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Ellegaard C, Levinsen MT. Experimental investigation of walking drops: Wave field and interaction with slit structures. Phys Rev E 2024; 109:035101. [PMID: 38632787 DOI: 10.1103/physreve.109.035101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/23/2023] [Accepted: 02/09/2024] [Indexed: 04/19/2024]
Abstract
While bouncing walking silicone oil droplets (walkers) do show many quantumlike phenomena, the original, most intriguing, double-slit experiment with walkers has been shown to be an overinterpretation of data. Several experiments and numerical simulations have proven that for at least some parameter region there is no randomness. Still, true randomness was claimed to be observed in an experiment on chaotically bouncing walkers. Also, most of the available phase space has not been investigated. The main goal of this paper is therefore to look for true interference and chaos in the entire phase space. Recently, we made an extensive investigation of drops interacting with slits, but still in a limited range. However, the outcome was always deterministic and only incidentally mimicked the statistics of the corresponding quantum case. We also showed that the extra interference, already seen by others, in the double-slit case was caused by reflection of waves from the outlet of the unused slit after passage and thus was not a true double-slit effect. A new theoretical treatment of bouncing drop dynamics has since given analytic relations for the associated wave field, leading to a proposal for criteria for the possible occurrence of true interference in the double-slit experiment. Satisfying these criteria, requires working close to the onset of the Faraday instability, with two spatial conditions favoring slow walkers, and a temporal condition favoring fast walkers. The regions of high velocity, where the walkers bounce periodically, and of very low velocity, with chaotically bouncing walkers, have not been comprehensively investigated so far. We have therefore looked at these regions, probing the limits for interaction with slits. Furthermore, noting that a short transit time is essential to fulfill the criteria, experiments were done using double-slit barriers only 0.5 and 2 mm broad. Nowhere was true interference or a chaotic response found. As the theory has implications for many of the observed quantumlike phenomena involving walkers as, e.g., tunneling and interaction between drops, we have measured the spatial and temporal decay of the wave field. A comparison with the theory shows very good agreement but leads to a reformulation of the above-mentioned criteria.
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Affiliation(s)
- Clive Ellegaard
- Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
| | - Mogens T Levinsen
- Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
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Choueiri G, Suri B, Merrin J, Serbyn M, Hof B, Budanur NB. Crises and chaotic scattering in hydrodynamic pilot-wave experiments. CHAOS (WOODBURY, N.Y.) 2022; 32:093138. [PMID: 36182399 DOI: 10.1063/5.0102904] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/13/2022] [Accepted: 08/29/2022] [Indexed: 06/16/2023]
Abstract
Theoretical foundations of chaos have been predominantly laid out for finite-dimensional dynamical systems, such as the three-body problem in classical mechanics and the Lorenz model in dissipative systems. In contrast, many real-world chaotic phenomena, e.g., weather, arise in systems with many (formally infinite) degrees of freedom, which limits direct quantitative analysis of such systems using chaos theory. In the present work, we demonstrate that the hydrodynamic pilot-wave systems offer a bridge between low- and high-dimensional chaotic phenomena by allowing for a systematic study of how the former connects to the latter. Specifically, we present experimental results, which show the formation of low-dimensional chaotic attractors upon destabilization of regular dynamics and a final transition to high-dimensional chaos via the merging of distinct chaotic regions through a crisis bifurcation. Moreover, we show that the post-crisis dynamics of the system can be rationalized as consecutive scatterings from the nonattracting chaotic sets with lifetimes following exponential distributions.
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Affiliation(s)
- George Choueiri
- Institute of Science and Technology Austria, 3400 Klosterneuburg, Austria
| | - Balachandra Suri
- Institute of Science and Technology Austria, 3400 Klosterneuburg, Austria
| | - Jack Merrin
- Institute of Science and Technology Austria, 3400 Klosterneuburg, Austria
| | - Maksym Serbyn
- Institute of Science and Technology Austria, 3400 Klosterneuburg, Austria
| | - Björn Hof
- Institute of Science and Technology Austria, 3400 Klosterneuburg, Austria
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Durey M, Bush JWM. Classical pilot-wave dynamics: The free particle. CHAOS (WOODBURY, N.Y.) 2021; 31:033136. [PMID: 33810713 DOI: 10.1063/5.0039975] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/08/2020] [Accepted: 02/18/2021] [Indexed: 06/12/2023]
Abstract
We present the results of a theoretical investigation into the dynamics of a vibrating particle propelled by its self-induced wave field. Inspired by the hydrodynamic pilot-wave system discovered by Yves Couder and Emmanuel Fort, the idealized pilot-wave system considered here consists of a particle guided by the slope of its quasi-monochromatic "pilot" wave, which encodes the history of the particle motion. We characterize this idealized pilot-wave system in terms of two dimensionless groups that prescribe the relative importance of particle inertia, drag and wave forcing. Prior work has delineated regimes in which self-propulsion of the free particle leads to steady or oscillatory rectilinear motion; it has further revealed parameter regimes in which the particle executes a stable circular orbit, confined by its pilot wave. We here report a number of new dynamical states in which the free particle executes self-induced wobbling and precessing orbital motion. We also explore the statistics of the chaotic regime arising when the time scale of the wave decay is long relative to that of particle motion and characterize the diffusive and rotational nature of the resultant particle dynamics. We thus present a detailed characterization of free-particle motion in this rich two-parameter family of dynamical systems.
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Affiliation(s)
- Matthew Durey
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - John W M Bush
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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Bush JWM, Oza AU. Hydrodynamic quantum analogs. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2020; 84:017001. [PMID: 33065567 DOI: 10.1088/1361-6633/abc22c] [Citation(s) in RCA: 26] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/29/2020] [Accepted: 10/16/2020] [Indexed: 06/11/2023]
Abstract
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.
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Affiliation(s)
- John W M Bush
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, United States of America
| | - Anand U Oza
- Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, United States of America
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Durey M. Bifurcations and chaos in a Lorenz-like pilot-wave system. CHAOS (WOODBURY, N.Y.) 2020; 30:103115. [PMID: 33138446 DOI: 10.1063/5.0020775] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/03/2020] [Accepted: 09/29/2020] [Indexed: 06/11/2023]
Abstract
A millimetric droplet may bounce and self-propel on the surface of a vertically vibrating fluid bath, guided by its self-generated wave field. This hydrodynamic pilot-wave system exhibits a vast range of dynamics, including behavior previously thought to be exclusive to the quantum realm. We present the results of a theoretical investigation of an idealized pilot-wave model, in which a particle is guided by a one-dimensional wave that is equipped with the salient features of the hydrodynamic system. The evolution of this reduced pilot-wave system may be simplified by projecting onto a three-dimensional dynamical system describing the evolution of the particle velocity, the local wave amplitude, and the local wave slope. As the resultant dynamical system is remarkably similar in form to the Lorenz system, we utilize established properties of the Lorenz equations as a guide for identifying and elucidating several pilot-wave phenomena, including the onset and characterization of chaos.
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Affiliation(s)
- Matthew Durey
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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Hubert M, Perrard S, Labousse M, Vandewalle N, Couder Y. Tunable bimodal explorations of space from memory-driven deterministic dynamics. Phys Rev E 2019; 100:032201. [PMID: 31639901 DOI: 10.1103/physreve.100.032201] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/30/2018] [Indexed: 06/10/2023]
Abstract
We present a wave-memory-driven system that exhibits intermittent switching between two propulsion modes in free space. The model is based on a pointlike particle emitting periodically cylindrical standing waves. Submitted to a force related to the local wave-field gradient, the particle is propelled, while the wave field stores positional information on the particle trajectory. For long memory, the linear motion is unstable and we observe erratic switches between two propulsive modes: linear motion and diffusive motion. We show that the bimodal propulsion and the stochastic aspect of the dynamics at long time are generated by a Shil'nikov chaos. The memory of the system controls the fraction of time spent in each phase. The resulting bimodal dynamics shows analogies with intermittent search strategies usually observed in living systems of much higher complexity.
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Affiliation(s)
- Maxime Hubert
- GRASP, Institute of Physics, Université de Liège, 4000 Liège, Belgium, European Union
| | - Stéphane Perrard
- Laboratoire de Physique de l'ENS, CNRS UMR 8550 ENS and PSL University, 75005 Paris, European Union
| | - Matthieu Labousse
- Gulliver, CNRS UMR 7083, ESPCI Paris and PSL University, 75005 Paris, France, European Union
| | - Nicolas Vandewalle
- GRASP, Institute of Physics, Université de Liège, 4000 Liège, Belgium, European Union
| | - Yves Couder
- Matière et Systèmes Complexes, CNRS UMR 7057, Université Paris Diderot, Sorbonne Paris Cité, 75013 Paris, France, European Union
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Bacot V, Perrard S, Labousse M, Couder Y, Fort E. Multistable Free States of an Active Particle from a Coherent Memory Dynamics. PHYSICAL REVIEW LETTERS 2019; 122:104303. [PMID: 30932640 DOI: 10.1103/physrevlett.122.104303] [Citation(s) in RCA: 17] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/08/2018] [Revised: 12/14/2018] [Indexed: 06/09/2023]
Abstract
We investigate the dynamics of a deterministic self-propelled particle endowed with coherent memory. We evidence experimentally and numerically that it exhibits several stable free states. The system is composed of a self-propelled drop bouncing on a vibrated liquid driven by the waves it emits at each bounce. This object possesses a propulsion memory resulting from the coherent interference of the waves accumulated along its path. We investigate here the transitory regime of the buildup of the dynamics which leads to velocity modulations. Experiments and numerical simulations enable us to explore unchartered areas of the phase space and reveal the existence of a self-sustained oscillatory regime. Finally, we show the coexistence of several free states. This feature emerges both from the spatiotemporal nonlocality of this path memory dynamics as well as the wave nature of the driving mechanism.
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Affiliation(s)
- V Bacot
- Matière et Systèmes Complexes, CNRS UMR 7057, Université Paris Diderot, Sorbonne Paris Cité, 75013 Paris, France
- Institut Langevin, CNRS UMR 7587, ESPCI Paris and PSL University, 75005 Paris, France
- LadHyX, CNRS UMR 7646, École Polytechnique, 91128 Palaiseau, France
| | - S Perrard
- Laboratoire de Physique Statistique, CNRS UMR 8550 ENS and PSL University, 75005 Paris, France
| | - M Labousse
- Gulliver, CNRS UMR 7083, ESPCI Paris and PSL University, 75005 Paris, France
| | - Y Couder
- Matière et Systèmes Complexes, CNRS UMR 7057, Université Paris Diderot, Sorbonne Paris Cité, 75013 Paris, France
| | - E Fort
- Institut Langevin, CNRS UMR 7587, ESPCI Paris and PSL University, 75005 Paris, France
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Budanur NB, Fleury M. State space geometry of the chaotic pilot-wave hydrodynamics. CHAOS (WOODBURY, N.Y.) 2019; 29:013122. [PMID: 30709150 DOI: 10.1063/1.5058279] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/18/2018] [Accepted: 12/18/2018] [Indexed: 06/09/2023]
Abstract
We consider the motion of a droplet bouncing on a vibrating bath of the same fluid in the presence of a central potential. We formulate a rotation symmetry-reduced description of this system, which allows for the straightforward application of dynamical systems theory tools. As an illustration of the utility of the symmetry reduction, we apply it to a model of the pilot-wave system with a central harmonic force. We begin our analysis by identifying local bifurcations and the onset of chaos. We then describe the emergence of chaotic regions and their merging bifurcations, which lead to the formation of a global attractor. In this final regime, the droplet's angular momentum spontaneously changes its sign as observed in the experiments of Perrard et al. [Phys. Rev. Lett.113(10), 104101 (2014)].
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Affiliation(s)
- Nazmi Burak Budanur
- Nonlinear Dynamics and Turbulence Group, IST Austria, 3400 Klosterneuburg, Austria
| | - Marc Fleury
- Freeside LLC, 3344 Peachtree Rd., Atlanta, Georgia 30326, USA
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Bush JWM, Couder Y, Gilet T, Milewski PA, Nachbin A. Introduction to focus issue on hydrodynamic quantum analogs. CHAOS (WOODBURY, N.Y.) 2018; 28:096001. [PMID: 30278632 DOI: 10.1063/1.5055383] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/07/2018] [Accepted: 09/09/2018] [Indexed: 06/08/2023]
Abstract
Hydrodynamic quantum analogs is a nascent field initiated in 2005 by the discovery of a hydrodynamic pilot-wave system [Y. Couder, S. Protière, E. Fort, and A. Boudaoud, Nature 437, 208 (2005)]. The system consists of a millimetric droplet self-propeling along the surface of a vibrating bath through a resonant interaction with its own wave field [J. W. M. Bush, Annu. Rev. Fluid Mech. 47, 269-292 (2015)]. There are three critical ingredients for the quantum like-behavior. The first is "path memory" [A. Eddi, E. Sultan, J. Moukhtar, E. Fort, M. Rossi, and Y. Couder, J. Fluid Mech. 675, 433-463 (2011)], which renders the system non-Markovian: the instantaneous wave force acting on the droplet depends explicitly on its past. The second is the resonance condition between droplet and wave that ensures a highly structured monochromatic pilot wave field that imposes an effective potential on the walking droplet, resulting in preferred, quantized states. The third ingredient is chaos, which in several systems is characterized by unpredictable switching between unstable periodic orbits. This focus issue is devoted to recent studies of and relating to pilot-wave hydrodynamics, a field that attempts to answer the following simple but provocative question: Might deterministic chaotic pilot-wave dynamics underlie quantum statistics?
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Affiliation(s)
- John W M Bush
- Department of Mathematics, MIT, Cambridge, Massachusetts 02139, USA
| | - Yves Couder
- Matière et Sytèmes Complexes, CNRS UMR 7057, Université Paris Diderot, Sorbonne Paris Cité, 75013 Paris, France
| | - Tristan Gilet
- Microfluidics Lab, Department of Mechanical and Aerospace Engineering, University of Liege, Allée de la Découverte 9, 4000 Liège, Belgium
| | - Paul A Milewski
- Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
| | - André Nachbin
- National Institute for Pure and Applied Mathematics (IMPA), Est. D. Castorina 110, Rio de Janeiro, RJ 22460-320, Brazil
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