PHYSICAL KINETICS

  Lecture 1.

 1. Consider a collision-less gas in one dimension. At time t=0   it
  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 the collision
  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?
 

     Lecture 2.

 1.  Find the equation, which describes the Maxwell relaxation of an
  extra charge in a conducting film in the presence of a perpendicular
  magnetic field.

 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 and
      thermoconductivity.

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) .
 

    Lecture 4.

 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 (  T  >> T). 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
  damping.

 3.  Consider a Maxwellian plasma and a beam of fast electrons in
  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 particle in
  a gas of light particles.

 2.  Plasma oscillations with the wave vector q result in
  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.
 

    Lecture 7.

  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 analysis of
  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 )

   Lecture 8.

   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
  down-conversion  cascade.

    Lectures   9-10.
 
 

     Lectures 11-12

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 resonance on
  conduction electrons under conditions of the anomalous
  skin-effect. (  Dyson; Azbel, Gerasimenko and  I.Lifshits )

  2.  Consider sound attenuation in metals in magnetic field  B   when
  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| -  vq . ( Andreev - Khalatnikov})

 3.  Consider oscillations in collision-less Fermi-liquid. Find
  conditions, when longitudinal sound, transverse sound and spin-waves
  exist.( Landau)