The Uncertain Trajectory of a Pilot-Wave

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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.
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