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Happy BDay JP!! The Uncertain Trajectory of a Pilot-Wave André Nachbin ! (IMPA, Brazil) in collaboration with ! ! John Bush (MIT/Math)! Paul Milewski (Bath/Math) ! Carlos Galeano Ríos (IMPA)! ! GEORGE 70: WAVE-PARTICLE PAIR = LAB VIDEOs by Couder’s (Paris 7) & Bush’s (MIT) groups = Fluid dynamics modeling of wave-droplet coupling = numerical simulations validating our reduced model JP 60: = repeat LAB VIDEOs = quickly highlight new modeling: VERTICAL DYNAMICS ! == NEW lab experiment where UNCERTAINTY gives rise to a remarkable long time property of the system ! == present NEW 1D simulations which displays: = a route from a HARMONIC oscillator to a DISORDERED oscillator = a TUNNELING-like experiment APPLICATION: = Connections bet ween CLASSICAL and QUANTUM MECHANICS = new Math modeling and theory arises from the novel dynamics ! Some Physics... John Bush (MIT/Math; Wet Lab), @Discovery Channel; You Tube >> waterdrop shot in 10000 frames a second “coalescence cascade” http://www.openculture.com/2010/12/water_drop_filmed_in_10000_frames_per_second.html ! video by John Bush: a bouncing (silicon oil) droplet (not a video loop!) discovered by Yves Couder, lab experiments (Physics, Paris 7). You Tube: search for Yves Couder video by John Bush et al. lab experiments (MIT/Math). https://www.youtube.com/watch?v=nmC0ygr08tE Faraday instability >>>>> the PILOT-WAVE Faraday 1831 Benjamin & Ursell 1954: Mathieu eqn. standing waves In real time: Y. Couder and E. Fort @ Paris 7 Lab https://www.youtube.com/watch?v=W9yWv5dqSKk The Sixth Brooke Benjamin Lecture on Fluid Dynamics Wednesday 17 October 2012 at 5pm Lecture Theatre L1 Mathematical Institute University of Oxford Yves Couder’s lab, Paris 7 Professor Yves Couder Laboratoire Matière et Systèmes Complexes Université Paris Diderot A fluid dynamical wave-particle duality Wave-particle duality is a quantum behaviour usually assumed to have no possible counterpart in classical physics. We revisited this question when we found that a droplet bouncing on a vibrated bath could become self-propelled by its coupling to the surface waves it excites. A dynamical wave-particle association is thus formed. Through several experiments we addressed the same general question. How can a localized and discrete droplet have a common dynamics with a continuous and spatially extended wave? Surprisingly several quantum-like behaviours emerge;; a form of uncertainty and a form of quantization are observed. I will show that both properties are related to the "path memory" contained in the wave field. The relation of this experiment with the pilot-wave models proposed by de Broglie and by Bohm for quantum mechanics will be discussed. All are warmly invited to attend the lecture and reception that follows. Please email [email protected] to register your attendance. http://www.maths.ox.ac.uk/events/brooke-benjamin-lecture A fluid dynamical wave-particle duality Wave-particle duality is a quantum behaviour usually assumed to have no possible counterpart in classical physics. We revisited this question when we found that a droplet bouncing on a vibrated bath could become self-propelled by its coupling to the surface waves it excites. A dynamical wave-particle association is thus formed. Through several experiments we addressed the same general question. How can a localized and discrete droplet have a common dynamics with a continuous and spatially extended wave? Surprisingly several quantum-like behaviours emerge;; a form of uncertainty and a form of quantization are observed. I will show that both properties are related to the "path memory" contained in the wave field. The relation of this experiment with the pilot-wave models proposed by de Broglie and by Bohm for quantum mechanics will be discussed. All are warmly invited to attend the lecture and reception that follows. Please email [email protected] to register your attendance. http://www.maths.ox.ac.uk/events/brooke-benjamin-lecture Couder et al. did the lab experiments for a SINGLE and DOUBLE SLIT. (Couder & Fort, PRL 2006) angle: great deal of uncertainty. (a) top view (b) side view (c) top view Couder et al. did the lab experiments for a SINGLE and DOUBLE SLIT. (Couder & Fort, PRL 2009) (video by Couder et al.) ! ODEs for droplet Droplet’s HORIZONTAL dynamics: the modelling Protière, Boudaoud & Couder, JFM ’06 Eddi, Sultan, Moukhtar, Fort, Rossi & Couder, JFM ‘11 JFM ‘13 Memory Bessel h = wave elevation Droplet’s dynamical system: Xp=(x,y) In simplest models F(t) comes from force balance, projected on (x,y). “trampolin” model: LINEAR SPRING (Bush & Gilet, PRL 09 & JFM 09) ↵ Z F = kZ Molacek & Bush (JFM 2013, A&B) DROPLET equations: ! VERTICAL dynamics in Z: g ⇤ = g(t) = g(1 Horizontal (X,Y) trajectory equation: sin(!0 t)) Bifurcation Diagram JFM13 S. Protière, A. Boudaoud and Y. Couder, JFM06! D B PDB with vertical dynamics: EXOTIC modes emerge. (m,n) =cycles of (container,droplet) W /g Faraday threshold Molacek & Bush (JFM 2013, A&B) DROPLET equations: ! VERTICAL dynamics in Z: Horizontal (X,Y) trajectory equation: OUR WORK: -- generate water waves/PDE model. DROPLET WAVE Reduced Model: Fluid g ⇤ = g(t) = g(1 sin(!0 t)) D DtN * Vertical “non-linear SPRING” W Horizontal Pressure: compact support in space and time G(t) G(t) depends on g(t) pilot-wave! guiding ! the particle Bifurcation Diagram (2,1) (m,n): m driving periods with n contacts.! MODEL responds to these modes. Fix Omega = 0.7, simulate Gamma = 1.6, 2.1, 2.5, 3.15, 3.8 A pair of walking droplets lock into orbit Video courtesy of Yves Couder our model ! The UNCERTAIN DROPLET TRAJECTORY Confined domain: corral video courtesy by J.Bush Droplet walking in a circular corral; courtesy of J. Bush “...the statistical behavior of the system can be described by a wave function that satisfies a linear wave equation.” “...the statistical behavior of the system can be described by a wave function that satisfies a linear wave equation.” side view of vessel CONFORMAL MAPPING of the 1D CORRAL vertical oscillation ⌦ = 50 periodic bound. cond. ⇣ CORRAL in PHYSICAL domain Dynamics done HERE 80Hz; DETAIL of CORRAL in CANONICAL domain ⇠ DtN operator computed HERE + oversampling v=-0.05 OSCILLATORs TIME = 4.2 2 = 4.8 + memory = 4.85 v=-0.05 = 4.82 1 “HARMONIC” OSCILLATOR + CONFINED SPIRAL “locked” standing waves ROUTE to UNCERTAINTY = 5.0 L = 1.40 L =1.60 v=-0.05 = 4.8 T= 3600*TF L = 2.0 large TIME BEHAVIOR ⇡1 F = 4.8 T= 3600*TF L = 2.0 = 4.8 T= 3600*TF L = 2.0 = 4.8 T= 3600*TF L = 2.0 a BARRIER Submerged frame video by Couder et al. lab experiments. = 5.5 TUNNELING { Wave-particle model: 1D version 1Itc = 15 TF wave as a “moving POTENTIAL” ~ dX Vd , t) 2 (t)) 3 (t)r⌘(xd , ydd⌘ ~d dV 2 d X( 1 + = dt m + c G(t) “HARMONIC” OSCILLATOR dt2 4 dt = 1Itc = 15 TF G(t) “DISORDERED” OSCILLATOR dx (X, t) QUESTIONS: to be investigated… Oscillatory states: frequency and amplitude dependence on cavity and Faraday wavelength? ! Quantized/discrete states of “harmonic-like” oscillators cavity + Faraday wavelength potential Will complete wave-function picture be captured? Tunneling: many things to play with * * barrier size… * phase dynamics within vertical dynamics….. * etc…. Thank you for your attention. http://www.impa.br/~nachbin IMPA, Rio de Janeiro, Brazil.