof) energies emerging from purely translational initial conditions.

• Collision rule and spin–orbit transfer: The change in angular velocities during nearly elastic collisions follows Δω ≈ −β(ω1+ω2) with β ≈ 3/8. Same-sign collisions (++, −−) convert spin angular momentum to orbital angular momentum (spin annihilation), while opposite-sign collisions (+−) largely preserve individual spins.
• Minority spin pumping: In mixed populations, the minority-handed spinners attain higher steady-state spin magnitudes than the majority. The toy model predicts ⟨ω±⟩ decreasing monotonically with increasing opposite-species count; experiments with N=18 (area fraction φ≈0.16) and simulations agree. Example from Eq. (1): a collision between +Ω and +ε (ε ≪ Ω) redistributes to +5/8 Ω and −3/8 Ω (smaller spin flips), illustrating non-intuitive sign changes and pumping.
• Energy partition: Average translational energy per spinner is approximately one-tenth of the rotational energy per spinner at φ≈0.16 (N=18), consistent with prior findings. Rotational energy per spinner shows a concave dependence centered at n_+=1/2, while orbital energy displays asymmetry vs n_+; trends captured by simple estimates E_rot,all ∝ Ω^2 φ^2[1−3(1/2−n_+)^2] and E_orb ∝ φ^2{[(3−2n_+)^2+ε]n_+}.
• Emergent spin currents and symmetry: Energy and spin-rate distributions depend on N_+/N, with mirror symmetry about n_+=1/2 in simulations. Experiments show slight asymmetry attributed to a small mechanical height difference (~1 cm) between + and − spinners.
• Entropy oscillations and vorticity: Positional mixing entropy S(τ) generally increases over time but exhibits pronounced oscillations when net handedness is strong (e.g., all spinners parallel), corresponding to transient global circulation and edge currents. Oscillations are absent for balanced initial spin states. For N=36 (φ≈0.32), the no-oscillation interval is |n_+−1/2| < Δ ≈ 1/4; for N=18 (φ≈0.16), Δ ≈ 1/2. A phase boundary in (n_+, φ) was identified using a dephasing time threshold τ_dephase ≈ 10 s. As n_+ approaches the no-oscillation region, the number of vortices increases and becomes more local/motile, slowing mixing.
• Mixing kinetics: The characteristic mixing time τ_mix peaks at even mixtures (n_+=1/2) and is minimal for single-species collectives, consistent with reduced spin-to-orbital transfer in anti-parallel collisions.
• Confinement and frustration: In a four-spinner ring (D ≈ 6 radii), topology controls energy states. Side-by-side parallel topology permits spin-annihilating collisions, yielding lower spin kinetic energies with a power-law-like distribution, while diagonal topology suppresses such losses, producing higher spin energies with a peaked distribution. The floating ring exhibits intermittent rotation bursts driven by internal frustration, demonstrating transmission of internal spin dynamics to external motion.

The findings demonstrate that in underdamped, memory-preserving collisions, angular momentum is conserved but redistributed between spin and orbital components in a handedness-dependent manner. Same-sign collisions efficiently convert spin to orbital motion, while opposite-sign collisions preserve spin, leading to systematic pumping of minority species to higher spin states and a breakdown of equipartition. This mechanism explains the observed concave dependence of rotational energy on composition and the higher spin rates of minority spinners. The time-oscillatory mixing entropy arises from the coherent conversion of net spin to global orbital angular momentum, producing transient edge currents and vorticity; balanced mixtures suppress these oscillations. Simulations calibrated to measured parameters reproduce experimental trends, indicate finite-size robustness, and suggest that geometry alone can mediate tangential interactions even without dissipation. Confinement experiments reveal topological control of energy flow and frustration, with internal configurations dictating energy distributions and intermittent transmission of torque to external structures. Altogether, the work links microscopic collision memory to macroscopic energy and entropy flows, extending active matter understanding beyond overdamped statistical mechanics.

Energy and entropy flow in an underdamped chiral spinner swarm are governed by memory-preserving elastic collisions that couple spin and orbital motion in a handedness-dependent fashion. The study uncovers a spin-pumping mechanism concentrating rotational energy in the minority-handed species, documents oscillatory mixing entropy tied to emergent vorticity and edge currents, and shows topology-dependent frustration and energy transmission under confinement. These results reveal behaviors beyond conventional equipartition and highlight design principles for chiral agents and gear-like active structures. Future directions include developing two-species continuum field theories to analyze critical spin-ratio dependence and thermodynamic limits, exploring larger, hierarchical (fractal) gear networks with multiple levels of coupling, and integrating onboard computation and sensing for interactive swarms with memory and predictive capabilities.

• Slight mechanical asymmetry between + and − spinners (e.g., battery height difference of ~1 cm) produces small deviations from mirror symmetry about n_+=1/2 in experiments.
• Experimental collision data exhibit noise and variance around theoretical predictions for Δω, introducing uncertainty in parameter estimates.
• Results are demonstrated at specific densities (area fractions ~0.16 and ~0.32) and finite sizes (N=18, 36); while simulations suggest scalability, full thermodynamic-limit behavior remains to be established.
• Simulations use phenomenological spring-dashpot parameters tuned to match restitution and collision patterns; although outcomes are reported to be insensitive within an order of magnitude, exact material properties may shift quantitative details.
• Geometry-specific concave spinner design and air-table boundary conditions may influence interaction details; generalization to other geometries or driving modalities requires validation.