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液体のずれ粘性とボース統計
高知大 国府 俊一郎
cond-mat / 0707.1931
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Two types of diffusion process
(1) Diffusion of matters,heat conduction
= diffusion of a scalarDensity, Temperature
always diffusive !
(2) Viscosity = diffusion of a vectorMomentum
A velocity field lead toa non-dissipative situation !
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Capillary flow Rotational flow
Dissipative flow Non-Dissipative flow
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Shear viscosity coefficient as a linear response
In a gas,
In a liquid, Poiseuille Flow
Reciprocal 1/η is a linear response coefficient
In the Perturbation expansion of 1/η with respect to U,a large U ⇒ a small η
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In a classical fluid
In a superfluid
normal fluid phase, superfluid phase
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Complex conductivity
: Kinetic equation
Generalized susceptibility to the external field v(ω)
Real Part = ω σ2(ω) Non-dissipative response
Rotation !
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Rotation as a response to the perturbation
The external fieldvd = r ×v the velocity field of the rigid-body rotation
The responseJ mass current
J = χ vdSusceptibility χ describes the rotational properties.Non-dissipative Transverse ・Response !
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Poiseuille Flow
σ1: dissipative flow, σ2: rotational flow
Kramers-Kronig relation
Susceptibility of Rotational flow
Conductivity in a capillary flow
Many-body version of
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In the normal liquid phase
In the superfluid phase
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weakly depends on ω than 1/ω at a small ω
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Conductivity ⇒ Shear viscosity
Kinematical shear viscosity
Kinematical superfluid density
Small ωs amplifies its effecton the shear viscosity
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Maxwell’s Relation in
Solid Liquid
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Relaxation time and Bose statistics
In Bose statistics, Displacements are reproduced by short arrows
The excited state lies close to the ground state in configuration space, but wave function oscillates
Spatial gradient in configuration space is large,
the excitation energy is high a small relaxation time
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Shear viscosity of a liquid helium 4In a superfluid, Landau-Khalatonikov model= a dilute Bose gas picture of phonons and rotons
Above the lambda point,
Dissipation mechanismof a liquid
+Influence of
Bose statistics
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The deference between L and T is extracted
The relation between the general and the classical moment of inertia
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Susceptibility in an ideal Bose gas within the linear response
In BEC phase, a macroscopic number of particlesaccumulate only in p=0 or p+q=0 terms.
The difference between both static susceptibilities has 1/q2 singularity.
In the normal phase, such a singularity disappears.BEC is a necessary and sufficient condition in an ideal Bose gas !
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The perturbation expansionof the current-current correlation function
As T approaches TBEC ,the growth of the coherent wave function must gradually include a new effect due to Bose statistics
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Susceptibility < J(p+q)J(p)>surrounded by Bubble excitations in the medium
The exchange of particles with the same momentum
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At a certain temperature T0 (>Tlambda), the denominator of thesusceptibility gets to begin with q2 due the cancellation.
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The cancellation occurs prior to the lamda transition
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Density ofliquid He4
kinematical shear viscosity
Mechanical superfluid density
液体のずれ粘性とボース統計Two types of diffusion processCapillary flow Rotational flow Dissipative flow Non-Dissipative flowShear viscosity coefficient as a linear responseIn a superfluidComplex conductivityRotation as a response to the perturbation Kramers-Kronig relationIn the normal liquid phaseweakly depends on than 1/ at a small Conductivity Shear viscosityMaxwell’s RelationShear viscosity of a liquid helium 4The relation between the general and the classical moment of inertiaSusceptibility in an ideal Bose gas within the linear responseThe exchange of particles with the same momentumAt a certain temperature T0 (>Tlambda), the denominator of the susceptibility gets to begin with q2 due the cancellation. The cancellation occurs prior to the lamda transitionDensity of liquid He4kinematical shear viscosity