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Bouncing droplets on a vibrating fluid bath Øistein Wind-Willassen (DTU COMPUTE) Mads Peter Sørensen (DTU COMPUTE) John Bush (MIT) Mathematical Colloquium DTU May 8 2013 Thursday, June 13, 13 What is this? Thursday, June 13, 13 Why is this interesting? • • Thursday, June 13, 13 Mathematics/physics viewpoint: • • Interesting dynamical system Complicated fluid physics Philosophical viewpoint: • • Particle-wave duality at macroscopic level Has features reminiscent of quantum mechanics This talk I will focus on the “mode” with which the drop bounces, and cover the following: • • • Thursday, June 13, 13 The general physics of the system • • Experimental setup Theoretical model The obtained results Some perspectives Faraday waves Thursday, June 13, 13 Faraday waves Consider a fluid tray oscillating vertically Thursday, June 13, 13 Faraday waves Consider a fluid tray oscillating vertically Amplitude = A Frequency = f = !/(2⇡) Forcing = Thursday, June 13, 13 = A! 2 Faraday waves Consider a fluid tray oscillating vertically Amplitude = A Frequency = f = !/(2⇡) Forcing = ⇤ = A! 2 g (t) = g + sin(2⇡f t) Thursday, June 13, 13 Faraday waves Consider a fluid tray oscillating vertically Amplitude = A Frequency = f = !/(2⇡) Forcing = ⇤ = A! 2 g (t) = g + sin(2⇡f t) Frequency is constant. Thursday, June 13, 13 Faraday waves Consider a fluid tray oscillating vertically Amplitude = A Frequency = f = !/(2⇡) Forcing = ⇤ = A! 2 g (t) = g + sin(2⇡f t) Frequency is constant. At = F the layer becomes unstable. Thursday, June 13, 13 Faraday waves Consider a fluid tray oscillating vertically Amplitude = A Frequency = f = !/(2⇡) Forcing = ⇤ = A! 2 g (t) = g + sin(2⇡f t) Frequency is constant. At = F the layer becomes unstable. Standing Faraday waves form, they follow standard dispersion relation. Thursday, June 13, 13 Faraday waves Consider a fluid tray oscillating vertically Amplitude = A Frequency = f = !/(2⇡) Forcing = = A! 2 ⇤ g (t) = g + sin(2⇡f t) Frequency is constant. At = F the layer becomes unstable. Standing Faraday waves form, they follow standard dispersion relation. In the experiment: Thursday, June 13, 13 < F , i.e. the interface is flat without a droplet! Faraday waves Consider a fluid tray oscillating vertically Amplitude = A Frequency = f = !/(2⇡) Forcing = = A! 2 ⇤ g (t) = g + sin(2⇡f t) Frequency is constant. At = F the layer becomes unstable. Standing Faraday waves form, they follow standard dispersion relation. In the experiment: Thursday, June 13, 13 < F , i.e. the interface is flat without a droplet! The drop locally excites Faraday waves. Qualitative behaviour • • • Coalescence for < B⇡g Bouncing in place for < W B < Walking horizontally for < F W < Thursday, June 13, 13 Remember: is the forcing Qualitative behaviour • • • Coalescence for < B⇡g Bouncing in place for < W B < Walking horizontally for < F W < Remember: is the forcing Particle–wave association on a fluid interface 89 1.4 1.2 Int Drop size 1.0 B = Bouncing PDB = Period doubling PDC = Period doubling cascade W = Walking Int = Intermittent F = Faraday waves D (mm) W B PDB 0.8 0.6 PDC 50cS - 50 Hz[Couder, 2006] Thursday, June 13, 13 F 0.4 Forcing 0 1 2 3 4 5 Qualitative behaviour • • • Coalescence for < B⇡g Bouncing in place for < W B < Walking horizontally for < F W < Remember: is the forcing Particle–wave association on a fluid interface 89 1.4 1.2 Int Drop size 1.0 B = Bouncing PDB = Period doubling PDC = Period doubling cascade W = Walking Int = Intermittent F = Faraday waves D (mm) W B PDB 0.8 0.6 PDC 50cS - 50 Hz[Couder, 2006] Thursday, June 13, 13 F 0.4 Forcing 0 1 2 3 4 5 Qualitative behaviour • • • Coalescence for < B⇡g Bouncing in place for < W B < Walking horizontally for < F W < Remember: is the forcing Particle–wave association on a fluid interface 89 1.4 1.2 Int Drop size 1.0 B = Bouncing PDB = Period doubling PDC = Period doubling cascade W = Walking Int = Intermittent F = Faraday waves D (mm) W B PDB 0.8 0.6 PDC 50cS - 50 Hz[Couder, 2006] Thursday, June 13, 13 F 0.4 Forcing 0 1 2 3 4 5 Qualitative behaviour • • • Coalescence for < B⇡g Bouncing in place for < W B < Walking horizontally for < F W < Remember: is the forcing Particle–wave association on a fluid interface 89 1.4 1.2 Int Drop size 1.0 B = Bouncing PDB = Period doubling PDC = Period doubling cascade W = Walking Int = Intermittent F = Faraday waves D (mm) W B PDB 0.8 0.6 PDC 50cS - 50 Hz[Couder, 2006] Thursday, June 13, 13 F 0.4 Forcing 0 1 2 3 4 5 Experimental setup Thursday, June 13, 13 Experimental setup • Thursday, June 13, 13 Shaker Experimental setup • • Thursday, June 13, 13 Shaker Fluid tray Experimental setup • • • Thursday, June 13, 13 Shaker Fluid tray High speed camera Experimental setup • • • • Thursday, June 13, 13 Shaker Fluid tray High speed camera CCD camera Experimental setup • • • • • Thursday, June 13, 13 Shaker Fluid tray High speed camera CCD camera Lamps with diffusers Experimental setup • • • • • • Thursday, June 13, 13 Shaker Fluid tray High speed camera CCD camera Lamps with diffusers Computer Examples - top view Thursday, June 13, 13 Examples - top view Thursday, June 13, 13 Examples - side view Thursday, June 13, 13 Examples - side view Thursday, June 13, 13 Experimental notation Thursday, June 13, 13 Experimental notation • p Proxy for drop size: the “vibration” number ⌦ = 2⇡f ⇢r03 / - Ratio of forcing frequency to natural oscillation of drop. Thursday, June 13, 13 Experimental notation • p Proxy for drop size: the “vibration” number ⌦ = 2⇡f ⇢r03 / - Ratio of forcing frequency to natural oscillation of drop. Bouncing “mode” is denoted (m, n)i Thursday, June 13, 13 Experimental notation • p Proxy for drop size: the “vibration” number ⌦ = 2⇡f ⇢r03 / - Ratio of forcing frequency to natural oscillation of drop. Bouncing “mode” is denoted (m, n)i • m: number of driving periods Thursday, June 13, 13 Experimental notation • p Proxy for drop size: the “vibration” number ⌦ = 2⇡f ⇢r03 / - Ratio of forcing frequency to natural oscillation of drop. Bouncing “mode” is denoted (m, n)i • • m: number of driving periods n: number of distinct impacts Thursday, June 13, 13 Experimental notation • p Proxy for drop size: the “vibration” number ⌦ = 2⇡f ⇢r03 / - Ratio of forcing frequency to natural oscillation of drop. Bouncing “mode” is denoted (m, n)i • • • m: number of driving periods n: number of distinct impacts i: mechanical energy (phase distinction) Thursday, June 13, 13 Experimental notation • p Proxy for drop size: the “vibration” number ⌦ = 2⇡f ⇢r03 / - Ratio of forcing frequency to natural oscillation of drop. Bouncing “mode” is denoted (m, n)i • • • m: number of driving periods n: number of distinct impacts i: mechanical energy (phase distinction) (2, 1) Thursday, June 13, 13 1 (2, 1) 2 7 Theoretical description Drops bouncing on a vibrating bath a) Z=0 Z=−1 b) ρ, ν v Z g σ Z=−1 v [Molacek & Bush, 2013] Figure 9. A schematic illustration of our choice of coordinates. The vertical position drop’s centre of mass Z is equal to 0 at the initiation of impact (a), and −1 at the equil level of the bath (b). In free flight: During contact: mg ⇤ (t) = mz̈ ⇤ mg (t) = 1+ c3 ln2 c1 r0 Z ! Fig. 8 shows the first two period-doubling thresholds. Smaller drops (Ω < 0.6) un a period-doubling cascade, so the first two thresholds correspond to (1, 1) → (2, 2 (2, 2) → (4, 4) transitions. Larger drops (Ω > 0.6) transition from (1, 1) to (2, 2) reduce the amplitude of their smaller bounce until a simple period-doubled bou mode (2, 1) is reached, and only then commence the period-doubling cascade (2 (4, 2) → (8, 4) → . . . . Note that the low frequency curves are shifted to the right o high frequency counterparts (60 Hz curve for 20 cS; 50 − 60 Hz for 50 cS), an effe to the influence of the standing waves created on the bath by previous drop impac lower frequencies, the Faraday threshold is closer to the period-doubling threshold the drop impacts create more slowly decaying standing waves on the bath surfa reducing the relative speed the drop and bath1at 0 impact, the standing 1 between 0 appear to stabilize the vertical motion, and so delay the period-doubling transitio The bounds of the frequency range explored were prescribed by experimenta straints. The presence of the Faraday threshold provides a lower limit on the ra frequencies over which the period-doubled modes can arise. For example, for 20 c cone oil, period-doubling occurs only for Γ > 1.58 (Fig. 8), while ΓF < 1.58 for f Hz. Thus, for f ≤ 45 Hz, the period-doubling transitions disappear. The upper lim the frequency range is imposed by the finite resolution of our camera. Since the w ! region of ultimate interest is given by Ω = 2πf ρR03 /σ ! 1.5 (see MBII ), the t size of a walker R0 ∼ f −2/3 . Thus, for higher frequencies, the constant error in size measurement leads to increasing relative error in Ω. Similarly, at high freque becomes increasingly difficult to distinguish between the different bouncing mod the motion itself happens over a distance of at most g(T /2)2 /2 ≤ gf −2 /2, which order 0.1mm for f = 200 Hz. 4⇡⌘r0 c2 (⌫) 2⇡ mz̈ + Ż + c r Z c r 3 ln Z Z 3. Vertical dynamics 3.1. Linear spring model We proceed by describing the simplest model of the drop’s vertical dynamics, analog works by Okumura et al. (2003), Gilet & Bush (2009b) and Terwagne (2011), in whi drop-impactor interaction is described in terms of a linear spring. We nondimensio the vertical displacement of the drop by its radius (see Table 2 for a list of dimensi ! variables) and time by the characteristic frequency of drop oscillations ωD = σ (Rayleigh 1879). We shall always consider the frame of reference fixed relative shaking platform, and place the origin so that the undisturbed bath surface is at Z Thursday, June 13, 13 Theoretical description Drops bouncing on a vibrating bath a) Z=0 Z=−1 b) ρ, ν v Z g σ Z=−1 v [Molacek & Bush, 2013] Figure 9. A schematic illustration of our choice of coordinates. The vertical position drop’s centre of mass Z is equal to 0 at the initiation of impact (a), and −1 at the equil level of the bath (b). In free flight: During contact: mg ⇤ (t) = mz̈ ⇤ mg (t) = kinetic energy (fluid motion in the two liquids) - this term is the added mass 1+ c3 ln2 c1 r0 Z ! Fig. 8 shows the first two period-doubling thresholds. Smaller drops (Ω < 0.6) un a period-doubling cascade, so the first two thresholds correspond to (1, 1) → (2, 2 (2, 2) → (4, 4) transitions. Larger drops (Ω > 0.6) transition from (1, 1) to (2, 2) reduce the amplitude of their smaller bounce until a simple period-doubled bou mode (2, 1) is reached, and only then commence the period-doubling cascade (2 (4, 2) → (8, 4) → . . . . Note that the low frequency curves are shifted to the right o high frequency counterparts (60 Hz curve for 20 cS; 50 − 60 Hz for 50 cS), an effe to the influence of the standing waves created on the bath by previous drop impac lower frequencies, the Faraday threshold is closer to the period-doubling threshold the drop impacts create more slowly decaying standing waves on the bath surfa reducing the relative speed the drop and bath1at 0 impact, the standing 1 between 0 appear to stabilize the vertical motion, and so delay the period-doubling transitio The bounds of the frequency range explored were prescribed by experimenta straints. The presence of the Faraday threshold provides a lower limit on the ra frequencies over which the period-doubled modes can arise. For example, for 20 c cone oil, period-doubling occurs only for Γ > 1.58 (Fig. 8), while ΓF < 1.58 for f Hz. Thus, for f ≤ 45 Hz, the period-doubling transitions disappear. The upper lim the frequency range is imposed by the finite resolution of our camera. Since the w ! region of ultimate interest is given by Ω = 2πf ρR03 /σ ! 1.5 (see MBII ), the t size of a walker R0 ∼ f −2/3 . Thus, for higher frequencies, the constant error in size measurement leads to increasing relative error in Ω. Similarly, at high freque becomes increasingly difficult to distinguish between the different bouncing mod the motion itself happens over a distance of at most g(T /2)2 /2 ≤ gf −2 /2, which order 0.1mm for f = 200 Hz. 4⇡⌘r0 c2 (⌫) 2⇡ mz̈ + Ż + c r Z c r 3 ln Z Z 3. Vertical dynamics 3.1. Linear spring model We proceed by describing the simplest model of the drop’s vertical dynamics, analog works by Okumura et al. (2003), Gilet & Bush (2009b) and Terwagne (2011), in whi drop-impactor interaction is described in terms of a linear spring. We nondimensio the vertical displacement of the drop by its radius (see Table 2 for a list of dimensi ! variables) and time by the characteristic frequency of drop oscillations ωD = σ (Rayleigh 1879). We shall always consider the frame of reference fixed relative shaking platform, and place the origin so that the undisturbed bath surface is at Z Thursday, June 13, 13 Theoretical description Drops bouncing on a vibrating bath a) Z=0 Z=−1 b) ρ, ν v Z g σ Z=−1 v [Molacek & Bush, 2013] Figure 9. A schematic illustration of our choice of coordinates. The vertical position drop’s centre of mass Z is equal to 0 at the initiation of impact (a), and −1 at the equil level of the bath (b). In free flight: During contact: mg ⇤ (t) = mz̈ ⇤ mg (t) = kinetic energy (fluid motion in the two liquids) - this term is the added mass 1+ c3 ln2 c1 r0 Z ! Fig. 8 shows the first two period-doubling thresholds. Smaller drops (Ω < 0.6) un a period-doubling cascade, so the first two thresholds correspond to (1, 1) → (2, 2 (2, 2) → (4, 4) transitions. Larger drops (Ω > 0.6) transition from (1, 1) to (2, 2) reduce the amplitude of their smaller bounce until a simple period-doubled bou mode (2, 1) is reached, and only then commence the period-doubling cascade (2 (4, 2) → (8, 4) → . . . . Note that the low frequency curves are shifted to the right o high frequency counterparts (60 Hz curve for 20 cS; 50 − 60 Hz for 50 cS), an effe to the influence of the standing waves created on the bath by previous drop impac lower frequencies, the Faraday threshold is closer to the period-doubling threshold the drop impacts create more slowly decaying standing waves on the bath surfa reducing the relative speed the drop and bath1at 0 impact, the standing 1 between 0 appear to stabilize the vertical motion, and so delay the period-doubling transitio The bounds of the frequency range explored were prescribed by experimenta straints. The presence of the Faraday threshold provides a lower limit on the ra frequencies over which the period-doubled modes can arise. For example, for 20 c cone oil, period-doubling occurs only for Γ > 1.58 (Fig. 8), while ΓF < 1.58 for f Hz. Thus, for f ≤ 45 Hz, the period-doubling transitions disappear. The upper lim the frequency range is imposed by the finite resolution of our camera. Since the w ! region of ultimate interest is given by Ω = 2πf ρR03 /σ ! 1.5 (see MBII ), the t size of a walker R0 ∼ f −2/3 . Thus, for higher frequencies, the constant error in size measurement leads to increasing relative error in Ω. Similarly, at high freque becomes increasingly difficult to distinguish between the different bouncing mod the motion itself happens over a distance of at most g(T /2)2 /2 ≤ gf −2 /2, which order 0.1mm for f = 200 Hz. 4⇡⌘r0 c2 (⌫) 2⇡ mz̈ + Ż + c r Z c r 3 ln Z Z Viscous dissipation 3. Vertical dynamics 3.1. Linear spring model We proceed by describing the simplest model of the drop’s vertical dynamics, analog works by Okumura et al. (2003), Gilet & Bush (2009b) and Terwagne (2011), in whi drop-impactor interaction is described in terms of a linear spring. We nondimensio the vertical displacement of the drop by its radius (see Table 2 for a list of dimensi ! variables) and time by the characteristic frequency of drop oscillations ωD = σ (Rayleigh 1879). We shall always consider the frame of reference fixed relative shaking platform, and place the origin so that the undisturbed bath surface is at Z Thursday, June 13, 13 Theoretical description Drops bouncing on a vibrating bath a) Z=0 Z=−1 b) ρ, ν v Z g σ Z=−1 v [Molacek & Bush, 2013] Figure 9. A schematic illustration of our choice of coordinates. The vertical position drop’s centre of mass Z is equal to 0 at the initiation of impact (a), and −1 at the equil level of the bath (b). In free flight: During contact: mg ⇤ (t) = mz̈ ⇤ mg (t) = kinetic energy (fluid motion in the two liquids) - this term is the added mass 1+ c3 ln2 c1 r0 Z ! Fig. 8 shows the first two period-doubling thresholds. Smaller drops (Ω < 0.6) un a period-doubling cascade, so the first two thresholds correspond to (1, 1) → (2, 2 (2, 2) → (4, 4) transitions. Larger drops (Ω > 0.6) transition from (1, 1) to (2, 2) reduce the amplitude of their smaller bounce until a simple period-doubled bou mode (2, 1) is reached, and only then commence the period-doubling cascade (2 (4, 2) → (8, 4) → . . . . Note that the low frequency curves are shifted to the right o high frequency counterparts (60 Hz curve for 20 cS; 50 − 60 Hz for 50 cS), an effe to the influence of the standing waves created on the bath by previous drop impac lower frequencies, the Faraday threshold is closer to the period-doubling threshold the drop impacts create more slowly decaying standing waves on the bath surfa reducing the relative speed the drop and bath1at 0 impact, the standing 1 between 0 appear to stabilize the vertical motion, and so delay the period-doubling transitio The bounds of the frequency range explored were prescribed by experimenta straints. The presence of the Faraday threshold provides a lower limit on the ra frequencies over which the period-doubled modes can arise. For example, for 20 c cone oil, period-doubling occurs only for Γ > 1.58 (Fig. 8), while ΓF < 1.58 for f Hz. Thus, for f ≤ 45 Hz, the period-doubling transitions disappear. The upper lim the frequency range is imposed by the finite resolution of our camera. Since the w ! region of ultimate interest is given by Ω = 2πf ρR03 /σ ! 1.5 (see MBII ), the t size of a walker R0 ∼ f −2/3 . Thus, for higher frequencies, the constant error in size measurement leads to increasing relative error in Ω. Similarly, at high freque becomes increasingly difficult to distinguish between the different bouncing mod the motion itself happens over a distance of at most g(T /2)2 /2 ≤ gf −2 /2, which order 0.1mm for f = 200 Hz. 4⇡⌘r0 c2 (⌫) 2⇡ mz̈ + Ż + c r Z c r 3 ln Z Z Viscous dissipation Nonlinearity of spring 3. Vertical dynamics 3.1. Linear spring model We proceed by describing the simplest model of the drop’s vertical dynamics, analog works by Okumura et al. (2003), Gilet & Bush (2009b) and Terwagne (2011), in whi drop-impactor interaction is described in terms of a linear spring. We nondimensio the vertical displacement of the drop by its radius (see Table 2 for a list of dimensi ! variables) and time by the characteristic frequency of drop oscillations ωD = σ (Rayleigh 1879). We shall always consider the frame of reference fixed relative shaking platform, and place the origin so that the undisturbed bath surface is at Z Thursday, June 13, 13 ⌦ = 2⇡f 1 0.9 Results 20 cS 80 Hz p ⇢r03 / (1,1) Walking 0.8 (2,1)1 (2,1)2 (4,2) 0.7 (2,2) 1 2 (2,1) & (2,1) (4,3) 0.6 0.4 C 2 (2,1) C 2 2.5 (2,1)1 (4,2) 3 2 = A! /g Thursday, June 13, 13 C (2,1)2 & chaos (4,4) 0.5 0.3 1 (2,1) 3.5 4 The mixed state Thursday, June 13, 13 The mixed state Thursday, June 13, 13 Thursday, June 13, 13 The mixed state The mixed state 11 Thursday, June 13, 13 Perspectives [Couder,2008] [Couder,2010] [Couder,2009] Thursday, June 13, 13 Perspectives [Couder,2008] Analogies with quantum mechanics [Couder,2010] [Couder,2009] Thursday, June 13, 13 Perspectives [Couder,2008] Analogies with quantum mechanics • Single/double slit FIG. 1. (a), (b) Sketch of the central region of the experimental cell (seen from above and in a cross section along the y axis, respectively). An individual trajectory is shown in (a) and the definitions of !i , yi , and " are given. The width of the slit being L, the parameter of impact is Yi' ! yi =L. (c) A photograph of the experiment lit with diffuse light showing the wave pattern as the walker crosses the aperture. The picture was taken at a time when the trajectory, initially perpendicular to the aperture, was deflected by the interference with reflected waves. ments, having the same walker cross the slit, with various initial motions. As shown on Fig. 1(a) each initial trajectory is defined by its incidence angle !i and by its normalized impact parameter in the slit Yi ! yi =L ("0:5 < Yi < 0:5) [Fig. 1(a)]. The walkers were always started far from the screen and only those impinging perpendicularly on the slit (!I ! 90# ) were retained. The plot "$Yi % [Fig. 2(a)] shows that there is no simple relation between the deviation and the parameter of impact. Thursday, June is 13, also 13 This observed on individual realizations: with ap- FIG. 2 (color online). (a) The measured deviations a of successive individual particles as a function of[Couder,2010] the parameter of impact Yi (with L=#F ! 2:86). (b) A superposition of three different trajectories of the same droplet passing 3 times through the slit with similar initial conditions !i ! 90# and Yi ! 0:1. (c) Experimental histogram of the deviation " as obtained with NT ! 125 single walkers with L=#F ! 2:11 (L ! 14:7 mm and #F ! 6:95 mm). Since each trajectory has a symmetrical counterpart with respect to the axis of the aperture, the statistic was improved by taking them into account so that the distributions correspond to 2NT realizations. The curve is the fit by Eq. (1) with L=#F ! 1:96. (d) Histogram obtained with L=#F ! 3:1 (L ! 14:7 mm and #F ! 4:75 mm). The curve is the fit by Eq. (1) with L=#F ! 2:86. [Couder,2009] #F =L and large amplitude lateral lobes are clearly observed. Hence, on Figs. 2(c) and 2(d) the general shapes of the histograms are compared with the amplitude diffraction pattern of a plane wave passing through a slit: ! ! ! ! ! ! sin$$L sin"=# % ! ! F ! ! f$"% ! A! !: (1) ! ! $L sin"=#F ! ! This amplitude of diffraction of a plane wave turns out to provide an approximate fit for these histograms. Hence, the ulus of the walkers velocity V W ðΩÞ is observed as the orbits become smaller. [Couder,2008] The possible orbits being discrete, we label the plateaus by an order n, with n ¼ 0 corresponding to the tightest orbit observed at large Ω. Each plateau crosses the continuous short path-memory curve given by Eq. 3 at a value Ω0n of the angular velocity (Fig. 2B). At these crossings, the observed orbits have the same radius exp 0 0 R0n ¼ Rexp n ðΩn Þ ¼ RC ðΩn Þ they would have had with short-term memory. A plateau extends on both sides of the curve given by relation 3 between two limiting values Ω−n < Ωn < Ωþ n . A hysteresis is observed: The abrupt transition from one plateau to the Perspectives Analogies with quantum mechanics • Single/double slit • Charged particle in magnetic field A FIG. 1. (a), (b) Sketch of the central region of the experimental cell (seen from above and in a cross section along the y axis, respectively). An individual trajectory is shown in (a) and the definitions of !i , yi , and " are given. The width of the slit being L, the parameter of impact is Yi' ! yi =L. (c) A photograph of the experiment lit with diffuse light showing the wave pattern as the walker crosses the aperture. The picture was taken at a time when the trajectory, initially perpendicular to the aperture, was deflected by the interference with reflected waves. ments, having the same walker cross the slit, with various initial motions. As shown on Fig. 1(a) each initial trajectory is defined by its incidence angle !i and by its normalized impact parameter in the slit Yi ! yi =L ("0:5 < Yi < 0:5) [Fig. 1(a)]. The walkers were always started far from the screen and only those impinging perpendicularly on the slit (!I ! 90# ) were retained. The plot "$Yi % [Fig. 2(a)] shows that there is no simple relation between the deviation and the parameter of impact. Thursday, June is 13, also 13 This observed on individual realizations: with ap- FIG. 2 (color online). (a) The measured deviations a of sucB cessive individual particles as a function of[Couder,2010] the parameter of impact Yi (with L=#F ! 2:86). (b) A superposition of three different trajectories of the same droplet passing 3 times through the slit with similar initial conditions !i ! 90# and Yi ! 0:1. (c) Experimental histogram of the deviation " as obtained with NT ! 125 single walkers with L=#F ! 2:11 (L ! 14:7 mm and #F ! 6:95 mm). Since each trajectory has a symmetrical counterpart with respect to the axis of the aperture, the statistic was improved by taking them into account so that the distributions correspond to 2NT realizations. The curve is the fit by Eq. (1) with L=#F ! 1:96. (d) Histogram obtained with L=#F ! 3:1 (L ! 14:7 mm and #F ! 4:75 mm). The curve is the fit by Eq. (1) with L=#F ! 2:86. C [Couder,2009] #F =L and large amplitude lateral lobes are clearly observed. Hence, on Figs. 2(c) and 2(d) the general shapes of the histograms are compared with the amplitude diffraction pattern of a plane wave passing through a slit: ! ! ! ! ! ! sin$$L sin"=# % ! ! F ! ! f$"% ! A! !: (1) ! ! $L sin"=#F ! ! This amplitude of diffraction of a plane wave turns out to provide an approximate fit for these histograms. Hence, the ulus of the walkers velocity V W ðΩÞ is observed as the orbits become smaller. [Couder,2008] The possible orbits being discrete, we label the plateaus by an order n, with n ¼ 0 corresponding to the tightest orbit observed at large Ω. Each plateau crosses the continuous short path-memory curve given by Eq. 3 at a value Ω0n of the angular velocity (Fig. 2B). At these crossings, the observed orbits have the same radius exp 0 0 R0n ¼ Rexp n ðΩn Þ ¼ RC ðΩn Þ they would have had with short-term memory. A plateau extends on both sides of the curve given by relation 3 between two limiting values Ω−n < Ωn < Ωþ n . A hysteresis is observed: The abrupt transition from one plateau to the Perspectives Analogies with quantum mechanics • Single/double slit • Charged particle in magnetic field FIG. 2 (color online). • Harmonic oscillator A FIG. 1. (a), (b) Sketch of the central region of the experimental cell (seen from above and in a cross section along the y axis, respectively). An individual trajectory is shown in (a) and the definitions of !i , yi , and " are given. The width of the slit being L, the parameter of impact is Yi' ! yi =L. (c) A photograph of the experiment lit with diffuse light showing the wave pattern as the walker crosses the aperture. The picture was taken at a time when the trajectory, initially perpendicular to the aperture, was deflected by the interference with reflected waves. ments, having the same walker cross the slit, with various initial motions. As shown on Fig. 1(a) each initial trajectory is defined by its incidence angle !i and by its normalized impact parameter in the slit Yi ! yi =L ("0:5 < Yi < 0:5) [Fig. 1(a)]. The walkers were always started far from the screen and only those impinging perpendicularly on the slit (!I ! 90# ) were retained. The plot "$Yi % [Fig. 2(a)] shows that there is no simple relation between the deviation and the parameter of impact. Thursday, June is 13, also 13 This observed on individual realizations: with ap- (a) The measured deviations a of sucB cessive individual particles as a function of[Couder,2010] the parameter of impact Yi (with L=#F ! 2:86). (b) A superposition of three different trajectories of the same droplet passing 3 times through the slit with similar initial conditions !i ! 90# and Yi ! 0:1. (c) Experimental histogram of the deviation " as obtained with NT ! 125 single walkers with L=#F ! 2:11 (L ! 14:7 mm and #F ! 6:95 mm). Since each trajectory has a symmetrical counterpart with respect to the axis of the aperture, the statistic was improved by taking them into account so that the distributions correspond to 2NT realizations. The curve is the fit by Eq. (1) with L=#F ! 1:96. (d) Histogram obtained with L=#F ! 3:1 (L ! 14:7 mm and #F ! 4:75 mm). The curve is the fit by Eq. (1) with L=#F ! 2:86. C [Couder,2009] #F =L and large amplitude lateral lobes are clearly observed. Hence, on Figs. 2(c) and 2(d) the general shapes of the histograms are compared with the amplitude diffraction pattern of a plane wave passing through a slit: ! ! ! ! ! ! sin$$L sin"=# % ! ! F ! ! f$"% ! A! !: (1) ! ! $L sin"=#F ! ! This amplitude of diffraction of a plane wave turns out to provide an approximate fit for these histograms. Hence, the ulus of the walkers velocity V W ðΩÞ is observed as the orbits become smaller. [Couder,2008] The possible orbits being discrete, we label the plateaus by an order n, with n ¼ 0 corresponding to the tightest orbit observed at large Ω. Each plateau crosses the continuous short path-memory curve given by Eq. 3 at a value Ω0n of the angular velocity (Fig. 2B). At these crossings, the observed orbits have the same radius exp 0 0 R0n ¼ Rexp n ðΩn Þ ¼ RC ðΩn Þ they would have had with short-term memory. A plateau extends on both sides of the curve given by relation 3 between two limiting values Ω−n < Ωn < Ωþ n . A hysteresis is observed: The abrupt transition from one plateau to the Perspectives Analogies with quantum mechanics • Single/double slit • Charged particle in magnetic field FIG. 2 (color online). (a) The measured deviations a of suc• Harmonic oscillator A B cessive individual particles as a function of[Couder,2010] the parameter of impact Y (with L=# ! 2:86). (b) A superposition of three • Lattice structures i F FIG. 1. (a), (b) Sketch of the central region of the experimental cell (seen from above and in a cross section along the y axis, respectively). An individual trajectory is shown in (a) and the definitions of !i , yi , and " are given. The width of the slit being L, the parameter of impact is Yi' ! yi =L. (c) A photograph of the experiment lit with diffuse light showing the wave pattern as the walker crosses the aperture. The picture was taken at a time when the trajectory, initially perpendicular to the aperture, was deflected by the interference with reflected waves. different trajectories of the same droplet passing 3 times through the slit with similar initial conditions !i ! 90# and Yi ! 0:1. (c) Experimental histogram of the deviation " as obtained with NT ! 125 single walkers with L=#F ! 2:11 (L ! 14:7 mm and #F ! 6:95 mm). Since each trajectory has a symmetrical counterpart with respect to the axis of the aperture, the statistic was improved by taking them into account so that the distributions correspond to 2NT realizations. The curve is the fit by Eq. (1) with L=#F ! 1:96. (d) Histogram obtained with L=#F ! 3:1 (L ! 14:7 mm and A. Eddi#et Fal.! 4:75 mm). The curve is the fit by Eq. (1) with L=#F ! 2:86. C [Couder,2009] ments, having the same walker cross the slit, with various initial motions. As shown on Fig. 1(a) each initial trajectory is defined by its incidence angle !i and by its normalized impact parameter in the slit Yi ! yi =L ("0:5 < Yi < 0:5) [Fig. 1(a)]. The walkers were always started far from the screen and only those impinging perpendicularly on the slit (!I ! 90# ) were retained. The plot "$Yi % [Fig. 2(a)] shows that there is no simple relation between the deviation and the parameter of impact. Thursday, June is 13, also 13 This observed on individual realizations: with ap- #F =L and large amplitude lateral lobes are clearly observed. Hence, on Figs. 2(c) and 2(d) the general shapes of the histograms are compared with the amplitude diffraction pattern of a plane wave passing through a slit: ! ! ! ! ! ! sin$$L sin"=# % ! ! F ! ! f$"% ! A! !: (1) ! ! $L sin"=#F ! ! Fig. 1: The crystalline organization of a large number of droplets = 61) with a triangular This amplitude of(Ndiffraction of lattice. a plane wave turns out to provide an approximate fit for these histograms. Hence, the ulus of the walkers velocity V W ðΩÞ is observed as the orbits comes shorter as the wall thickness is decreased [Fig. 4(c)]. become smaller. [Couder,2008] possible orbits being discrete, we label the plateaus by an Ultimately the limit cycle is never reachedThe [Fig. 4(d)] [10– order n, with n ¼ 0 corresponding to the tightest orbit observed at 12]. The trajectories inside the cavity and the escapes are large Ω. Each plateau crosses the continuous short path-memory correlated. As long as the droplet follows limit curvethe given by Eq.cycle, 3 at a value Ω0n of the angular velocity (Fig. 2B). At thesebecome crossings, disthe observed orbits have the same radius no escape is observed. When the trajectories exp exp 0 0 Rn ¼ Rn ðΩn Þ ¼ RC ðΩ0n Þ they would have had with short-term ordered, the collisions with the walls show a larger variety memory. A plateau extends on both sides of the curve given by relation 3 between two limiting values Ω−n < Ωn < Ωþ n . A hysteresis is observed: The abrupt transition from one plateau to the Perspectives Analogies with quantum mechanics • Single/double slit • Charged particle in magnetic field FIG. 2 (color online). (a) The measured deviations a of suc• Harmonic oscillator A B cessive individual particles as a function of[Couder,2010] the parameter of impact Y (with L=# ! 2:86). (b) A superposition of three • Lattice structures different trajectories of the same droplet passing 3 times through the slit with similar initial conditions ! ! 90 and Y ! 0:1. effects • FIG.Tunneling 1. (a), (b) Sketch of the central region of the experimental i F i # i (c) Experimental histogram of the deviation " as obtained with cell (seen from above and in a cross section along the y axis, NT ! 125 single walkers with L=#F ! 2:11 (L ! 14:7 mm and respectively). An individual trajectory is shown in (a) and the #F ! 6:95 mm). Since each trajectory has a symmetrical coundefinitions of !i , yi , and " are given. The width of the slit being terpart with respect to the axis of the aperture, the statistic was L, the parameter of impact is Yi' ! yi =L. (c) A photograph of the improved by taking them into account so that the distributions experiment lit with diffuse light showing the wave pattern as the correspond to 2NT realizations. The curve is the fit by Eq. (1) walker crosses the aperture. The picture was taken at a time with L=#F ! 1:96. (d) Histogram obtained with L=#F ! 3:1 when the trajectory, initially perpendicular to the aperture, was (L ! 14:7 mm and A. Eddi#et Fal.! 4:75 mm). The curve is the fit by [Couder,2008] deflected by the interference with reflected waves. Eq. (1) with L=#F ! 2:86. C [Couder,2009] ments, having the same walker cross the slit, with various initial motions. As shown on Fig. 1(a) each initial trajectory is defined by its incidence angle !i and by its normalized impact parameter in the slit Yi ! yi =L ("0:5 < Yi < 0:5) [Fig. 1(a)]. The walkers were always started far from the screen and only those impinging perpendicularly on the slit (!I ! 90# ) were retained. The plot "$Yi % [Fig. 2(a)] shows that there is no simple relation between the deviation and the parameter of impact. Thursday, June is 13, also 13 This observed on individual realizations: with ap- #F =L and large amplitude lateral lobes are clearly observed. Hence, on Figs. 2(c) and 2(d) the general shapes of the histograms are compared with the amplitude diffraction pattern of a plane wave passing through a slit: ! ! ! ! FIG. 5 (color online). (a) Sup ! ! sin$$L sin"=# % ! ! F ! ! f$"% ! A! ! : (1) ! ! ! ! walker velocity V ¼ 12 mm $Lof sin"=# F a barrier with e ¼ 3 mm leadin Fig. 1: The crystalline organization of a large number of droplets = 61) with a triangular This amplitude of(Ndiffraction of lattice. a plane wave turns out to offit10forconsecutive collisions of provide an approximate these histograms. Hence, the ulus of the walkers velocity V W ðΩÞ is observed as the orbits comes shorter as the wall thickness is decreased [Fig. 4(c)]. become smaller. [Couder,2008] possible orbits being discrete, we label the plateaus by an Ultimately the limit cycle is never reachedThe [Fig. 4(d)] [10– order n, with n ¼ 0 corresponding to the tightest orbit observed at 12]. The trajectories inside the cavity and the escapes are large Ω. Each plateau crosses the continuous short path-memory correlated. As long as the droplet follows limit curvethe given by Eq.cycle, 3 at a value Ω0n of the angular velocity (Fig. 2B). At thesebecome crossings, disthe observed orbits have the same radius no escape is observed. When the trajectories exp exp 0 0 Rn ¼ Rn ðΩn Þ ¼ RC ðΩ0n Þ they would have had with short-term ordered, the collisions with the walls show a larger variety memory. A plateau extends on both sides of the curve given by relation 3 between two limiting values Ω−n < Ωn < Ωþ n . A hysteresis is observed: The abrupt transition from one plateau to the Perspectives Analogies with quantum mechanics • Single/double slit • Charged particle in magnetic field FIG. 2 (color online). (a) The measured deviations a of suc• Harmonic oscillator A B cessive individual particles as a function of[Couder,2010] the parameter of impact Y (with L=# ! 2:86). (b) A superposition of three • Lattice structures different trajectories of the same droplet passing 3 times through the slit with similar initial conditions ! ! 90 and Y ! 0:1. effects • FIG.Tunneling 1. (a), (b) Sketch of the central region of the experimental (c) Experimental histogram of the deviation " as obtained with cell (seen from above and in a cross section along the y axis, N ! 125 single walkers with L=# ! 2:11 (L ! 14:7 mm and Wave-like statistics in confined geometries • respectively). An individual trajectory is shown in (a) and the i F i # i 3 T F #F ! 6:95 mm). Since each trajectory has a symmetrical coundefinitions of !i , yi , and " are given. The width of the slit being terpart with respect to the axis of the aperture, the statistic was L, the parameter of impact is Yi' ! yi =L. (c) A photograph of the improved by taking them into account so that the distributions experiment lit with diffuse light showing the wave pattern as the correspond to 2NT realizations. The curve is the fit by Eq. (1) walker crosses the aperture. The picture was taken at a time with L=#F ! 1:96. (d) Histogram obtained with L=#F ! 3:1 when the trajectory, initially perpendicular to the aperture, was (L ! 14:7 mm and A. Eddi#et Fal.! 4:75 mm). The curve is the fit by [Couder,2008] deflected by the interference withc reflected waves. Eq. (1) with L=#F ! 2:86. b C [Harris,2013] [Couder,2009] a 1 30 0.5 y/R 20 0 10 −0.5 −1 −1 0 −0.5 0 x/R 0.5 1 4000 3000 n/r ments, having the same walker cross the slit, with various initial motions. As shown on Fig. 1(a) each initial trajectory is defined by its incidence angle !i and by its normalA B ized impact parameter in the slit Yi ! yi =L ("0:5 < Yi < 0:5) [Fig. 1(a)]. The walkers were always started far from the screen and only those impinging perpendicularly on the slit (!I ! 90# ) were retained. The plot "$Yi % [Fig. 2(a)] shows that there is no simple relation between the deviation and the parameter of impact. Thursday, June(a)is 13, 13 FIG. 3.This (color) Trajectories of a droplet of diameter = 0.67 mm walkingrealizations: in a circular corral with with radius 14.3apmm and also observed on Dindividual 2000 1000 0 velocity difference from mean (%) 50 0 −50 0 0.2 0.4 0.6 r/R 0.8 1 #F =L and large amplitude lateral lobes are clearly observed. Hence, on Figs. 2(c) and 2(d) the general shapes of the histograms are compared with the amplitude diffraction pattern of a plane wave passing through a slit: ! ! ! ! FIG. 5 (color online). (a) Sup ! ! sin$$L sin"=# % ! ! F ! ! f$"% ! A! ! : (1) ! ! ! ! walker velocity V ¼ 12 mm $Lof sin"=# F a barrier with e ¼ 3 mm leadin Fig. 1: The crystalline organization of a large number of droplets = 61) with a triangular This amplitude of(Ndiffraction of lattice. a plane wave turns out to offit10forconsecutive collisions of provide an approximate these histograms. Hence, the ulus of the walkers velocity V W ðΩÞ is observed as the orbits comes shorter as the wall thickness is decreased [Fig. 4(c)]. become smaller. [Couder,2008] possible orbits being discrete, we label the plateaus by an Ultimately the limit cycle is never reachedThe [Fig. 4(d)] [10– order n, with n ¼ 0 corresponding to the tightest orbit observed at 12]. The trajectories inside the cavity and the escapes are large Ω. Each plateau crosses the continuous short path-memory correlated. As long as the droplet follows limit curvethe given by Eq.cycle, 3 at a value Ω0n of the angular velocity (Fig. 2B). At thesebecome crossings, disthe observed orbits have the same radius no escape is observed. When the trajectories exp exp 0 0 Rn ¼ Rn ðΩn Þ ¼ RC ðΩ0n Þ they would have had with short-term ordered, the collisions with the walls show a larger variety memory. A plateau extends on both sides of the curve given by relation 3 between two limiting values Ω−n < Ωn < Ωþ n . A hysteresis is observed: The abrupt transition from one plateau to the Perspectives Analogies with quantum mechanics • Single/double slit • Charged particle in magnetic field FIG. 2 (color online). (a) The measured deviations a of suc• Harmonic oscillator A B cessive individual particles as a function of[Couder,2010] the parameter of impact Y (with L=# ! 2:86). (b) A superposition of three • Lattice structures different trajectories of the same droplet passing 3 times through the slit with similar initial conditions ! ! 90 and Y ! 0:1. effects • FIG.Tunneling 1. (a), (b) Sketch of the central region of the experimental (c) Experimental histogram of the deviation " as obtained with cell (seen from above and in a cross section along the y axis, N ! 125 single walkers with L=# ! 2:11 (L ! 14:7 mm and Wave-like statistics in confined geometries • respectively). An individual trajectory is shown in (a) and the # ! 6:95 mm). Since each trajectory has a symmetrical coundefinitions of ! , y , and " are given. The width of the slit being terpart with respect to the axis of the aperture, the statistic was • L,Scattering? the parameter of impact is Y ! y =L. (c) A photograph of the i F i # i 3 a 1 30 i 0.5 T F F i i' 20 i y/R improved by taking them into account so that the distributions experiment lit with diffuse light showing the wave pattern as the correspond to 2NT realizations. The curve is the fit by Eq. (1) walker crosses the aperture. The picture was taken at a time with L=#F ! 1:96. (d) Histogram obtained with L=#F ! 3:1 when the trajectory, initially perpendicular to the aperture, was (L ! 14:7 mm and A. Eddi#et Fal.! 4:75 mm). The curve is the fit by [Couder,2008] deflected by the interference withc reflected waves. Eq. (1) with L=#F ! 2:86. b C [Harris,2013] [Couder,2009] 0 10 −0.5 −1 −1 0 −0.5 0 x/R 0.5 1 4000 3000 n/r ments, having the same walker cross the slit, with various initial motions. As shown on Fig. 1(a) each initial trajectory is defined by its incidence angle !i and by its normalA B ized impact parameter in the slit Yi ! yi =L ("0:5 < Yi < 0:5) [Fig. 1(a)]. The walkers were always started far from the screen and only those impinging perpendicularly on the slit (!I ! 90# ) were retained. The plot "$Yi % [Fig. 2(a)] shows that there is no simple relation between the deviation and the parameter of impact. Thursday, June(a)is 13, 13 FIG. 3.This (color) Trajectories of a droplet of diameter = 0.67 mm walkingrealizations: in a circular corral with with radius 14.3apmm and also observed on Dindividual 2000 1000 0 velocity difference from mean (%) 50 0 −50 0 0.2 0.4 0.6 r/R 0.8 1 #F =L and large amplitude lateral lobes are clearly observed. Hence, on Figs. 2(c) and 2(d) the general shapes of the histograms are compared with the amplitude diffraction pattern of a plane wave passing through a slit: ! ! ! ! FIG. 5 (color online). (a) Sup ! ! sin$$L sin"=# % ! ! F ! ! f$"% ! A! ! : (1) ! ! ! ! walker velocity V ¼ 12 mm $Lof sin"=# F a barrier with e ¼ 3 mm leadin Fig. 1: The crystalline organization of a large number of droplets = 61) with a triangular This amplitude of(Ndiffraction of lattice. a plane wave turns out to offit10forconsecutive collisions of provide an approximate these histograms. Hence, the ulus of the walkers velocity V W ðΩÞ is observed as the orbits comes shorter as the wall thickness is decreased [Fig. 4(c)]. become smaller. [Couder,2008] possible orbits being discrete, we label the plateaus by an Ultimately the limit cycle is never reachedThe [Fig. 4(d)] [10– order n, with n ¼ 0 corresponding to the tightest orbit observed at 12]. The trajectories inside the cavity and the escapes are large Ω. Each plateau crosses the continuous short path-memory correlated. As long as the droplet follows limit curvethe given by Eq.cycle, 3 at a value Ω0n of the angular velocity (Fig. 2B). At thesebecome crossings, disthe observed orbits have the same radius no escape is observed. When the trajectories exp exp 0 0 Rn ¼ Rn ðΩn Þ ¼ RC ðΩ0n Þ they would have had with short-term ordered, the collisions with the walls show a larger variety memory. A plateau extends on both sides of the curve given by relation 3 between two limiting values Ω−n < Ωn < Ωþ n . A hysteresis is observed: The abrupt transition from one plateau to the Perspectives Analogies with quantum mechanics • Single/double slit • Charged particle in magnetic field FIG. 2 (color online). (a) The measured deviations a of suc• Harmonic oscillator A B cessive individual particles as a function of[Couder,2010] the parameter of impact Y (with L=# ! 2:86). (b) A superposition of three • Lattice structures different trajectories of the same droplet passing 3 times through the slit with similar initial conditions ! ! 90 and Y ! 0:1. effects • FIG.Tunneling 1. (a), (b) Sketch of the central region of the experimental (c) Experimental histogram of the deviation " as obtained with cell (seen from above and in a cross section along the y axis, N ! 125 single walkers with L=# ! 2:11 (L ! 14:7 mm and Wave-like statistics in confined geometries • respectively). An individual trajectory is shown in (a) and the # ! 6:95 mm). Since each trajectory has a symmetrical coundefinitions of ! , y , and " are given. The width of the slit being terpart with respect to the axis of the aperture, the statistic was • L,Scattering? the parameter of impact is Y ! y =L. (c) A photograph of the improved by taking them into account so that the distributions experiment lit with diffuse light showing the wave pattern as the Entanglement? correspond to 2N realizations. The curve is the fit by Eq. (1) • walker crosses the aperture. The picture was taken at a time i F i # i 3 a 1 30 i y/R 0.5 F F i i' 0 T 20 i 10 −0.5 T with L=#F ! 1:96. (d) Histogram obtained with L=#F ! 3:1 when the trajectory, initially perpendicular to the aperture, was (L ! 14:7 mm and A. Eddi#et Fal.! 4:75 mm). The curve is the fit by [Couder,2008] deflected by the interference withc reflected waves. Eq. (1) with L=#F ! 2:86. b C [Harris,2013] [Couder,2009] −1 −1 0 −0.5 0 x/R 0.5 1 4000 3000 n/r ments, having the same walker cross the slit, with various initial motions. As shown on Fig. 1(a) each initial trajectory is defined by its incidence angle !i and by its normalA B ized impact parameter in the slit Yi ! yi =L ("0:5 < Yi < 0:5) [Fig. 1(a)]. The walkers were always started far from the screen and only those impinging perpendicularly on the slit (!I ! 90# ) were retained. The plot "$Yi % [Fig. 2(a)] shows that there is no simple relation between the deviation and the parameter of impact. Thursday, June(a)is 13, 13 FIG. 3.This (color) Trajectories of a droplet of diameter = 0.67 mm walkingrealizations: in a circular corral with with radius 14.3apmm and also observed on Dindividual 2000 1000 0 velocity difference from mean (%) 50 0 −50 0 0.2 0.4 0.6 r/R 0.8 1 #F =L and large amplitude lateral lobes are clearly observed. Hence, on Figs. 2(c) and 2(d) the general shapes of the histograms are compared with the amplitude diffraction pattern of a plane wave passing through a slit: ! ! ! ! FIG. 5 (color online). (a) Sup ! ! sin$$L sin"=# % ! ! F ! ! f$"% ! A! ! : (1) ! ! ! ! walker velocity V ¼ 12 mm $Lof sin"=# F a barrier with e ¼ 3 mm leadin Fig. 1: The crystalline organization of a large number of droplets = 61) with a triangular This amplitude of(Ndiffraction of lattice. a plane wave turns out to offit10forconsecutive collisions of provide an approximate these histograms. Hence, the