Equilibrium statistical physics
Topics planned :
1. Microscopic approach to statistical mechanics via ergodicity theory. Ergodic and central limit theorems for systems in equilibrium. Steady non-equilibrium states as represented by phase space density concentrated on a multifractal attractor.
2. Derivation of reduced descriptions by elimination of fast degrees of freedom and irreversibility. Slow variables view of thermodynamics and some "proofs" of the second law.
3. Brownian motion and diffusion. Einstein-Stokes relation as an example of fluctuation-dissipation theorem. Fluctuations near equlibrium and Langevin equation.
4. General Langevin and Fokker-Planck equations. Growth of entropy. Functional integral formulation and Nelson formulation of quantum mechanics.
5. Kinetic description of dilute gases. Boltzmann equation and H-theorem. Some applications of Boltzmann equation.
6. Hydrodynamics and hydrodynamic modes. Hierarchy of descriptions of a dilute gas and derivation of hydrodynamics from Boltzmann equation. Generalized hydrodynamic modes.
7. Stochastic modeling.
The students will learn fundamental characteristics of non-equilibrium steady states of an open system using a framework that allows making calculations. They will acquire tools that are indispensable in studies of complex systems, most of all stochastic differential equations containing random forces. The crucial role of time scale separation in passing between different levels of description of the same system will be stressed, developing a way of thinking of how closed frameworks arise at different levels, as in the case of the gas that can be described mechanically, via Boltzmann equation, via compressible hydrodynamics and finally via incompressible hydrodynamics.
Landau and Lifshitz, Statistical Physics and Kinetic Theory
Ma Statistical Physics
Risken "Fokker-Planck equation"
More will be provided