Collective Effects. Equilibrium and Nonequilibrium Physics

Transcription

1 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Collective Effects in Equilibrium and Nonequilibrium Physics Website: Caltech Mirror:

2 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Today s Lecture Superfluids and superconductors What are superfluidity and superconductivity? Review of phase dynamics Description in terms of a macroscopic phase Supercurrents that flow for ever Josephson effect Four sounds

3 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, The Amazing World of Superfluidity and Superconductivity Electric currents in loops that flow for ever (measured for decade) Beakers of fluid that empty themselves Fluids that flow without resistance through tiny holes Flow in surface films less than an atomic layer thick Flow driven by temperature differences (fountain effect)

4 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, History of Superfluidity and Superconductivity 1908 Liquefaction of 4 He by Kamerlingh Onnes 1911 Discovery of superconductivity by Onnes (resistance drops to zero) 1933 Meissner effect: superconductors expel magnetic field 1937 Discovery of superfluidity in 4 He by Allen and Misener 1938 Connection of superfluidity with Bose-Einstein condensation by London 1955 Feynman s theory of quantized vortices 1956 Onsager and Penrose identify the broken symmetry in superfluidity ODLRO 1957 BCS theory of superconductivity 1962 Josephson effect 1973 Discovery of superfluidity in 3 He at 2mK by Osheroff, Lee, and Richardson 1986 Discovery of high-t c superconductors by Bednorz and Müller Study of superfluidity in ultracold trapped dilute gases

5 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Review of Phase Dynamics with a Conserved Quantity S z Θ S Rotational symmetry in the XY plane (angle ) The XY and Z components of the spin have different properties: S z = 1 s iz is a conserved quantity S = 1 i in i in s i is the XY order parameter

6 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Review of Phase Dynamics with a Conserved Quantity S z Θ S Rotational symmetry in the XY plane (angle ) The XY and Z components of the spin have different properties: S z = 1 s iz is a conserved quantity S = 1 i in i in s i is the XY order parameter S z and are canonically conjugate variables, so that with the free energy [ ] F = d d 1 x 2 K ( )2 + S2 z 2χ S zb z we get Ṡ z = δf δ giving Ṡ z = j Sz with j Sz = K = δf δs z giving = χ 1 (S z χb z )

7 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Spin Current Ṡ z = j Sz

8 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Spin Current Ṡ z = j Sz This is a conservation law with a current j Sz of the conserved quantity S z given by a phase gradient j Sz = K

9 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Spin Current Ṡ z = j Sz This is a conservation law with a current j Sz of the conserved quantity S z given by a phase gradient For example j Sz = K Θ(r) L

10 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Phase Dynamics = χ 1 (S z χb z )

11 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Phase Dynamics = χ 1 (S z χb z ) No dynamics in full thermodynamic equilibrium: S z = χb 0z

12 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Phase Dynamics = χ 1 (S z χb z ) No dynamics in full thermodynamic equilibrium: S z = χb 0z Add an additional external field b 1z = γ B 1z = b 1z = γ B 1z the usual precession of a magnetic moment in an applied field (Larmor precession).

13 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Phase Dynamics = χ 1 (S z χb z ) No dynamics in full thermodynamic equilibrium: S z = χb 0z Add an additional external field b 1z = γ B 1z = b 1z = γ B 1z the usual precession of a magnetic moment in an applied field (Larmor precession). Note that this is an equilibrium state: S z = χ(b 0z + b 1z ) but is a conserved quantity

14 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Phase Dynamics = χ 1 (S z χb z ) No dynamics in full thermodynamic equilibrium: S z = χb 0z Add an additional external field b 1z = γ B 1z = b 1z = γ B 1z the usual precession of a magnetic moment in an applied field (Larmor precession). Note that this is an equilibrium state: S z = χ(b 0z + b 1z ) but is a conserved quantity no approximations

15 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Phase Dynamics = χ 1 (S z χb z ) No dynamics in full thermodynamic equilibrium: S z = χb 0z Add an additional external field b 1z = γ B 1z = b 1z = γ B 1z the usual precession of a magnetic moment in an applied field (Larmor precession). Note that this is an equilibrium state: S z = χ(b 0z + b 1z ) but is a conserved quantity no approximations For formal proof see Halperin and Saslow, Phys. Rev. B 16, 2154 (1977), Appendix: the Larmor precession theorem

16 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Hydrodynamic Approach Hydrodynamics: a formal derivation of long wavelength dynamics of conserved quantities and broken symmetry variables in a thermodynamic approach

17 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Hydrodynamic Approach Hydrodynamics: a formal derivation of long wavelength dynamics of conserved quantities and broken symmetry variables in a thermodynamic approach Starting points generalized rigidity: extra contribution to the energy density from gradients of the broken symmetry variable thermodynamic identity ε = 1 2 K ( )2 dε = Tds+ µ z ds z + d( ) with = K equilibrium phase dynamics (Larmor precession theorem) = µ z

18 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Hydrodynamic Approach Hydrodynamics: a formal derivation of long wavelength dynamics of conserved quantities and broken symmetry variables in a thermodynamic approach Starting points generalized rigidity: extra contribution to the energy density from gradients of the broken symmetry variable thermodynamic identity ε = 1 2 K ( )2 dε = Tds+ µ z ds z + d( ) with = K equilibrium phase dynamics (Larmor precession theorem) = µ z

19 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Hydrodynamic Approach Hydrodynamics: a formal derivation of long wavelength dynamics of conserved quantities and broken symmetry variables in a thermodynamic approach Starting points generalized rigidity: extra contribution to the energy density from gradients of the broken symmetry variable thermodynamic identity ε = 1 2 K ( )2 dε = Tds+ µ z ds z + d( ) with = K equilibrium phase dynamics (Larmor precession theorem) Derive = µ z dynamical equations for conserved quantities and broken symmetry variables for slowly varying disturbances

20 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Rigidity and the Thermodynamic Identity In terms of the energy density dε = Tds + µ z ds z + d( ) conjugate fields are µ z = ( ) ε s z s, and = ( ε ) s,s z

21 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Rigidity and the Thermodynamic Identity In terms of the energy density dε = Tds + µ z ds z + d( ) conjugate fields are µ z = ( ) ε s z s, and = ( ) ε s,s z Or with the free energy density f = ε Ts df = sdt + µ z ds z + d( ) conjugate fields are µ z = ( ) f s z T, and = ( f ) T,s z

22 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Rigidity and the Thermodynamic Identity In terms of the energy density dε = Tds + µ z ds z + d( ) conjugate fields are µ z = ( ) ε s z s, and = ( ) ε s,s z Or with the free energy density f = ε Ts df = sdt + µ z ds z + d( ) conjugate fields are µ z = ( ) f s z T, and = ( ) f T,s z These give µ z = χ 1 (S z χb z ) and = K

23 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Entropy Production Tds = dε µ z ds z d( )

24 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Entropy Production Tds = dε µ z ds z d( ) Form time derivative of entropy density ds dt = 1 T dε dt µ z T ds z dt T d( ) dt

25 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Entropy Production Tds = dε µ z ds z d( ) Form time derivative of entropy density ds dt = 1 T dε dt µ z T ds z dt T d( ) dt Conservation laws and dynamics of broken symmetry variable (j ε, j s z unknown) ds dt = 1 T jε + µ z T js z T µ z

26 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Entropy Production Tds = dε µ z ds z d( ) Form time derivative of entropy density ds dt = 1 T dε dt µ z T ds z dt T d( ) dt Conservation laws and dynamics of broken symmetry variable (j ε, j s z unknown) Entropy production equation ds dt = 1 T jε + µ z T js z T µ z ds dt = js + R with R 0

27 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Entropy Production Tds = dε µ z ds z d( ) Form time derivative of entropy density ds dt = 1 T dε dt µ z T ds z dt T d( ) dt Conservation laws and dynamics of broken symmetry variable (j ε, j s z unknown) Entropy production equation ds dt = 1 T jε + µ z T js z T µ z ds dt = js + R with R 0 Identify the entropy current and production j s = T 1 (j ε µ z j s z ) RT = T 1 (j ε µ z j s z ) T (j s z + ) µ z

28 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Equilibrium Dynamics Entropy Production RT = T 1 (j ε µ z j s z ) T (j s z + ) µ z (strategy: R should be a function of gradients of the conjugate variables)

29 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Equilibrium Dynamics Entropy Production RT = T 1 (j ε µ z j s z ) T (j s z + ) µ z (strategy: R should be a function of gradients of the conjugate variables) In the absence of dissipation the rate of entropy production must be zero.

30 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Equilibrium Dynamics Entropy Production RT = T 1 (j ε µ z j s z ) T (j s z + ) µ z (strategy: R should be a function of gradients of the conjugate variables) In the absence of dissipation the rate of entropy production must be zero. Spin current j s z =

31 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Equilibrium Dynamics Entropy Production RT = T 1 (j ε µ z j s z ) T (j s z + ) µ z (strategy: R should be a function of gradients of the conjugate variables) In the absence of dissipation the rate of entropy production must be zero. Spin current j s z = = K

32 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Equilibrium Dynamics Entropy Production RT = T 1 (j ε µ z j s z ) T (j s z + ) µ z (strategy: R should be a function of gradients of the conjugate variables) In the absence of dissipation the rate of entropy production must be zero. Spin current Energy current j s z = = K j ε = µ z j s z

33 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Equilibrium Dynamics Entropy Production RT = T 1 (j ε µ z j s z ) T (j s z + ) µ z (strategy: R should be a function of gradients of the conjugate variables) In the absence of dissipation the rate of entropy production must be zero. Spin current Energy current j s z = = K j ε = µ z j s z = µ z K

34 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Equilibrium Dynamics Entropy Production RT = T 1 (j ε µ z j s z ) T (j s z + ) µ z (strategy: R should be a function of gradients of the conjugate variables) In the absence of dissipation the rate of entropy production must be zero. Spin current Energy current Entropy current j s z = = K j ε = µ z j s z = µ z K j s = 0

35 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Equilibrium Dynamics Entropy Production RT = T 1 (j ε µ z j s z ) T (j s z + ) µ z (strategy: R should be a function of gradients of the conjugate variables) In the absence of dissipation the rate of entropy production must be zero. Spin current Energy current Entropy current j s z = = K j ε = µ z j s z = µ z K j s = 0 We will consider adding dissipation later.

36 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Superfluidity

37 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Superfluidity superfluidity occurs due to Bose condensation

38 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Superfluidity superfluidity occurs due to Bose condensation the order parameter is the expectation value of the quantum field operator for destroying a particle = ψ

39 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Superfluidity superfluidity occurs due to Bose condensation the order parameter is the expectation value of the quantum field operator for destroying a particle = ψ is a complex variable: = e i

40 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Superfluidity superfluidity occurs due to Bose condensation the order parameter is the expectation value of the quantum field operator for destroying a particle = ψ is a complex variable: = e i 2 gives the condensate density n 0 : the fraction of particles in the zero momentum state is n 0 /n is the phase of the condensate wave function

41 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Superfluidity superfluidity occurs due to Bose condensation the order parameter is the expectation value of the quantum field operator for destroying a particle = ψ is a complex variable: = e i 2 gives the condensate density n 0 : the fraction of particles in the zero momentum state is n 0 /n is the phase of the condensate wave function There are a macroscopic number of particles in a single wave function and so is a macroscopic thermodynamic variable, andis the broken symmetry variable.

42 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Broken Phase (Gauge) Symmetry

43 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Broken Phase (Gauge) Symmetry Any phase gives an equivalent state; the ordered state is characterized by a particular phase

44 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Broken Phase (Gauge) Symmetry Any phase gives an equivalent state; the ordered state is characterized by a particular phase There is an energy cost for gradients of the phase E = 1 2 n h 2 s ( ) 2 d d x m Stiffness constant K is written as n s ( h 2 /m) and n s is called the superfluid density Stiffness constant not the same as the condensate density n s = n 0

45 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Broken Phase (Gauge) Symmetry Any phase gives an equivalent state; the ordered state is characterized by a particular phase There is an energy cost for gradients of the phase E = 1 2 n h 2 s ( ) 2 d d x m Stiffness constant K is written as n s ( h 2 /m) and n s is called the superfluid density Stiffness constant not the same as the condensate density n s = n 0 Conjugate variable to the phase is the number of particles N

46 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Broken Phase (Gauge) Symmetry Any phase gives an equivalent state; the ordered state is characterized by a particular phase There is an energy cost for gradients of the phase E = 1 2 n h 2 s ( ) 2 d d x m Stiffness constant K is written as n s ( h 2 /m) and n s is called the superfluid density Stiffness constant not the same as the condensate density n s = n 0 Conjugate variable to the phase is the number of particles N Currents and dynamics of the phase are coupled to the density, i.e., mass or electric currents

47 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Broken Phase (Gauge) Symmetry Any phase gives an equivalent state; the ordered state is characterized by a particular phase There is an energy cost for gradients of the phase E = 1 2 n h 2 s ( ) 2 d d x m Stiffness constant K is written as n s ( h 2 /m) and n s is called the superfluid density Stiffness constant not the same as the condensate density n s = n 0 Conjugate variable to the phase is the number of particles N Currents and dynamics of the phase are coupled to the density, i.e., mass or electric currents Currents are present in equilibrium, and so are supercurrents

48 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Supercurrents by Analogy One-to-one correspondence at the quantum operator level hn S z and phase spin (e.g., particle, no particle)

49 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Supercurrents by Analogy One-to-one correspondence at the quantum operator level hn S z and phase spin (e.g., particle, no particle) Gradient of the phase gives a flow of particles n t = j with j = n s( h/m)

50 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Supercurrents by Analogy One-to-one correspondence at the quantum operator level hn S z and phase spin (e.g., particle, no particle) Gradient of the phase gives a flow of particles n t = j with j = n s( h/m) Often associate a flow with a velocity: introduce superfluid velocity v s = ( h/m)

51 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Supercurrents by Analogy One-to-one correspondence at the quantum operator level hn S z and phase spin (e.g., particle, no particle) Gradient of the phase gives a flow of particles n t = j with j = n s( h/m) Often associate a flow with a velocity: introduce superfluid velocity v s = ( h/m) and then j = n s v s

52 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Supercurrents by Analogy One-to-one correspondence at the quantum operator level hn S z and phase spin (e.g., particle, no particle) Gradient of the phase gives a flow of particles n t = j with j = n s( h/m) Often associate a flow with a velocity: introduce superfluid velocity v s = ( h/m) and then j = n s v s Or write in terms of flow of mass ρ t = g with g = ρ sv s,ρ s = mn s

53 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Hydrodynamic Derivation Free energy expression: generalized rigidity and energy in external potential V (K is bulk modulus) f = h 2 n s 2m ( ) Kn2 + Vn

54 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Hydrodynamic Derivation Free energy expression: generalized rigidity and energy in external potential V (K is bulk modulus) f = h 2 n s 2m ( ) Kn2 + Vn Equilibrium phase dynamics (Larmor precession theorem) from dynamics with added constant potential δv : (V, t) = (0, t)e i hnδvt gives h = δv or in general h = ( ) f n T = µ

55 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Hydrodynamic Derivation Free energy expression: generalized rigidity and energy in external potential V (K is bulk modulus) f = h 2 n s 2m ( ) Kn2 + Vn Equilibrium phase dynamics (Larmor precession theorem) from dynamics with added constant potential δv : (V, t) = (0, t)e i hnδvt gives ( ) f h = δv or in general h = n T Entropy production argument from the thermodynamic identity = µ dε = Tds+ µdn + d( ) with = ( h 2 n s /m) gives the current of particles ṅ = j with j = n s ( h/m)

56 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Currents that Flow Forever Θ Θ(r) L g = ρ s ( h/m)(π/l) g = ρ s ( h/m)(2π/l) vs dl = h m quantum of circulation

57 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Josephson Effect y x 1 2 Energy depends on phase difference. For weak coupling E = J c cos( 2 1 ) Change in number of particles: current I = dn 2 /dt I = de d 2 d.c. Josephson effect I = J c sin( 2 1 ) Time dependence of phase is given by the potential h i = µ i a.c. Josephson effect h( 2 1 ) = µ

58 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Breakdown of Superfluidity n µ = s T + P v s h = µ, v s = ( h/m)n s /L pressure or temperature difference accelerates superflow constant superflow does not require pressure drop

59 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Breakdown of Superfluidity n µ = s T + P v s h = µ, v s = ( h/m)n s /L pressure or temperature difference accelerates superflow constant superflow does not require pressure drop n µ = s T + P ÿ v s pressure drop (dissipation) requires passage of vortex topological defects ( quantized vortex lines ) across flow channel presence of dissipation depends on whether vortices can be produced by thermal activation or other mechanism

60 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Josephson Effect for a Superconductor is phase of pair wave function expressions must be gauge invariant in presence of vector potential

61 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Josephson Effect for a Superconductor is phase of pair wave function expressions must be gauge invariant in presence of vector potential For bulk material Supercurrent: j = n s h 2m Josephson equation: h = 2eV ( (x)+ 2e ) hc A

62 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Josephson Effect for a Superconductor is phase of pair wave function expressions must be gauge invariant in presence of vector potential For bulk material Supercurrent: j = n s h 2m Josephson equation: h = 2eV ( (x)+ 2e ) hc A For Josephson junction, current is I = j (y, z) dydz with and j (y, z) = j c sin ( e hc V = ( h/2e)( 2 1 ) 2 1 ) A x dx

63 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Josephson Junction in a Magnetic Field y B x 1 2 Josephson current density: j (y, z) = j c sin ( e hc 2 1 ) A x dx

64 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Josephson Junction in a Magnetic Field y B x 1 2 Josephson current density: j (y, z) = j c sin ( e hc For field Bẑ in junction the vector potential is A = Byˆx, so that I dy j c sin[ 2 1 (2e/ hc)byd] 2 1 ) A x dx giving I = I c (B) sin( 2 1 ) with I c (B) = sin(πφ/φ 0) πφ/φ 0 where φ = Bld is the flux through the junction and φ 0 = hc/2e is the flux quantum ( gauss cm 2 )

65 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Experimental Discovery of the dc Josephson Effect Rowell, Phys. Rev. Lett. 11, 200 (1963)

66 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, SQUID Superconducting Quantum Interference Device B δθ 1 δθ 2

67 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, SQUID Superconducting Quantum Interference Device B δθ 1 δθ 2 Integrate j = n s h 2m ( ) (x)+ 2e hc A around whole loop using fact that current j is small δ 1 δ 2 = 2e A dl = 2πφ/φ 0 hc with φ = B area, the flux through the loop.

68 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, SQUID Superconducting Quantum Interference Device B δθ 1 δθ 2 Integrate j = n s h 2m ( ) (x)+ 2e hc A around whole loop using fact that current j is small δ 1 δ 2 = 2e A dl = 2πφ/φ 0 hc with φ = B area, the flux through the loop. Total current I = J c [sin δ 1 + sin δ 2 ] = 2J c sin(πφ/φ 0 ) sin[ 1 2 (δ 1 + δ 2 )] Maximum current varies periodically with applied field very sensitive magnetometer.

69 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Four Sounds in a Superfluid Equations of motion for conserved quantities ρ = g ġ = P ṡ = 0 and the dynamics of the broken symmetry variable h = µ which can be written as ρ v s = s T P

70 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Four Sounds in a Superfluid Equations of motion for conserved quantities ρ = g ġ = P ṡ = 0 and the dynamics of the broken symmetry variable h = µ which can be written as ρ v s = s T P Need to connect the momentum density to the superfluid velocity.

71 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Galilean Invariance

72 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Galilean Invariance Transform to frame with a velocity v n :

73 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Galilean Invariance Transform to frame with a velocity v n : Momentum density g = ρ s v s

74 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Galilean Invariance Transform to frame with a velocity v n : Momentum density g = ρ s v s (0)

75 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Galilean Invariance Transform to frame with a velocity v n : Momentum density g = ρ s v s (0) + ρv n

76 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Galilean Invariance Transform to frame with a velocity v n : Momentum density g = ρ s v s (0) + ρv n Define the normal fluid density ρ n = ρ ρ s and write the transformed superfluid velocity v s = v (0) s + v n g = ρ s v s + ρ n v n

77 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Galilean Invariance Transform to frame with a velocity v n : Momentum density g = ρ s v s (0) + ρv n Define the normal fluid density ρ n = ρ ρ s and write the transformed superfluid velocity v s = v (0) s + v n g = ρ s v s + ρ n v n Entropy current j s = sv n

78 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Galilean Invariance Transform to frame with a velocity v n : Momentum density g = ρ s v s (0) + ρv n Define the normal fluid density ρ n = ρ ρ s and write the transformed superfluid velocity v s = v (0) s + v n g = ρ s v s + ρ n v n Entropy current j s = sv n Momentum equation can be transformed to ρ s v s + ρ n v n = P and using the equation for v s in the form (ρ s + ρ n ) v s = s T P gives ρ n ( v s v n ) = s T

79 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, First Sound Usual coupled density and momentum equations ρ = g ġ = P and the pressure-density relationship (K is the bulk modulus) δ P = K δρ/ρ

80 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, First Sound Usual coupled density and momentum equations ρ = g ġ = P and the pressure-density relationship (K is the bulk modulus) δ P = K δρ/ρ These give first sound waves e i(q r ωt) propagating with the usual sound speed ω = c 1 q with K c 1 = ρ

81 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Second Sound Coupled counterflow and entropy wave. Use c 2 c 1 density constant, g = 0 ρ s v s + ρ n v n = 0 v s v n = (ρ/ρ s )v n (remember ρ s + ρ n = ρ).

82 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Second Sound Coupled counterflow and entropy wave. Use c 2 c 1 density constant, g = 0 ρ s v s + ρ n v n = 0 v s v n = (ρ/ρ s )v n (remember ρ s + ρ n = ρ). Entropy equation: ṡ = s v n

83 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Second Sound Coupled counterflow and entropy wave. Use c 2 c 1 density constant, g = 0 ρ s v s + ρ n v n = 0 v s v n = (ρ/ρ s )v n (remember ρ s + ρ n = ρ). Entropy equation: ṡ = s v n Entropy-temperature relationship (C is the specific heat): δs = CδT/T CṪ = st(ρ s /ρ) (v s v n ) Counterflow equation ρ n ( v s v n ) = s T

84 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Second Sound Coupled counterflow and entropy wave. Use c 2 c 1 density constant, g = 0 ρ s v s + ρ n v n = 0 v s v n = (ρ/ρ s )v n (remember ρ s + ρ n = ρ). Entropy equation: ṡ = s v n Entropy-temperature relationship (C is the specific heat): δs = CδT/T CṪ = st(ρ s /ρ) (v s v n ) Counterflow equation ρ n ( v s v n ) = s T These give propagating second sound waves with the speed ρ s s c 2 = 2 T ρ n ρc

85 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Fourth Sound Fluid confined in porous media: no conserved momentum, no Galilean invariance (no v n ), temperature constant

86 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Fourth Sound Fluid confined in porous media: no conserved momentum, no Galilean invariance (no v n ), temperature constant ρ = g g = ρ s v s ρ v s = P δ P = Kδρ/ρ

87 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Fourth Sound Fluid confined in porous media: no conserved momentum, no Galilean invariance (no v n ), temperature constant ρ = g g = ρ s v s ρ v s = P δ P = K δρ/ρ These gives a fourth sound wave propagating with the speed ρ s K c 4 = ρ ρ

88 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Third Sound Wave propagating in thin films down to atomic layer thickness. Like fourth sound, but involve changes of thickness rather than density, and effective compressibility depends on strength of interaction with surface.

89 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Sounds in Helium-4 first Speed c fourth second c 1 = Temperature T K ρ, c 2 = ρ s s 2 T ρ n ρc, c 4 = ρ s ρ K ρ T c

90 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, Next Lecture Onsager theory and the fluctuation-dissipation theorem Derivation and discussion Application to nanomechanics and biodetectors