## Statistical mechanics of irreversible processes; 5A1351

Course for F4, 4p.

Course may be given in English.

Course will be given in period 4, spring  2001 if there is sufficient interest.

First lecture, Monday 12 March, 10-12, Q12.

Lecturer: Clas Blomberg, Theoretical Physics, phone. 08-790 7176; email cob@theophys.kth.se

#### Aim

The basic courses in statistical mechanics deal with systems in equilibrium and questions about how the energy in equilibrium is distributed among the molecules in the system, and how thermodynamic properties can be calculated from such knowledge. Here, this is generalized to include fluxes and time processes when one does not have equilibrium. Among other things, we take up questions about irreversible processes when systems approach equilibrium. This is made along two directions. One starts from a molecular description of the interaction between atoms and molecules (the Boltzmann equation) from which we can describe transport properties and fluxes. This provides possibilities to calculate quantities such as viscosity and heat conductivity. The other direction is to start from a statistical view where main aspects of the system are described by random variations, fluctuations describe Time courses of important features are considered as stochastic processes. Important applications are relaxation processes in systems where particles and energies are randomly distributed. Brownian motion is a central concept.

#### Contents

The Boltzmann-equation. Relations between conservation laws and transport equations. Derivation of viscsity and heat conductivity. Electrical conduction.

Brownian motion. Stochastic equations: The Langevin-equation, Master equations, Fokker-Planck-equations. Time series analysis. Wiener-Khinchin's theorem. Fluxes and forces, linear relations. The fluctuation-dissipation theorem. Irreversible thermodynamics.

Fluctuations i nonlinear systems with corresponding problems.

#### Prerequisites

Basic mathematical statistics and statistical mechanics corresponding to that part in quantum physics, 5A1450 and preferably (but not necessary) 5A1350.

#### Course requirements

Solution of home works (INL1;4p).
Exercises (as PostScriptfiles):  Exercise1

#### Literature

Primarily : own material

F. Reif, Fundamentals of statistical and thermal physics, McGrawHill, 1965, chapter 13-15
K. Huang, Statistical Mechanics, John Wiley Paperback, 1987.
N G van Kampen: Stochastic Processes in Physics and Chemistry, North Holland, paperback 1992
(A more extensive presentation of reference litterature will be given at the lectures).

TENTATIVE SCHEDULE FOR THE SECOND PART (spring 2000):

6. Th 30/3 Stochastic description introduction. Examples of physical stochastic processes. Brownian Motion. Liouville Equation. (Lect. Notes 6.)

7. Mo 3/4  Basic features of stochastic processes. Markov processes. Master equations. Examples. (Lect. Notes 7)

8. Th 6/4 Examples, continuation. Methods for solving Master equations. Random walk.
Time series analysis. Spectral analysis. Characterization of noise. (Finish lect. notes 7, lecture notes 8).

9. Mo 10/4 Equations for stochastic processes. The Fokker-Planck equation. Solutions. Brownian motion again. (Lect. Notes 9)

10. Th 13/4 Linear Response. The Fluctuation - Dissipation Theorem. Start non-linear effects. (Lect. Notes 9, finish, 10)

EASTER INTERRUPTION

11. Mo  8/5 Linear response principles revisited. The basics of irreversible thermodynamics. Onsager relations.  Entropy production and entropy flows. Principle of minimum entropy production. (Lect. Notes 11)

12. Th 11/5 Non-Equilibrium Thermodynamics, continuation. Non-linear effects. (Lect. Notes 12, continuation 10).

13. Mo 15/5 Summing up. Actual problems concerning fluctuations and non-linear effects. (Lect. Notes 10 end, extra material).

Home exercises are given every Thursday. They are to be made before a date which will be determined during the course, 2-3 weeks after the course is finished.

For being  approved as graduate student course or grade 5 as 'civilingenjörskurs', you shall try all exercises (you may avoid the most difficult ones), and most (but not necessarily everything) shall be correct.

For grade 3, you should have a little more than half of the exercises correct, an you need not have solutions to all. For grade 4, more shall be correct, but you need not try all exercises.