Megastable quantization in generalized pilot-wave hydrodynamics.

Area of Science:

• Fluid Dynamics
• Quantum Mechanics Analogs
• Nonlinear Dynamics

Background:

• Classical harmonic oscillators have continuous energy spectra, unlike quantum counterparts with discrete energy levels.
• Walking droplets, classical non-Markovian wave-particle entities, demonstrate hydrodynamic quantum analogs.
• Previous studies showed walking droplets exhibiting quantization in harmonic potentials with few limit cycles.

Purpose of the Study:

• To model a classical harmonic oscillator using a truncated-memory pilot-wave model.
• To investigate the emergence of countably infinite quantized energy levels (megastability) in this classical system.
• To derive analytical approximations for the properties of these megastable states.

Main Methods:

• Utilized a truncated-memory stroboscopic pilot-wave model for walking droplets.
• Applied averaging techniques in the low-memory regime.
• Analyzed a classical harmonic oscillator perturbed by oscillatory nonconservative forces.

Main Results:

• Identified countably infinite coexisting limit-cycle states (megastability) in the classical system.
• Derived analytical approximations for orbital radii, frequency, and Lyapunov energy function.
• Demonstrated average energy conservation along these quantized states.

Conclusions:

• The classical pilot-wave model successfully reproduces quantized energy levels observed in walking droplets.
• The derived megastable spectrum offers a classical explanation for quantum-like phenomena.
• The formalism is generalizable to other self-excited oscillators for constructing similar energy-frequency relations.