Quantum particle statistics in classical shallow water waves

We present a new hydrodynamic analogy of nonrelativistic quantum particles in potential wells. Similarities between a real variant of the Schrödinger equation and gravity-capillary shallow water waves are reported and analyzed. We show that when locally oscillating particles are guided by real wave gradients, particles may exhibit trajectories of alternating periodic or chaotic dynamics while increasing the wave potential. The particle probability distribution function of this analogy reveals the quantum statistics of the standard solutions of the Schrödinger equation and thus manifests as a classical deterministic interpretation of Born’s rule. Finally, a classical mechanism for the transition between quasi-stationary states is proposed.