Irrotational motion of a compressible inviscid fluid

Transcription

1 Irrotational motion of a compressible inviscid fluid University of Cambridge Computer Laboratory JJ Thomson Avenue, Cambridge CB3 0FD, UK robert.brady@cl.cam.ac.uk University of Warwick, January 2013 u.dl = 0 Warwick / 29

2 Irrotational motion of a compressible inviscid fluid 1 Introduction 2 Irrotational solutions Linear eddy Sonon quasiparticles Spin symmetry 3 Equations of motion Experimental analogue Lorentz covariance Wavefunction Forces between quasiparticles 4 Possible interpretation 5 Further work 6 Summary u.dl = 0 Warwick / 29

3 Introduction Compressible inviscid fluid eg air with no viscosity or thermal conductivity Equations by Leonhard Euler Taught to undergraduates No special knowledge needed All equations completely classical no quantum mechanics u.dl = 0 Warwick / 29

4 Euler s equation for a compressible fluid Euler s equation describes the air without viscosity u t + (u. )u = 1 ρ P Reduces to wave equation at low amplitude where c 2 = P/ ρ 2 ρ t 2 c2 2 ρ = 0 u.dl = 0 Warwick / 29

5 Rotational solutions Rotational - defined by u.dl 0 We will examine an irrotational equivalent of these u.dl = 0 Warwick / 29

6 Irrotational motion of a compressible inviscid fluid 1 Introduction 2 Irrotational solutions Linear eddy Sonon quasiparticles Spin symmetry 3 Equations of motion Experimental analogue Lorentz covariance Wavefunction Forces between quasiparticles 4 Possible interpretation 5 Further work 6 Summary u.dl = 0 Warwick / 29

7 Solutions to wave equation (low amplitude) (J m cylindrical Bessel function) ρ = A cos(ω o t + mθ) J m (k r r) u.dl = 0 Warwick / 29

8 Solutions to wave equation (low amplitude) (J m cylindrical Bessel function) ρ = A cos(ω o t + mθ) J m (k r r) m=0 (approximate, 2D) u.dl = 0 Warwick / 29

9 Solutions to wave equation (low amplitude) (J m cylindrical Bessel function) ρ = A cos(ω o t + mθ) J m (k r r) - + m=0 (approximate, 2D) (animation) m = 1 Irrotational u.dl = 0 u.dl = 0 Warwick / 29

10 Sonon quasiparticles Curve the eddy into a ring Quasiparticle solution similar to Dolphin air ring sonon u.dl = 0 Warwick / 29

11 Sonon quasiparticles Curve the eddy into a ring Quasiparticle solution similar to Dolphin air ring sonon z ϕ σ θ' R 10 R o (animation) ξ = A e iωot R mn (x) R mn (x) = e i(mθ nφ) j m (k r σ) R o dφ j m spherical Bessel function Solution for m = 0, 1 (need Legendre polynomials for m > 1) u.dl = 0 Warwick / 29

12 Sonon quasiparticles Curve the eddy into a ring Quasiparticle solution similar to Dolphin air ring sonon z ϕ σ θ' R 10 R o R11 (animation) (animation) ξ = A e iωot R mn (x) R mn (x) = e i(mθ nφ) j m (k r σ) R o dφ j m spherical Bessel function Solution for m = 0, 1 (need Legendre polynomials for m > 1) u.dl = 0 Warwick / 29

13 Density pattern at large r R 11 z R o ϕ σ θ' e i(mθ nφ) j m (k r σ) R o dφ factorise at large r [B r.φ r (r)] [B φ.φ φ (φ)] [B θ.φ θ (θ)] u.dl = 0 Warwick / 29

14 Density pattern at large r R 11 z R o ϕ σ θ' e i(mθ nφ) j m (k r σ) R o dφ factorise at large r [B r.φ r (r)] [B φ.φ φ (φ)] [B θ.φ θ (θ)] Φ r = 1 r [sin(k r r), cos(k r r)] Φ φ = [e inφ, e inφ ] Φ θ = [e imθ +e imθ, e imθ e imθ ] u.dl = 0 Warwick / 29

15 Density pattern at large r R 11 z R o ϕ σ θ' e i(mθ nφ) j m (k r σ) R o dφ factorise at large r [B r.φ r (r)] [B φ.φ φ (φ)] [B θ.φ θ (θ)] Φ r = 1 r [sin(k r r), cos(k r r)] ( 0 i i i 0 ). Φ r = k r Φ Φ φ = [e inφ, e inφ ] i( ). χ Φ = nφ Φ θ = [e imθ +e imθ, e imθ e imθ ] Pauli spin matrix ( 0 1 i 1 0 ). Φ θ = mφ u.dl = 0 Warwick / 29

16 Irrotational motion of a compressible inviscid fluid 1 Introduction 2 Irrotational solutions Linear eddy Sonon quasiparticles Spin symmetry 3 Equations of motion Experimental analogue Lorentz covariance Wavefunction Forces between quasiparticles 4 Possible interpretation 5 Further work 6 Summary u.dl = 0 Warwick / 29

17 Experimental analogue in 2D (particle drawn just for illustration) Droplet bouncing on the surface of the same liquid (Bath vibrated vertically) S Protiere, A Badaoud, Y Couder Particle wave association on a fluid interface J Fluid Mech (2006) u.dl = 0 Warwick / 29

18 Experimental analogue in 2D (particle drawn just for illustration) Droplet bouncing on the surface of the same liquid (Bath vibrated vertically) Main difference from sonons 2D driven dissipative system Effective c reduced near droplet Droplet itself does not obey Euler s equation S Protiere, A Badaoud, Y Couder Particle wave association on a fluid interface J Fluid Mech (2006) u.dl = 0 Warwick / 29

19 Lorentz covariance Sonon quasiparticle ξ = A e iωot R mn (x) u.dl = 0 Warwick / 29

20 Lorentz covariance Sonon quasiparticle ξ = A e iωot R mn (x) A 1 Obeys the wave equation The wave equation is unchanged by Lorentz transformation If ξ(x, t) is a solution then so is ξ(x, t ) ξ(x, t ) moves at velocity v, Lorentz-Fitzgerald contraction u.dl = 0 Warwick / 29

21 Lorentz covariance Sonon quasiparticle ξ = A e iωot R mn (x) A 1 Obeys the wave equation The wave equation is unchanged by Lorentz transformation If ξ(x, t) is a solution then so is ξ(x, t ) ξ(x, t ) moves at velocity v, Lorentz-Fitzgerald contraction Finite amplitude perturbed by ɛ = (v. )u if moving at constant velocity v but the sonon is oscillatory: udt = 0 over a cycle Therefore ɛdt = 0, ie perturbation vanishes over a cycle u.dl = 0 Warwick / 29

22 Lorentz covariance Sonon quasiparticle ξ = A e iωot R mn (x) A 1 Obeys the wave equation The wave equation is unchanged by Lorentz transformation If ξ(x, t) is a solution then so is ξ(x, t ) ξ(x, t ) moves at velocity v, Lorentz-Fitzgerald contraction Finite amplitude perturbed by ɛ = (v. )u if moving at constant velocity v but the sonon is oscillatory: udt = 0 over a cycle Therefore ɛdt = 0, ie perturbation vanishes over a cycle Expectation values converge on long term average Lorentz covariant at all amplitudes u.dl = 0 Warwick / 29

23 Experimental analogue - walker Walker Increase amplitude Droplet bounces higher Frequency reduces Velocity v increases v = c A A o A where c < c A Eddi et al Information stored in Faraday waves: the origin of a path memory J Fluid Mech (2011) u.dl = 0 Warwick / 29

24 Experimental analogue - walker Walker Increase amplitude Droplet bounces higher Frequency reduces Velocity v increases v = c A A o A Rearrange where c < c Approximate τ A ω ωo γ where 1 γ = 1 v 2 c 2 Time dilation A Eddi et al Information stored in Faraday waves: the origin of a path memory J Fluid Mech (2011) u.dl = 0 Warwick / 29

25 Equation for ψ ξ = A e iωot R mn (x) = ψ o χ u.dl = 0 Warwick / 29

26 Equation for ψ ξ = A e iωot R mn (x) = ψ o χ 2 ψ o = 0, 2 t 2 ψ o = ω 2 ψ o Must be Lorentz covariant 2 ψ t 2 c2 2 ψ = ω 2 oψ Klein-Gordon equation u.dl = 0 Warwick / 29

27 Equation for ψ ξ = A e iωot R mn (x) = ψ o χ 2 ψ o = 0, 2 t 2 ψ o = ω 2 ψ o Must be Lorentz covariant 2 ψ t 2 c2 2 ψ = ω 2 oψ Klein-Gordon equation Low velocity ψ = exp( iω o t)ψ, neglect 2 t 2 ψ i t ψ + 2 2m 2 ψ = V ψ Schrödinger equation units where m = ω o /c 2 u.dl = 0 Warwick / 29

28 Equation of motion for particle Quasiparticle aligned with wave troughs (n.b. easily disrupted) Speed v of the wave troughs k = ω v c 2 v = ( ) ψ m Im ψ u.dl = 0 Warwick / 29

29 Equation of motion for particle Quasiparticle aligned with wave troughs (n.b. easily disrupted) Speed v of the wave troughs k = ω v c 2 v = ( ) ψ m Im ψ Bohm rearranged Schrödinger s ψ 2 t = ( ψ 2 v) Continuity equation for ψ 2 ψ(x, t) 2 = probability of reaching (x, t) (averaged over nearby trajectories) u.dl = 0 Warwick / 29

30 Equation of motion for particle Quasiparticle aligned with wave troughs (n.b. easily disrupted) Speed v of the wave troughs k = ω v c 2 v = ( ) ψ m Im ψ Bohm rearranged Schrödinger s ψ 2 t = ( ψ 2 v) Continuity equation for ψ 2 ψ(x, t) 2 = probability of reaching (x, t) (averaged over nearby trajectories) Bohmian trajectories u.dl = 0 Warwick / 29

31 Aditional (superfluous) assumption Sonon changes state or delocalizes in some undefined way u.dl = 0 Warwick / 29

32 Aditional (superfluous) assumption Sonon changes state or delocalizes in some undefined way Suddenly reappears before measurement, without changing path u.dl = 0 Warwick / 29

33 Aditional (superfluous) assumption Sonon changes state or delocalizes in some undefined way Suddenly reappears before measurement, without changing path Probability of reappearing at (x, t) is ψ(x, t) 2 u.dl = 0 Warwick / 29

34 Aditional (superfluous) assumption Sonon changes state or delocalizes in some undefined way Suddenly reappears before measurement, without changing path Probability of reappearing at (x, t) is ψ(x, t) 2 Analogue of Copenhagen interpretation same equations for ψ and collapse assumption superfluous in the case of sonons u.dl = 0 Warwick / 29

35 Aditional (superfluous) assumption Sonon changes state or delocalizes in some undefined way Suddenly reappears before measurement, without changing path Probability of reappearing at (x, t) is ψ(x, t) 2 Analogue of Copenhagen interpretation same equations for ψ and collapse assumption superfluous in the case of sonons Reminder Euler s equation and all sonon motion is completely classical u.dl = 0 Warwick / 29

36 Classical experiment diffraction Single slit Y Couder, E Fort Single-Particle Diffraction and Interference at a Macroscopic Scale PRL (2006) u.dl = 0 Warwick / 29

37 Classical experiment diffraction Single slit Double slit Y Couder, E Fort Single-Particle Diffraction and Interference at a Macroscopic Scale PRL (2006) u.dl = 0 Warwick / 29

38 Classical experiment tunnelling A Eddi, E Fort, F Moisi, Y Couder Unpredictable tunneling of a classical wave-particle association PRL 102, (2009) u.dl = 0 Warwick / 29

39 Classical experiment Landau levels E Fort et al Path-memory induced quantization of classical orbits PNAS (2010) u.dl = 0 Warwick / 29

40 Classical experiment incapable of quantum collapse u.dl = 0 Warwick / 29

41 Classical experiment incapable of quantum collapse Wavefunction collapse superfluous u.dl = 0 Warwick / 29

42 Classical experiment incapable of quantum collapse Wavefunction collapse superfluous ξ = A e i(ωt kx) R mn (x ) = ψ χ ψ obeys same equations as quantum mechanical wavefunction ψ modulates χ (usually omitted) localisation u.dl = 0 Warwick / 29

43 Classical experiment incapable of quantum collapse Wavefunction collapse superfluous ξ = A e i(ωt kx) R mn (x ) = ψ χ modulation carrier Modulation of a carrier wave amplitude and phase complex valued ψ obeys same equations as quantum mechanical wavefunction ψ modulates χ (usually omitted) localisation u.dl = 0 Warwick / 29

44 Forces between quasiparticles 2D Meniscus at boundary Image droplet antiphase Repulsive force Stroboscopic photograph S Protiere, A Badaoud, Y Couder Particle wave association on a fluid interface J Fluid Mech (2006) u.dl = 0 Warwick / 29

45 Forces between quasiparticles 2D Meniscus at boundary Image droplet antiphase Repulsive force Stroboscopic photograph In-phase attraction S Protiere, A Badaoud, Y Couder Particle wave association on a fluid interface J Fluid Mech (2006) u.dl = 0 Warwick / 29

46 Forces between quasiparticles 3D Ultrasonic degassing of oil (5 seconds) Switch on ultrasonic transducer Bubbles expand and contract in phase Fluid dynamic attraction Inverse square force u.dl = 0 Warwick / 29

47 Forces between quasiparticles 3D Ultrasonic degassing of oil (5 seconds) Fluid dynamic calculation for R 11 (results) Opposite chiralities attract Like chiralities repel Inverse square force Lorentz covariant same equations to electromagnetism Switch on ultrasonic transducer Bubbles expand and contract in phase Fluid dynamic attraction Inverse square force u.dl = 0 Warwick / 29

48 Forces between quasiparticles 3D Ultrasonic degassing of oil (5 seconds) Switch on ultrasonic transducer Bubbles expand and contract in phase Fluid dynamic attraction Inverse square force Fluid dynamic calculation for R 11 (results) Opposite chiralities attract Like chiralities repel Inverse square force Lorentz covariant same equations to electromagnetism Magnitude of the force Characterised by dimensionless number α 1 49 Could be calculated more accurately by computer simulation Experimental fine structure constant α = u.dl = 0 Warwick / 29

49 Irrotational motion of a compressible inviscid fluid 1 Introduction 2 Irrotational solutions Linear eddy Sonon quasiparticles Spin symmetry 3 Equations of motion Experimental analogue Lorentz covariance Wavefunction Forces between quasiparticles 4 Possible interpretation 5 Further work 6 Summary u.dl = 0 Warwick / 29

50 Extends field of analogue gravity Irrotational motion of a compressible inviscid fluid Acoustic metric like general relativity 1981 Unruh proposed sonic experiment: Hawking radiation Acoustic black hole C Barceló et al Analogue Gravity arxiv:gr-qc/ v3 (2011) (review) W G Unruh. experimental black hole evaporation?. Phys Rev Lett, 46: , (1981) u.dl = 0 Warwick / 29

51 Extends field of analogue gravity Irrotational motion of a compressible inviscid fluid Acoustic metric like general relativity 1981 Unruh proposed sonic experiment: Hawking radiation Acoustic black hole 2003 Volovik, model based on superfluid Helium-3 symmetries of general relativity and standard model C Barceló et al Analogue Gravity arxiv:gr-qc/ v3 (2011) (review) W G Unruh. experimental black hole evaporation?. Phys Rev Lett, 46: , (1981) G E Volovik. The Universe in a Helium Droplet. Clarendon Press, Oxford, (2003) u.dl = 0 Warwick / 29

52 Extends field of analogue gravity Irrotational motion of a compressible inviscid fluid Acoustic metric like general relativity 1981 Unruh proposed sonic experiment: Hawking radiation Acoustic black hole 2003 Volovik, model based on superfluid Helium-3 symmetries of general relativity and standard model 2010 Experimental black hole analogue in Bose-Einstein condensate C Barceló et al Analogue Gravity arxiv:gr-qc/ v3 (2011) (review) W G Unruh. experimental black hole evaporation?. Phys Rev Lett, 46: , (1981) G E Volovik. The Universe in a Helium Droplet. Clarendon Press, Oxford, (2003) O Lahav et al realization of a sonic black hole analog in a Bose-Einstein condensate Phys. Rev. Lett, 105(24): (2010) u.dl = 0 Warwick / 29

53 Possible interpretation Why do gyroscopes remain aligned with the fixed stars? Superconducting gyroscope (and me with hair) A Einstein, Ether and the Theory of Relativity Sidelights on Relativity Methuen pp 3 24 (1922) H u.dl = 0 Warwick / 29

54 Possible interpretation Why do gyroscopes remain aligned with the fixed stars? Einstein s answer (1920) All interactions are local Correlation due to a substance occupying the space between them medium for the effects of inertia Can t be solid if consistent with special relativity Superconducting gyroscope (and me with hair) A Einstein, Ether and the Theory of Relativity Sidelights on Relativity Methuen pp 3 24 (1922) H u.dl = 0 Warwick / 29

55 Possible interpretation Why do gyroscopes remain aligned with the fixed stars? Einstein s answer (1920) All interactions are local Correlation due to a substance occupying the space between them medium for the effects of inertia Can t be solid if consistent with special relativity Euclidian space, compressible fluid Spin- 12 quasiparticles Obey special relativity Superconducting gyroscope (and me with hair) Diffract like quantum particles Electromagnetic force between them A Einstein, Ether and the Theory of Relativity Sidelights on Relativity Methuen pp 3 24 (1922) H u.dl = 0 Warwick / 29

56 Irrotational motion of a compressible inviscid fluid 1 Introduction 2 Irrotational solutions Linear eddy Sonon quasiparticles Spin symmetry 3 Equations of motion Experimental analogue Lorentz covariance Wavefunction Forces between quasiparticles 4 Possible interpretation 5 Further work 6 Summary u.dl = 0 Warwick / 29

57 Limits to coherence Coherence between particles needs coherence with carrier wave One correlation per dimension = 3 independent correlations u.dl = 0 Warwick / 29

58 Limits to coherence Coherence between particles needs coherence with carrier wave One correlation per dimension = 3 independent correlations Quantum computing major multi-year research effort Relies on multiple correlations (or possibly entanglement) R Anderson, R Brady Why quantum computing is hard in preparation u.dl = 0 Warwick / 29

59 Limits to coherence Coherence between particles needs coherence with carrier wave One correlation per dimension = 3 independent correlations Quantum computing major multi-year research effort Relies on multiple correlations (or possibly entanglement) Proxies for qubits, but no calculations with > 3 qubits R Anderson, R Brady Why quantum computing is hard in preparation u.dl = 0 Warwick / 29

60 Irrotational motion of a compressible inviscid fluid 1 Introduction 2 Irrotational solutions Linear eddy Sonon quasiparticles Spin symmetry 3 Equations of motion Experimental analogue Lorentz covariance Wavefunction Forces between quasiparticles 4 Possible interpretation 5 Further work 6 Summary u.dl = 0 Warwick / 29

61 Irrotational solutions to Euler s equation R 10 R 11 u.dl = 0 Warwick / 29

62 Irrotational solutions to Euler s equation R 10 R 11 Lorentz covariant spin- 1 2 quasiparticles Move and diffract like quantum particles of mass ω/c 2 Maxwell s equations with α 1/45 u.dl = 0 Warwick / 29

63 Irrotational solutions to Euler s equation R 10 R 11 Lorentz covariant spin- 1 2 quasiparticles Move and diffract like quantum particles of mass ω/c 2 Maxwell s equations with α 1/45 Just ordinary Newton s equations applied to a fluid No wavefunction collapse, multiverses, cats or dice No distortion of space and time Lorentz covariance a symmetry of Euler s equation No action at a distance ordinary forces mediated by the fluid u.dl = 0 Warwick / 29