Yves Couder

Developmental Patterning by Mechanical Signals in Arabidopsis

A central question in developmental biology is whether and how mechanical forces serve as cues for cellular behavior and thereby regulate morphogenesis. We found that morphogenesis at the Arabidopsis shoot apex depends on the microtubule cytoskeleton, which in turn is regulated by mechanical stress. A combination of experiments and modeling shows that a feedback loop encompassing tissue morphology, stress patterns, and microtubule-mediated cellular properties is sufficient to account for the coordinated patterns of microtubule arrays observed in epidermal cells, as well as for patterns of apical morphogenesis.

Dynamical phenomena: Walking and orbiting droplets

Small drops can bounce indefinitely on a bath of the same liquid if the container is oscillated vertically at a sufficiently high acceleration. Here we show that bouncing droplets can be made to ‘walk’ at constant horizontal velocity on the liquid surface by increasing this acceleration. This transition yields a new type of localized state with particle–wave duality: surface capillary waves emanate from a bouncing drop, which self-propels by interaction with its own wave and becomes a walker. When two walkers come close, they interact through their waves and this ‘collision’ may cause the two walkers to orbit around each other.

Particle–wave association on a fluid interface

A small liquid drop can be kept bouncing on the surface of a bath of the same fluid for an unlimited time when this substrate oscillates vertically. With fluids of low viscosity the repeated collisions generate a surface wave at the bouncing frequency. The various dynamical regimes of the association of the drop with its wave are investigated first. The drop, usually a simple ‘bouncer’, undergoes a drift bifurcation when the forcing amplitude is increased. It thus becomes a ‘walker’ propagating at a constant velocity on the interface. This transition occurs just below the Faraday instability threshold, when the drop becomes a local emitter of a parametrically forced wave. A model of the particle–wave interaction accounts for this drift bifurcation. The self-organization of several identical bouncers is also investigated. At low forcing, bouncers form bound states or crystal-like aggregates. At larger forcing, the collisions between walkers reveal that their interaction can be either repulsive or attractive, depending on their distance apart. The attraction leads to the spontaneous formation of orbiting pairs, the possible orbit diameters forming a discrete set. A theoretical model of the non-local interaction resulting from the interference of the waves is given. The nature of the interaction is thus clarified and the various types of self-organization recovered.

Path-memory induced quantization of classical orbits

A droplet bouncing on a liquid bath can self-propel due to its interaction with the waves it generates. The resulting “walker” is a dynamical association where, at a macroscopic scale, a particle (the droplet) is driven by a pilot-wave field. A specificity of this system is that the wave field itself results from the superposition of the waves generated at the points of space recently visited by the particle. It thus contains a memory of the past trajectory of the particle. Here, we investigate the response of this object to forces orthogonal to its motion. We find that the resulting closed orbits present a spontaneous quantization. This is observed only when the memory of the system is long enough for the particle to interact with the wave sources distributed along the whole orbit. An additional force then limits the possible orbits to a discrete set. The wave-sustained path memory is thus demonstrated to generate a quantization of angular momentum. Because a quantum-like uncertainty was also observed recently in these systems, the nonlocality generated by path memory opens new perspectives.

Information stored in Faraday waves: the origin of a path memory

On a vertically vibrating fluid interface, a droplet can remain bouncing indefinitely.When approaching the Faraday instability onset, the droplet couples to the wave itgenerates and starts propagatinghorizontally.Theresultingwave–particleassociation,called a walker, was shown previously to have remarkable dynamical properties,reminiscent of quantum behaviours. In the present article, the nature of a walker’swave field is investigated experimentally, numerically and theoretically. It is shownto result from the superposition of waves emitted by the droplet collisions with theinterface.Asingleimpactisstudiedexperimentallyandinafluidmechanicstheoreticalapproach. It is shown that each shock emits a radial travelling wave, leaving behinda localized mode of slowly decaying Faraday standing waves. As it moves, the walkerkeeps generating waves and the global structure of the wave field results from thelinear superposition of the waves generated along the recent trajectory. For rectilineartrajectories, this results in a Fresnel interference pattern of the global wave field. Sincethe droplet moves due to its interaction with the distorted interface, this means that itis guided by a pilot wave that contains a path memory. Through this wave-mediatedmemory, the past as well as the environment determines the walker’s present motion.Key words: drops, Faraday waves, pattern formation

Self-organization into quantized eigenstates of a classical wave-driven particle

A growing number of dynamical situations involve the coupling of particles or singularities with physical waves. In principle these situations are very far from the wave particle duality at quantum scale where the wave is probabilistic by nature. Yet some dual characteristics were observed in a system where a macroscopic droplet is guided by a pilot wave it generates. Here we investigate the behaviour of these entities when confined in a two-dimensional harmonic potential well. A discrete set of stable orbits is observed, in the shape of successive generalized Cassinian-like curves (circles, ovals, lemniscates, trefoils and so on). Along these specific trajectories, the droplet motion is characterized by a double quantization of the orbit spatial extent and of the angular momentum. We show that these trajectories are intertwined with the dynamical build-up of central wave-field modes. These dual self-organized modes form a basis of eigenstates on which more complex motions are naturally decomposed.

Wave propelled ratchets and drifting rafts

Several droplets, bouncing on a vertically vibrated liquid bath, can form various types of bound states, their interaction being due to the waves emitted by their bouncing. Though they associate droplets which are individually motionless, we show that these bound states are self-propelled when the droplets are of uneven size. The driving force is linked to the assymetry of the emitted surface waves. The direction of this ratchet-like displacement can be reversed, by varying the amplitude of forcing. This direction reversal occurs when the bouncing of one of the drops becomes sub-harmonic. As a generalization, a larger number of bouncing droplets form crystalline rafts which are also shown to drift or rotate when assymetrical.

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.

Archimedean lattices in the bound states of wave interacting particles

The possible periodic arrangements of droplets bouncing on the surface of a vibrated liquid are investigated. Because of the nature of the interaction through waves, the possible distance of binding of nearest neighbors is multi-valued. For large amplitude of the forcing, the bouncing becomes sub-harmonic and the droplets can have two different phases. This effect increases the possible distances of binding and the formation of various polygonal clusters is observed. From these elements it is possible to assemble crystalline structures related to the Archimedean tilings of the plane, the periodic tesselations which tile uniformly the 2D plane with convex polygons. Eight of the eleven possible configurations are observed. They are stabilized by the coupling of two sub-lattices of droplets of different phase, both contributing to sustain a common wave field.

Build-up of macroscopic eigenstates in a memory-based constrained system

A bouncing drop and its associated accompanying wave forms a walker. Based on previous works, we show in this article that it is possible to formulate a simple theoretical framework for the walker dynamics. It relies on a time scale decomposition corresponding to the effects successively generated when the memory effects increase. While the short time scale effect is simply responsible for the walkers propulsion, the intermediate scale generates spontaneously pivotal structures endowed with angular momentum. At an even larger memory scale, if the walker is spatially confined, the pivots become the building blocks of a self-organization into a global structure. This new theoretical framework is applied in the presence of an external harmonic potential, and reveals the underlying mechanisms leading to the emergence of the macroscopic spatial organization reported by Perrard et al. (2014, Nature Commun. 5, 3219)