1
|
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.
Collapse
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
| |
Collapse
|
2
|
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.
Collapse
|
3
|
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.
Collapse
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
| |
Collapse
|
4
|
Tadrist L, Gilet T, Schlagheck P, Bush JWM. Predictability in a hydrodynamic pilot-wave system: Resolution of walker tunneling. Phys Rev E 2020; 102:013104. [PMID: 32795022 DOI: 10.1103/physreve.102.013104] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/20/2019] [Accepted: 06/21/2020] [Indexed: 06/11/2023]
Abstract
A walker is a macroscopic coupling of a droplet and a capillary wave field that exhibits several quantumlike properties. In 2009, Eddi et al. [Phys. Rev. Lett. 102, 240401 (2009)PRLTAO0031-900710.1103/PhysRevLett.102.240401] showed that walkers may cross a submerged barrier in an unpredictable manner and named this behavior "unpredictable walker tunneling." In quantum mechanics, tunneling is one of the simplest arrangements where similar unpredictability occurs. In this paper, we investigate how unpredictability can be unveiled for walkers through an experimental study of walker tunneling with precision. We refine both time and position measurements to take into account the fast bouncing dynamics of the system. Tunneling is shown to be unpredictable until a distance of 2.6 mm from the barrier center, where we observe the separation of reflected and transmitted trajectories in the position-velocity phase-space. The unpredictability is unlikely to be attributable to either uncertainty in the initial conditions or to the noise in the experiment. It is more likely due to changes in the drop's vertical dynamics arising when it interacts with the barrier. We compare this macroscopic system to a tunneling quantum particle that is subjected to repeated measurements of its position and momentum. We show that, despite the different theoretical treatments of these two disparate systems, similar patterns emerge in the position-velocity phase space.
Collapse
Affiliation(s)
- Loïc Tadrist
- Microfluidics Lab, Aerospace and Mechanical Engineering, University of Liege, Allée de la découverte 9, 4000 Liège, Belgium
| | - Tristan Gilet
- Microfluidics Lab, Aerospace and Mechanical Engineering, University of Liege, Allée de la découverte 9, 4000 Liège, Belgium
| | - Peter Schlagheck
- IPNAS, CESAM research unit, University of Liege, Allée du 6 Août 15, 4000 Liège, Belgium
| | - John W M Bush
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| |
Collapse
|
5
|
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.
Collapse
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
| |
Collapse
|
6
|
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.
Collapse
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
| |
Collapse
|
7
|
Nachbin A. Walking droplets correlated at a distance. CHAOS (WOODBURY, N.Y.) 2018; 28:096110. [PMID: 30278623 DOI: 10.1063/1.5050805] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/02/2018] [Accepted: 09/04/2018] [Indexed: 06/08/2023]
Abstract
Bouncing fluid droplets can walk on the surface of a vibrating bath forming a wave-particle association. Walking droplets have many quantum-like features. Research efforts are continuously exploring quantum analogues and respective limitations. Here, we demonstrate that two oscillating particles (millimetric droplets) confined to separate potential wells exhibit correlated dynamical features, even when separated by a large distance. A key feature is the underlying wave mediated dynamics. The particles' phase space dynamics is given by the system as a whole and cannot be described independently. Numerical phase space histograms display statistical coherence; the particles' intricate distributions in phase space are statistically indistinguishable. However, removing one particle changes the phase space picture completely, which is reminiscent of entanglement. The model here presented also relates to nonlinearly coupled oscillators where synchronization can break out spontaneously. The present oscillator-coupling is dynamic and can change intensity through the underlying wave field as opposed to, for example, the Kuramoto model where the coupling is pre-defined. There are some regimes where we observe phase-locking or, more generally, regimes where the oscillators are statistically indistinguishable in phase-space, where numerical histograms display their (mutual) most likely amplitude and phase.
Collapse
Affiliation(s)
- André Nachbin
- National Institute for Pure and Applied Mathematics (IMPA), Est. D. Castorina 110, Rio de Janeiro, RJ 22460-320, Brazil
| |
Collapse
|
8
|
Valani RN, Slim AC, Simula T. Hong-Ou-Mandel-like two-droplet correlations. CHAOS (WOODBURY, N.Y.) 2018; 28:096104. [PMID: 30278625 DOI: 10.1063/1.5032114] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/02/2018] [Accepted: 07/05/2018] [Indexed: 06/08/2023]
Abstract
We present a numerical study of two-droplet pair correlations for in-phase droplets walking on a vibrating bath. Two such walkers are launched toward a common point of intersection. As they approach, their carrier waves may overlap and the droplets have a non-zero probability of forming a two-droplet bound state. The likelihood of such pairing is quantified by measuring the probability of finding the droplets in a bound state at late times. Three generic types of two-droplet correlations are observed: promenading, orbiting, and chasing pair of walkers. For certain parameters, the droplets may become correlated for certain initial path differences and remain uncorrelated for others, while in other cases, the droplets may never produce droplet pairs. These observations pave the way for further studies of strongly correlated many-droplet behaviors in the hydrodynamical quantum analogs of bouncing and walking droplets.
Collapse
Affiliation(s)
- Rahil N Valani
- School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
| | - Anja C Slim
- School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia
| | - Tapio Simula
- School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
| |
Collapse
|
9
|
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?
Collapse
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
| |
Collapse
|