1. Consider a collision-less gas in one dimension. At time
has a zero density at positive x and a constant density n at
negative x. Distribution function of the atoms is Maxwellian with the
temperature T . This gas expands into vacuum. Find its distribution
function, density n(x,t), current j (x,t) and energy density
E (x,t) .
2. A satellite has a shape of a disc and moves in the rare,
collision-less gas. Find density distribution of this gas
in the trace of the satellite. Velocity of the satelite is larger,
than the thermal velocity of the atams.
3. Electrons are interacting with phonons. Write
integral for the electron phonon collisions in the Boltzmann equation
for electrons and for phonon-electron collisions in the equation for
phonons. How the conditions of the detailed balance look like for both
integrals? Which integrals of the motion these detailed balances preserve?
1. Find the equation, which describes the Maxwell
relaxation of an
extra charge in a conducting film in the presence of a perpendicular
2 . Find the frequency dependence of electric conductivity
3. Consider an electron gas with the energy dependent
collision rate .
Calculate the current j in the presence of temperature gradient and electric field
E. Setting the current to zero, find the thermo-power.
4. Consider a gas with binary collisions and find its viscosity
5. Consider a uniform neutral mixture of negative and positive
particles (electrons and holes), colliding with irregularities of the
background and with each other. Finds the conductivity, its dependence
on magnetic field and the Hall resistance.
6. Consider thermo-conductivity of electron gas.Compare it with
electric conductivity and find their ratio. (Wiedemann-Franz Law and the
Lorentz factor) .
1. Find static dielectric function from Eq
(4.10). Find electrostatic potential
of an external charge, located at the origin.
2. Consider a neutral plasma with electron
temperature Te much
lager, than that Ti of the ions ( Te >> Ti ). Find dielectric
function at v(Te) >> vph >> v(Ti) and show that there
is a low lying branch of collective excitations (ionic sound). Find its
3. Consider a Maxwellian plasma and a beam of fast
it. Show that under these conditions plasma waves have a negative
damping at certain wave vectors (the beam instability).
4. Consider a cloud of electrons and ions in a medium. Electrons
diffuse in it with the diffusion coefficient De and ions - with
Di . Find electrostatic potential and effective diffusion coefficient
of whole cloud.
5. Consider sound attenuation in metals.
Lecture 5 - 6
1. Using Fokker-Planck equation, find mobility of a heavy
a gas of light particles.
2. Plasma oscillations with the wave vector q
oscillations of the distribution function f(v t) ~ exp [ -i q v t ] those survive the Landau
damping (see Lectur 4). Assuming that electron-electron collisions are characterized
by the small momentum transfer, find relaxation rate of this oscillations.
1. Calculate conductance of a thin film of thickness d
>> l ( l is
the mean free path of electrons in the material) with
a diffusive scattering of electron on inner boundary of the film (Klaus Fuchs problem ).
2. Consider absorption of infra-red light on a metal
surface at so
high frequency that the quasi-static condition is not valid.
3. Use Pippard's efficiency concept for a qualitative
resonant high frequency field absorption by a metal in magnetic field
parallel to its surface ( V.Heine )
4. Neutrons are diffusing in a nuclear reactor
close to its
plane well. Escaped neutrons are never comming back. Consider kinetic
problem and derive effective boundary condition for the diffusion
equation. ( Milne problem )
1. Consider phonon thermo-conductivity
in a dielectric with
elastic mean free path l being invercely proportional to fourth power of the frequency.
Derive effective equation for spreading a hot spot.
2. Analyse fluctuation of distribution function
in the process of space uniform
1. Factor Lande of electrons in a metal depends on its
position p on the Fermi surface
g( p ) .
Find how the line shape
of electron spin resonance (ESR) depends upon disorder in a metal, that
causes a momentum relaxation.
Problems for whole course
1. Using consept of effectiveness, consider electron spin
conduction electrons under conditions of the anomalous
skin-effect. ( Dyson; Azbel, Gerasimenko and I.Lifshits )
2. Consider sound attenuation in metals in magnetic field
the sound wave vector q is perpendicular to the field direction
( B q = 0 ). Show that the attenuation coefficient exhibits
the so called ``Pippard geometric resonance'', i.e. resonance
enhancemment of attenuation at sound wave lengthes commensurable with
the Larmor radius of electrons.
3. Find high frequency sound attenuation
in the Bose-liquid, keeping
in mind that its dispersion relation in the frame, moving with velocity
v is given by the Galilean law w( q) = s| q| - vs q . ( Andreev - Khalatnikov})
3. Consider oscillations in collision-less Fermi-liquid.
conditions, when longitudinal sound, transverse sound and spin-waves