Introduction to focus issue on hydrodynamic quantum analogs

Infinite-memory classical wave-particle entities, attractor-driven active particles, and the diffusionless Lorenz equations

A classical wave-particle entity (WPE) can materialize as a millimeter-sized droplet walking horizontally on the free surface of a vertically vibrating liquid bath. This WPE comprises a particle (droplet) that shapes its environment by locally exciting decaying standing waves, which, in turn, guides the particle motion. At high amplitude of bath vibrations, the particle-generated waves decay very slowly in time and the particle motion is influenced by the history of waves along its trajectory. In this high-memory regime, WPEs exhibit hydrodynamic quantum analogs where quantum-like statistics arise from underlying chaotic dynamics. Exploration of WPE dynamics in the very high-memory regime requires solving an integrodifferential equation of motion. By using an idealized one-dimensional WPE model where the particle generates sinusoidal waves, we show that in the limit of infinite memory, the system dynamics reduce to a 3D nonlinear system of ordinary differential equations (ODEs) known as the diffusionless Lorenz equations (DLEs). We use our algebraically simple ODE system to explore in detail, theoretically and numerically, the rich set of periodic and chaotic dynamical behaviors exhibited by the WPE in the parameter space. Specifically, we link the geometry and dynamics in the phase-space of the DLE system to the dynamical and statistical features of WPE motion, paving a way to understand hydrodynamic quantum analogs using phase-space attractors. Our system also provides an alternate interpretation of an attractor-driven particle, i.e., an active particle driven by internal state-space variables of the DLE system. Hence, our results might also provide new insights into modeling active particle locomotion.

Pilot-wave dynamics: Using dynamic mode decomposition to characterize bifurcations, routes to chaos, and emergent statistics

We develop a data-driven characterization of the pilot-wave hydrodynamic system in which a bouncing droplet self-propels along the surface of a vibrating bath. We consider drop motion in a confined one-dimensional geometry and apply the dynamic mode decomposition (DMD) in order to characterize the evolution of the wave field as the bath's vibrational acceleration is increased progressively. Dynamic mode decomposition provides a regression framework for adaptively learning a best-fit linear dynamics model over snapshots of spatiotemporal data. Thus, DMD reduces the complex nonlinear interactions between pilot waves and droplet to a low-dimensional linear superposition of DMD modes characterizing the wave field. In particular, it provides a low-dimensional characterization of the bifurcation structure of the pilot-wave physics, wherein the excitation and recruitment of additional modes in the linear superposition models the bifurcation sequence. This DMD characterization yields a fresh perspective on the bouncing-droplet problem that forges valuable new links with the mathematical machinery of quantum mechanics. Specifically, the analysis shows that as the vibrational acceleration is increased, the pilot-wave field undergoes a series of Hopf bifurcations that ultimately lead to a chaotic wave field. The established relation between the mean pilot-wave field and the droplet statistics allows us to characterize the evolution of the emergent statistics with increased vibrational forcing from the evolution of the pilot-wave field. We thus develop a numerical framework with the same basic structure as quantum mechanics, specifically a wave theory that predicts particle statistics.

The Stability of a Hydrodynamic Bravais Lattice

We present the results of a theoretical investigation of the stability and collective vibrations of a two-dimensional hydrodynamic lattice comprised of millimetric droplets bouncing on the surface of a vibrating liquid bath. We derive the linearized equations of motion describing the dynamics of a generic Bravais lattice, as encompasses all possible tilings of parallelograms in an infinite plane-filling array. Focusing on square and triangular lattice geometries, we demonstrate that for relatively low driving accelerations of the bath, only a subset of inter-drop spacings exist for which stable lattices may be achieved. The range of stable spacings is prescribed by the structure of the underlying wavefield. As the driving acceleration is increased progressively, the initially stationary lattices destabilize into coherent oscillatory motion. Our analysis yields both the instability threshold and the wavevector and polarization of the most unstable vibrational mode. The non-Markovian nature of the droplet dynamics renders the stability analysis of the hydrodynamic lattice more rich and subtle than that of its solid state counterpart.

A classical analog of the quantum Zeeman effect

We extend a recent classical mechanical analog of Bohr's atom consisting of a scalar field coupled to a massive point-like particle [P. Jamet and A. Drezet, "A mechanical analog of Bohr's atom based on de Broglie's double-solution approach," Chaos 31, 103120 (2021)] by adding and studying the contribution of a uniform weak magnetic field on their dynamics. In doing so, we are able to recover the splitting of the energy levels of the atom called Zeeman's effect within the constraints of our model and in agreement with the semiclassical theory of Sommerfeld. This result is obtained using Larmor's theorem for both the field and the particle, associating magnetic effects with inertial Coriolis forces in a rotating frame of reference. Our work, based on the old "double solution" theory of de Broglie, shows that a dualistic model involving a particle guided by a scalar field can reproduce the normal Zeeman effect.

Lorenz-like systems emerging from an integro-differential trajectory equation of a one-dimensional wave-particle entity

Vertically vibrating a liquid bath can give rise to a self-propelled wave-particle entity on its free surface. The horizontal walking dynamics of this wave-particle entity can be described adequately by an integro-differential trajectory equation. By transforming this integro-differential equation of motion for a one-dimensional wave-particle entity into a system of ordinary differential equations (ODEs), we show the emergence of Lorenz-like dynamical systems for various spatial wave forms of the entity. Specifically, we present and give examples of Lorenz-like dynamical systems that emerge when the wave form gradient is (i) a solution of a linear homogeneous constant coefficient ODE, (ii) a polynomial, and (iii) a periodic function. Understanding the dynamics of the wave-particle entity in terms of Lorenz-like systems may prove to be useful in rationalizing emergent statistical behavior from underlying chaotic dynamics in hydrodynamic quantum analogs of walking droplets. Moreover, the results presented here provide an alternative physical interpretation of various Lorenz-like dynamical systems in terms of the walking dynamics of a wave-particle entity.

A mechanical analog of Bohr's atom based on de Broglie's double-solution approach

Motivated by recent developments of hydrodynamical quantum mechanical analogs [J. W. M. Bush, Annu. Rev. Fluid Mech. 47, 269-292 (2015)], we provide a relativistic model for a classical particle coupled to a scalar wave field through a holonomic constraint. In the presence of an external Coulomb field, we define a regime where the particle is guided by the wave in a way similar to the old de Broglie phase-wave proposal. Moreover, this dualistic mechanical analog of the quantum theory is reminiscent of the double-solution approach suggested by de Broglie in 1927 and is able to reproduce the Bohr-Sommerfeld semiclassical quantization formula for an electron moving in an atom.

Discrete and periodic complex Ginzburg-Landau equation for a hydrodynamic active lattice

A discrete and periodic complex Ginzburg-Landau equation, coupled to a mean equation, is systematically derived from a driven and dissipative lattice oscillator model, close to the onset of a supercritical Andronov-Hopf bifurcation. The oscillator model is inspired by recent experiments exploring active vibrations of quasi-one-dimensional lattices of self-propelled millimetric droplets bouncing on a vertically vibrating fluid bath. Our systematic derivation provides a direct link between the constitutive properties of the lattice system and the coefficients of the resultant amplitude equations, paving the way to compare the emergent nonlinear dynamics-namely, the onset and formation of discrete dark solitons, breathers, and traveling waves-against experiments. The framework presented herein is expected to be applicable to a wider class of oscillators characterized by the presence of a dynamic coupling potential between particles. More broadly, our results point to deeper connections between nonlinear oscillators and the physics of active and driven matter.

Quantum hydrodynamic theory of quantum fluctuations in dipolar Bose-Einstein condensate

Traditional quantum hydrodynamics of Bose-Einstein condensates (BECs) is restricted by the continuity and Euler equations. The quantum Bohm potential (the quantum part of the momentum flux) has a nontrivial part that can evolve under quantum fluctuations. The quantum fluctuations are the effect of the appearance of particles in the excited states during the evolution of BEC mainly consisting of the particles in the quantum state with the lowest energy. To cover this phenomenon in terms of hydrodynamic methods, we need to derive equations for the momentum flux and the current of the momentum flux. The current of the momentum flux evolution equation contains the interaction leading to the quantum fluctuations. In the dipolar BECs, we deal with the long-range interaction. Its contribution is proportional to the average macroscopic potential of the dipole-dipole interaction (DDI) appearing in the mean-field regime. The current of the momentum flux evolution equation contains the third derivative of this potential. It is responsible for the dipolar part of quantum fluctuations. Higher derivatives correspond to the small scale contributions of the DDI. The quantum fluctuations lead to the existence of the second wave solution. The quantum fluctuations introduce the instability of the BECs. If the dipole-dipole interaction is attractive, but being smaller than the repulsive short-range interaction presented by the first interaction constant, there is the long-wavelength instability. There is a more complex picture for the repulsive DDI. There is the small area with the long-wavelength instability that transits into a stability interval where two waves exist.

Hydrodynamic quantum analogs

The walking droplet system discovered by Yves Couder and Emmanuel Fort presents an example of a vibrating particle self-propelling through a resonant interaction with its own wave field. It provides a means of visualizing a particle as an excitation of a field, a common notion in quantum field theory. Moreover, it represents the first macroscopic realization of a form of dynamics proposed for quantum particles by Louis de Broglie in the 1920s. The fact that this hydrodynamic pilot-wave system exhibits many features typically associated with the microscopic, quantum realm raises a number of intriguing questions. At a minimum, it extends the range of classical systems to include quantum-like statistics in a number of settings. A more optimistic stance is that it suggests the manner in which quantum mechanics might be completed through a theoretical description of particle trajectories. We here review the experimental studies of the walker system, and the hierarchy of theoretical models developed to rationalize its behavior. Particular attention is given to enumerating the dynamical mechanisms responsible for the emergence of robust, structured statistical behavior. Another focus is demonstrating how the temporal nonlocality of the droplet dynamics, as results from the persistence of its pilot wave field, may give rise to behavior that appears to be spatially nonlocal. Finally, we describe recent explorations of a generalized theoretical framework that provides a mathematical bridge between the hydrodynamic pilot-wave system and various realist models of quantum dynamics.

Speed oscillations in classical pilot-wave dynamics

We present the results of a theoretical investigation of a dynamical system consisting of a particle self-propelling through a resonant interaction with its own quasi-monochromatic pilot-wave field. We rationalize two distinct mechanisms, arising in different regions of parameter space, that may lead to a wavelike statistical signature with the pilot-wavelength. First, resonant speed oscillations with the wavelength of the guiding wave may arise when the particle is perturbed from its steady self-propelling state. Second, a random-walk-like motion may set in when the decay rate of the pilot-wave field is sufficiently small. The implications for the emergent statistics in classical pilot-wave systems are discussed.

A hydrodynamic analog of Friedel oscillations

We present a macroscopic analog of an open quantum system, achieved with a classical pilot-wave system. Friedel oscillations are the angstrom-scale statistical signature of an impurity on a metal surface, concentric circular modulations in the probability density function of the surrounding electron sea. We consider a millimetric drop, propelled by its own wave field along the surface of a vibrating liquid bath, interacting with a submerged circular well. An ensemble of drop trajectories displays a statistical signature in the vicinity of the well that is strikingly similar to Friedel oscillations. The droplet trajectories reveal the dynamical roots of the emergent statistics. Our study elucidates a new mechanism for emergent quantum-like statistics in pilot-wave hydrodynamics and so suggests new directions for the nascent field of hydrodynamic quantum analogs.