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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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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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Valani RN. Anomalous transport of a classical wave-particle entity in a tilted potential. Phys Rev E 2022; 105:L012101. [PMID: 35193237 DOI: 10.1103/physreve.105.l012101] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/25/2021] [Accepted: 12/17/2021] [Indexed: 11/07/2022]
Abstract
A classical wave-particle entity in the form of a millimetric walking droplet can emerge on the free surface of a vertically vibrating liquid bath. Such wave-particle entities have been shown to exhibit hydrodynamic analogs of quantum systems. Using an idealized theoretical model of this wave-particle entity in a tilted potential, we explore its transport behavior. The integro-differential equation of motion governing the dynamics of the wave-particle entity transforms to a Lorenz-like system of ordinary differential equations that drives the particle's velocity. Several anomalous transport regimes such as absolute negative mobility, differential negative mobility, and lock-in regions corresponding to force-independent mobility are observed. These observations motivate experiments in the hydrodynamic walking-droplet system for the experimental realizations of anomalous transport phenomena.
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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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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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Sáenz PJ, Pucci G, Turton SE, Goujon A, Rosales RR, Dunkel J, Bush JWM. Emergent order in hydrodynamic spin lattices. Nature 2021; 596:58-62. [PMID: 34349289 DOI: 10.1038/s41586-021-03682-1] [Citation(s) in RCA: 11] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/06/2020] [Accepted: 06/01/2021] [Indexed: 02/06/2023]
Abstract
Macroscale analogues1-3 of microscopic spin systems offer direct insights into fundamental physical principles, thereby advancing our understanding of synchronization phenomena4 and informing the design of novel classes of chiral metamaterials5-7. Here we introduce hydrodynamic spin lattices (HSLs) of 'walking' droplets as a class of active spin systems with particle-wave coupling. HSLs reveal various non-equilibrium symmetry-breaking phenomena, including transitions from antiferromagnetic to ferromagnetic order that can be controlled by varying the lattice geometry and system rotation8. Theoretical predictions based on a generalized Kuramoto model4 derived from first principles rationalize our experimental observations, establishing HSLs as a versatile platform for exploring active phase oscillator dynamics. The tunability of HSLs suggests exciting directions for future research, from active spin-wave dynamics to hydrodynamic analogue computation and droplet-based topological insulators.
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Affiliation(s)
- Pedro J Sáenz
- Department of Mathematics, University of North Carolina, Chapel Hill, NC, USA. .,Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA.
| | - Giuseppe Pucci
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA.,Institut de Physique de Rennes, CNRS, Université de Rennes, UMR 6251, Rennes, France
| | - Sam E Turton
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA
| | - Alexis Goujon
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA.,School of Engineering, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
| | - Rodolfo R Rosales
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA
| | - Jörn Dunkel
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA
| | - John W M Bush
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA.
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Valani RN, Slim AC, Simula TP. Stop-and-go locomotion of superwalking droplets. Phys Rev E 2021; 103:043102. [PMID: 34005929 DOI: 10.1103/physreve.103.043102] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/06/2020] [Accepted: 03/19/2021] [Indexed: 11/07/2022]
Abstract
Vertically vibrating a liquid bath at two frequencies, f and f/2, having a constant relative phase difference can give rise to self-propelled superwalking droplets on the liquid surface. We have numerically investigated such superwalking droplets in the regime where the phase difference varies slowly with time. We predict the emergence of stop-and-go motion of droplets, consistent with experimental observations [Valani et al. Phys. Rev. Lett. 123, 024503 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.024503]. Our simulations in the parameter space spanned by the droplet size and the rate of traversal of the phase difference uncover three different types of droplet motion: back-and-forth, forth-and-forth, and irregular stop-and-go motion, which we explore in detail. Our findings lay a foundation for further studies of dynamically driven droplets, whereby the droplet's motion may be guided by engineering arbitrary time-dependent phase difference functions.
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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
| | - Tapio P Simula
- Optical Sciences Centre, Swinburne University of Technology, Melbourne 3122, Australia
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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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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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