Marder M.P. Condensed Matter Physics

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Problems

443

superfluidity, while Osheroff (1997), Richardson (1997), Lee (1997), and Leggett

(2006) provide more recent overviews.

Problems

Viscosity obtained from kinetic theory: Consider a shear flow where dv

/dy

—

v'. Viscosity may be calculated by drawing a plane at y = 0 in Figure 15.1

and finding the rate σ^ at which momentum p

in the x direction travels across

it.

(a) Argue that

σ^ = -(ρ

δ{?)) . (15.129)

(b) The average is not an equilibrium average, because the fluid is flowing, and

therefore is not in equilibrium, and Eq. (15.129) would vanish in equilibrium.

The Boltzmann equation, derived in Section 17.2, deals with this problem by

defining g(7, v, t) to be the normalized probability that a particle has position

r, and velocity v at time /. The time evolution of

is given by

dg d _ g-g

-^τ = --^-

8 -> (15.130)

dt or T

where r is a characteristic time for a nonequilibrium state to approach equi-

librium once driving forces are removed, and g

is the distribution function

*"·

(Μ)=

^2^

eXP

3 Γ

n.. ../i2 , „.2 , „.2

{[v

-v'yY + vf,+v

) (15.131)

Here n is density, and v'

—

/dy. At every point in space, this distribution

takes the form that equilibrium would demand for the local conditions. So

Boltzmann's equation says that the system always tends to drift toward local

equilibrium. Show that the solution of

Eq.

(15.130) is

* —

y/)/v

■

(15.132)

VyT

θχγ = rirkeTv', (15.133)

and that the viscosity can therefore be written as

η=^ηηιΙτν, (15.134)

where v is the root mean square velocity in equilibrium at temperature T, and

is the mean free path.

444

Chapter 15. Fluid Mechanics

Flow resistance of a sphere: Suppose that a sphere of radius R moves at ve-

locity

= uz in a viscous incompressible liquid. The velocity u is sufficiently

slow that the nonlinear term (v

■

V)v is to be neglected everywhere. The goal

is to calculate the force that must be applied to the sphere to keep it moving

at constant velocity, as originally determined by Stokes (1851).

Move to a reference frame traveling with the sphere; the sphere is stationary,

and the fluid moves at velocity — u far away. The flow pattern is steady in this

reference frame and obeys

v = VP. (15.135)

(a) Because the liquid is incompressible, V

■

v = 0. Therefore, v results from the

curl of a vector potential A, v = V x A. Move to spherical polar coordinates.

Assume that the solution v(r, θ, φ) is independent of φ and that

νψ

= 0. Show

that only

Αψ

needs to be considered, and express v in terms of Αψ.

(b) From Eq. (15.135) it follows that

V x V

t5 = 0.

(15.136)

Show that

i d

rdr

]_d_

sin

= 0.

(15.137)

Αψ

ex r sin

Θ,

assume that

= f(r) sin

Θ.

Show that

]_c?_

_2_

rdr

Ar)=0.

(15.138)

(15.139)

(d) Show that

with

)=Ar

+ßr+^+D,

(15.140)

A = 0, B = -u/2, C = -R

u/4, and D = 3Ru/4.

(15.141)

(e) Show that the pressure is given by

3 R

P = PQ~\ uri-x COS Θ. Po is a constant background pressure. (15.142)

2 r

Problems

445

(f) The force on the sphere is given by integrating the tangential component of

the stress tensor in an appropriate fashion over its surface, including contribu-

tions both from pressure and viscosity. Expressing the relevant components

of Eq. (15.13) in spherical coordinates as

σ„ = 2η^-Ρ (15.143)

fìdvr Ove ν

= η

-θθ

^-7)

(15

144)

show that the downward force F on the sphere is

F = 6^Ru (15.145)

Vortex I: The vorticity

of a flow is defined to be ώ = V x v. A vortex is a

swirling flow pattern containing a large amount of vorticity in its core.

(a) Show that the vorticity of a two-dimensional flow v living in the x-y plane

points entirely in the z direction.

(b) Consider the flow v that in cylindrical coordinates (r, φ) has the form

= 0, ν

= -. (15.146)

Show that the vorticity of this flow vanishes.

origin is constant, and find the constant.

(d) Using Stokes' theorem, relate this integral to an integral over ω, showing

that the vorticity has a delta function singularity at the origin.

Vortex II: It is possible to construct solutions of Euler's equation that behave

as a vortex but without any singularities in the flow field. Thus the previous

problem provides a rough description of a vortex seen from far away, while

this one focuses upon a possible structure for the core.

(a) By taking the curl of Euler's equation, show that the vorticity obeys

-£- + (ΰ·ν)ώ = (ώ·ν)ΰ. (15.147)

(b) Consider a time independent, incompressible, two-dimensional flow v. Ar-

gue that because the flow is incompressible, there exists a stream function ψ

with

-ΤΓ'ΤΓ)-

(15

148)

oy ox

in favor of the stream function ψ.

446

Chapter 15. Fluid Mechanics

(d) Show that the preceding equation is satisfied whenever ω

f{\b),

where /

is an arbitrary function, and rewrite this condition entirely in terms of φ.

(e) Construct a vortex by moving to cylindrical coordinates and by requiring for

r < a:

= —1<

'φ. ί is a constant. (15.149)

Find a solution of Eq. (15.149) that is proportional to sin φ and to a Bessel

function of kr.

(f) For r > a require

φ= (r-—

)t/sin0.

(15.150)

Find the conditions at r = a that make velocity and vorticity continuous, and

employ these conditions to complete the solution for the flow field near the

center of a vortex.

Chain dynamics:

(a) Using Eq. (15.26), verify Eq. (15.35).

(b) Verify Eq. (15.40).

6. Frequency dependence of viscosity: Assume that

Wjy

in Eq. (15.38) varies

cos ujt. Show that the shear stress is given by Eq. (15.49).

Weakly interacting Bose gas: Bogoliubov (1947) treated the problem of

superfluidity in a fashion quite similar to his method for superconductivity,

presented in Section 27.3.4. Consider the Hamiltonian for Bose particles

= Σ

?4**

+ 2V

Σ 4

-A+f~

' (15-151)

Ά kk'q

e^ = H

/2m is the kinetic energy of a noninteracting Bose particle, and U

has dimensions of energy times an interaction volume and will be assumed to

be very small. In the ground state, a macroscopic number of particles inhabits

the single-particle state q = 0. The occupation number of q = 0 should be

very high for low-lying excited states as well.

Whenever

âo

act on a state, they will change the occupation number of

the q = 0 state, it is true, but the change will be negligible compared with the

particles already in this state. Therefore, it should be accurate to replace

âo and â

by y/Nô whenever they appear. Furthermore make the mean-field

approximation of excluding from the interaction term of

Eq.

(15.151) all terms

except those where two or more of the indices equal zero.

(a) Show that the Hamiltonian becomes

= Σ

<?4^

^W + ^W Y^âl^ +

â-qâq

+ Aâ^]. (15.152)

q q-φθ

References

447

(b) Using the fact that the total number of particles

Ν =

+ Σ

'ί (15.153)

=Σ>*+^]φ?+^ Σ[^4+«-?«?]·

(15

154)

is a conserved quantity, rewrite Eq. (15.152), up to a constant, as

NU,^„ NU

^ =

Μ<?7<?

+ V^-? (15.155a)

\ =

ql\ + Vql-q- (15.155b)

Show that for 7^ to preserve the Bose commutation relations of

Eq.

(C.5), one

must have ui — v~= 1.

(d) Set out to diagonalize the Hamiltonian. The task can be accomplished if

Uq =

~q and vq = ν^ξ, so assume these relations. Putting Eqs. (15.155) into

(15.154),

find the condition on

iig-

and v$ needed to ensure that the coeffi-

cient of %T-3 vanishes. Imposing this condition is enough to diagonalize the

Hamiltonian.

(e) Note for later reference that u^v^ must be less than zero. Solve for vl and w|.

(f) Insert the expressions for

and u^ into Eq. (15.154), remembering that

q < 0, putting the Hamiltonian into the form

^j7? +

const

· (15.156)

(g) In the limit q —» 0, £^ =

Hcq.

Thus even weak interactions are able to change

the low-energy excitations from free particle form to sound. Find the sound

speed c.

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Part IV

ELECTRON TRANSPORT

Condensed Matter

Physics,

Second Edition

by Michael P. Marder