Table 1 List of symbols and physical terms used in this study.

Symbol	Description	Physical Meaning / Role
Ti	Temperature of particle i	Local surface temperature (K) of each particle.
Tm	Melting temperature	Reference melting point of water (0◦C).
∆Tnuc	Supercooling threshold	Temperature difference below Tm required for nucleation.
Tnuc	Nucleation temperature	Tm − ∆Tnuc, the critical temperature for seed formation.
αi	Liquid fraction	Ratio of liquid phase for particle i (α = 1: liquid, α = 0: solid).
Ji, Jw	Evaporation flux (ice/water)	Mass flux of vaporization from ice and water surfaces, used in impulse computation.
ni	Surface normal	Outward unit normal vector at particle i, defines direction of flux.
Jacc	Accumulated linear impulse	Time-integrated linear momentum imbalance causing self-jump.
τacc	Accumulated angular impulse	Integrated torque resulting from asymmetric flux distribution.
Fjump	Jumping force	Time-averaged force generated from accumulated impulse Jacc.
τjump	Jumping torque	Torque applied during detachment, computed from τacc.
θseed	Nucleation angle	Angular position of the first seed where Ti < Tnuc.
θpos/neg	Front propagation angles	Angular coverage of recalescence front in +/− directions.
wm	Mushy-band width	Temperature band width for thermal clamping around Tm.
kα	Phase transition rate	Rate constant controlling liquid-to-solid conversion.
vpos/neg	Front propagation speed	Angular velocity of the recalescence front.
∆tJ	Impulse accumulation window	Time interval over which impulses are integrated to produce force.
ξ	Arc half-width parameter	Angular width of arc region used for directional impulse integration.
impulseGain	Impulse amplification factor	Scaling constant controlling jump strength in simulation.
ri	Position vector	Relative position of particle i from the droplet’s center of mass.
xcm	Center of mass	Position of the droplet’s center; main point of force application.
M	Total mass	Sum of all particle masses; used in rigid-body motion computation.
g	Gravity	Gravitational acceleration (9.81 m/s2).