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

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 !

Capillary flow Rotational flow

Dissipative flow Non-Dissipative flow

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

In a classical fluid

In a superfluid

normal fluid phase, superfluid phase

Complex conductivity

: Kinetic equation

Generalized susceptibility to the external field v(ω)

Real Part = ω σ2(ω) Non-dissipative response

Rotation !

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 !

Poiseuille Flow

σ1: dissipative flow, σ2: rotational flow

Kramers-Kronig relation

Susceptibility of Rotational flow

Conductivity in a capillary flow

Many-body version of

In the normal liquid phase

In the superfluid phase

weakly depends on ω than 1/ω at a small ω

Conductivity ⇒ Shear viscosity

Kinematical shear viscosity

Kinematical superfluid density

Small ωs amplifies its effecton the shear viscosity

Maxwell’s Relation in

Solid Liquid

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

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

The deference between L and T is extracted

The relation between the general and the classical moment of inertia

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 !

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

Susceptibility < J(p+q)J(p)>surrounded by Bubble excitations in the medium

The exchange of particles with the same momentum

At a certain temperature T0 (>Tlambda), the denominator of thesusceptibility gets to begin with q2 due the cancellation.

The cancellation occurs prior to the lamda transition

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