1
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Kutz JN, Nachbin A, Baddoo PJ, Bush JWM. Pilot-wave dynamics: Using dynamic mode decomposition to characterize bifurcations, routes to chaos, and emergent statistics. Phys Rev E 2023; 108:034213. [PMID: 37849115 DOI: 10.1103/physreve.108.034213] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/14/2022] [Accepted: 08/29/2023] [Indexed: 10/19/2023]
Abstract
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.
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Affiliation(s)
- J Nathan Kutz
- Department of Applied Mathematics and Electrical and Computer Engineering, University of Washington, Seattle, Washington 98195, USA
| | - André Nachbin
- Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, Massachusetts 01742, USA
| | - Peter J Baddoo
- 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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2
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Rahman A. Damped-driven system of bouncing droplets leading to deterministic diffusive behavior. Phys Rev E 2023; 108:035103. [PMID: 37849082 DOI: 10.1103/physreve.108.035103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/24/2022] [Accepted: 08/24/2023] [Indexed: 10/19/2023]
Abstract
Damped-driven systems are ubiquitous in science, however, the damping and driving mechanisms are often quite convoluted. This paper presents an experimental and theoretical investigation of a fluidic droplet on a vertically vibrating fluid bath as a damped-driven system. We study a fluidic droplet in an annular cavity with the fluid bath forced above the Faraday wave threshold. We model the droplet as a kinematic point particle in air and as inelastic collisions during impact with the bath. In both experiments and the model, the droplet is observed to chaotically change velocity with a Gaussian distribution. Finally, the statistical distributions from experiments and theory are analyzed. Incredibly, this simple deterministic interaction of damping and driving of the droplet leads to more complex Brownian-like and Levy-like behavior.
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Affiliation(s)
- Aminur Rahman
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195-3925, USA
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3
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Hélias A, Labousse M. Statistical self-organization of an assembly of interacting walking drops in a confining potential. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2023; 46:29. [PMID: 37058179 DOI: 10.1140/epje/s10189-023-00288-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/26/2022] [Accepted: 03/31/2023] [Indexed: 06/19/2023]
Abstract
A drop bouncing on a vertically vibrated surface may self-propel forward by standing waves and travels along a fluid interface. This system called walking drop forms a non-quantum wave-particle association at the macroscopic scale. The dynamics of one particle has triggered many investigations and has resulted in spectacular experimental results in the last decade. We investigate numerically the dynamics of an assembly of walkers, i.e., a large number of walking drops evolving on a unbounded fluid interface in the presence of a confining potential acting on the particles. We show that even if the individual trajectories are erratic, the system presents a well-defined ordered internal structure that remains invariant to parameter variations such as the number of drops, the memory time and the bath radius. We rationalize such non-stationary self-organization in terms of the symmetry of the waves and show that oscillatory pair potentials form a wavy collective state of active matter.
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Affiliation(s)
- Adrien Hélias
- Gulliver, UMR CNRS 7083, ESPCI Paris, Université PSL, 75005, Paris, France
- Department of Physics and Astronomy, Western University, 1151 Richmond St, London, N6A 3K7, Canada
| | - Matthieu Labousse
- Gulliver, UMR CNRS 7083, ESPCI Paris, Université PSL, 75005, Paris, France.
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4
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Perks J, Valani RN. Dynamics, interference effects, and multistability in a Lorenz-like system of a classical wave-particle entity in a periodic potential. CHAOS (WOODBURY, N.Y.) 2023; 33:033147. [PMID: 37003812 DOI: 10.1063/5.0125727] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/13/2022] [Accepted: 03/03/2023] [Indexed: 06/19/2023]
Abstract
A classical wave-particle entity (WPE) can be realized experimentally as a droplet walking on the free surface of a vertically vibrating liquid bath, with the droplet's horizontal walking motion guided by its self-generated wave field. These self-propelled WPEs have been shown to exhibit analogs of several quantum and optical phenomena. Using an idealized theoretical model that takes the form of a Lorenz-like system, we theoretically and numerically explore the dynamics of such a one-dimensional WPE in a sinusoidal potential. We find steady states of the system that correspond to a stationary WPE as well as a rich array of unsteady motions, such as back-and-forth oscillating walkers, runaway oscillating walkers, and various types of irregular walkers. In the parameter space formed by the dimensionless parameters of the applied sinusoidal potential, we observe patterns of alternating unsteady behaviors suggesting interference effects. Additionally, in certain regions of the parameter space, we also identify multistability in the particle's long-term behavior that depends on the initial conditions. We make analogies between the identified behaviors in the WPE system and Bragg's reflection of light as well as electron motion in crystals.
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Affiliation(s)
- J Perks
- School of Computer and Mathematical Sciences, University of Adelaide, Adelaide, South Australia 5005, Australia
| | - R N Valani
- School of Computer and Mathematical Sciences, University of Adelaide, Adelaide, South Australia 5005, Australia
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5
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Overload wave-memory induces amnesia of a self-propelled particle. Nat Commun 2022; 13:4357. [PMID: 35896544 PMCID: PMC9329294 DOI: 10.1038/s41467-022-31736-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/24/2021] [Accepted: 06/30/2022] [Indexed: 11/17/2022] Open
Abstract
Information storage is a key element of autonomous, out-of-equilibrium dynamics, especially for biological and synthetic active matter. In synthetic active matter however, the implementation of internal memory in self-propelled systems is often absent, limiting our understanding of memory-driven dynamics. Recently, a system comprised of a droplet generating its guiding wavefield appeared as a prime candidate for such investigations. Indeed, the wavefield, propelling the droplet, encodes information about the droplet trajectory and the amount of information can be controlled by a single scalar experimental parameter. In this work, we show numerically and experimentally that the accumulation of information in the wavefield induces the loss of time correlations, where the dynamics can then be described by a memory-less process. We rationalize the resulting statistical behavior by defining an effective temperature for the particle dynamics where the wavefield acts as a thermostat of large dimensions, and by evidencing a minimization principle of the generated wavefield. Memory and information storage play an important role in biological systems, however challenging to implement in synthetic active matter. The authors show that the wave field, propelling the particle, acts as a memory repository, and an excess of memory leads to a memory-less particle dynamics.
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6
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Valani RN. Lorenz-like systems emerging from an integro-differential trajectory equation of a one-dimensional wave-particle entity. CHAOS (WOODBURY, N.Y.) 2022; 32:023129. [PMID: 35232028 DOI: 10.1063/5.0076162] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/21/2021] [Accepted: 01/28/2022] [Indexed: 06/14/2023]
Abstract
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.
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Affiliation(s)
- Rahil N Valani
- School of Mathematical Sciences, University of Adelaide, Adelaide, South Australia 5005, Australia
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7
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Valani RN, Slim AC, Paganin DM, Simula TP, Vo T. Unsteady dynamics of a classical particle-wave entity. Phys Rev E 2021; 104:015106. [PMID: 34412331 DOI: 10.1103/physreve.104.015106] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/17/2020] [Accepted: 06/08/2021] [Indexed: 11/07/2022]
Abstract
A droplet bouncing on the surface of a vertically vibrating liquid bath can walk horizontally, guided by the waves it generates on each impact. This results in a self-propelled classical particle-wave entity. By using a one-dimensional theoretical pilot-wave model with a generalized wave form, we investigate the dynamics of this particle-wave entity. We employ different spatial wave forms to understand the role played by both wave oscillations and spatial wave decay in the walking dynamics. We observe steady walking motion as well as unsteady motions such as oscillating walking, self-trapped oscillations, and irregular walking. We explore the dynamical and statistical aspects of irregular walking and show an equivalence between the droplet dynamics and the Lorenz system, as well as making connections with the Langevin equation and deterministic diffusion.
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Affiliation(s)
- Rahil N Valani
- School of Physics and Astronomy, Monash University, Victoria 3800, Australia
| | - Anja C Slim
- School of Mathematics, Monash University, Victoria 3800, Australia.,School of Earth, Atmosphere and Environment, Monash University, Victoria 3800, Australia
| | - David M Paganin
- School of Physics and Astronomy, Monash University, Victoria 3800, Australia
| | - Tapio P Simula
- Optical Sciences Centre, Swinburne University of Technology, Melbourne 3122, Australia
| | - Theodore Vo
- School of Mathematics, Monash University, Victoria 3800, Australia
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8
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Montes J, Revuelta F, Borondo F. Bohr-Sommerfeld-like quantization in the theory of walking droplets. Phys Rev E 2021; 103:053110. [PMID: 34134206 DOI: 10.1103/physreve.103.053110] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2019] [Accepted: 04/29/2021] [Indexed: 11/07/2022]
Abstract
Recent experiments have shown that self-propelled millimetric walking droplets bouncing on a vibrating liquid surface exhibit phenomena, such as interference or tunneling, that so far were thought to be possible only in the microscopic realm. Here we present calculations showing that the surface wave satisfies, in the long-memory limit, a Bohr-Sommerfeld quantization-like relation. This strongly suggest the possibility of a novel fundamental type of quantization in these experiments, which can simultaneously explain their emulation of the quantum behavior and, more importantly, shed light into some of the interpretational difficulties of the standard quantum theory.
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Affiliation(s)
- J Montes
- Instituto de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049 Madrid, Spain.,Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
| | - F Revuelta
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, Avda. Puerta de Hierro 2-4, 28040 Madrid, Spain
| | - F Borondo
- Instituto de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049 Madrid, Spain.,Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
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9
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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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10
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Andreev PA. Quantum hydrodynamic theory of quantum fluctuations in dipolar Bose-Einstein condensate. CHAOS (WOODBURY, N.Y.) 2021; 31:023120. [PMID: 33653039 DOI: 10.1063/5.0036511] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2020] [Accepted: 01/27/2021] [Indexed: 06/12/2023]
Abstract
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.
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Affiliation(s)
- Pavel A Andreev
- Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russian Federation and Faculty of Physics, Mathematics and Natural Sciences, Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street, Moscow 117198, Russian Federation
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11
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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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12
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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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13
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Durey M, Turton SE, Bush JWM. Speed oscillations in classical pilot-wave dynamics. Proc Math Phys Eng Sci 2020; 476:20190884. [PMID: 32831603 DOI: 10.1098/rspa.2019.0884] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2019] [Accepted: 06/24/2020] [Indexed: 11/12/2022] Open
Abstract
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.
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Affiliation(s)
- Matthew Durey
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
| | - Sam E Turton
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
| | - John W M Bush
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
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14
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Sáenz PJ, Cristea-Platon T, Bush JWM. A hydrodynamic analog of Friedel oscillations. SCIENCE ADVANCES 2020; 6:eaay9234. [PMID: 32440541 PMCID: PMC7228752 DOI: 10.1126/sciadv.aay9234] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/29/2019] [Accepted: 03/09/2020] [Indexed: 06/01/2023]
Abstract
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.
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Affiliation(s)
- Pedro J. Sáenz
- Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
| | - Tudor Cristea-Platon
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
| | - John W. M. Bush
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
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15
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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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16
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Cristea-Platon T, Sáenz PJ, Bush JWM. Walking droplets in a circular corral: Quantisation and chaos. CHAOS (WOODBURY, N.Y.) 2018; 28:096116. [PMID: 30278624 DOI: 10.1063/1.5034123] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/09/2018] [Accepted: 08/12/2018] [Indexed: 06/08/2023]
Abstract
A millimetric liquid droplet may walk across the surface of a vibrating liquid bath through a resonant interaction with its self-generated wavefield. Such walking droplets, or "walkers," have attracted considerable recent interest because they exhibit certain features previously believed to be exclusive to the microscopic, quantum realm. In particular, the intricate motion of a walker confined to a closed geometry is known to give rise to a coherent wave-like statistical behavior similar to that of electrons confined to quantum corrals. Here, we examine experimentally the dynamics of a walker inside a circular corral. We first illustrate the emergence of a variety of stable dynamical states for relatively low vibrational accelerations, which lead to a double quantisation in angular momentum and orbital radius. We then characterise the system's transition to chaos for increasing vibrational acceleration and illustrate the resulting breakdown of the double quantisation. Finally, we discuss the similarities and differences between the dynamics and statistics of a walker inside a circular corral and that of a walker subject to a simple harmonic potential.
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Affiliation(s)
- Tudor Cristea-Platon
- Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
| | - Pedro J Sáenz
- Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
| | - John W M Bush
- Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
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