Probabilities and trajectories in a classical wave-particle duality

Several recent experiments were devoted to walkers, structures that associate a droplet bouncing on a vibrated liquid with the surface waves it excites. They reveal that a form of wave-particle duality exists in this classical system with the emergence of quantum-like behaviours. Here we revisit the single particle diffraction experiment and show the coexistence of two waves. The measured probability distributions are ruled by the diffraction of a quantumlike probability wave. But the observation of a single walker reveals that the droplet is driven by a pilot wave of different spatial structure that determines its trajectory in real space. The existence of two waves of these types had been proposed by de Broglie in his "double solution" model of quantum mechanics. A difference with the latter is that the pilot-wave is, in our experiment, endowed with a "path memory". When intrusive measurements are performed, this memory effect induces transient chaotic individual trajectories that generate the resulting statistical behaviour.