Physics of non-equlibrium.


Preliminary Plan of the course.

1.       Classical theory

A.      Stochastic processes

B.      Fokker-Planck equation

C.      Liouville equation

D.      Kinetic equation and its application to classical gases.  

E.       Foundation of classical theory.

F.       Quasiclassical approach to transport in metals and insulators

G.     Conductivity, Hall effect and other kinetic coefficients of normal metals.

2.       Modern approach to classical problems.

A.      Path integral formulation of Langevin dynamics.

B.      Nucleation processes in path integral formulation.

3.       Quantum non-equilibrium problems.

A.      Quantization of classical chaos. Level statistics.

B.      Quantum analogue of Boltzman equation: Keldysh formalism.

C.      Simplest example: 1D wire between two reservoirs.

D.      Path integral formulation of Keldysh technique, thermally assisted tunneling.

E.       Zero bias anomaly in dirty metals.

F.       Hydrodynamics of quantum fluids.    

G.     Small metallic grains: Coulomb blockade and Kondo regime.

H.      Coherent states of small superconducting devices.



1. Introduction and the plan of the course.

2. General stochastic processes driven by external noise, Markov processes, master equation. [1]

3. H-theorem for stochastic processes. Langevin equation. [1]

4. Fokker-Planck equation, its equivalence to Langevin dynamics. Derivations of diffusion equation. [1]

5. Iterative solution of Fokker-Planck equation. [1]

6.  Field theoretical approach to Langevin dynamics: path integral. Derivation of the escape (nucleation) rate in path integral formalism. [2]

7. Boltzman kinetic equation for semi-ideal gases. [3]

8. Hydrodynamics of gases as a consequence of Boltzman equation. [3]

9. Kinetic coefficients of semi-ideal gases. Approximation of a single relaxation time and solution in a general case (Chapman method). [3] 

10. Foundations of classical theory: ergodic behavior, approximate integrals of motion. [4]




1.       R. Kubo, M. Toda and N. Hashitsume “Statistical Physics II”

2.       J. Zinn-Justin “Quantum field theory and critical phenomena”.

3.       L. D. Landau and E. M. Lifshitz, “Physical Kinetics”.

4.       V. Arnold “Mathematical methods of classical mechanics”; M. Tabor, “Chaos and Integrability in Nonlinear Dynamics”