260008 VO Advanced Statistical Physics and Soft Matter Physics (2023W)

Non-equilibrium thermodynamics and statistical mechanics

Statistical mechanics has been very successful at describing physical systems at equilibrium. However, in nature and technology many phenomena occur under non-equilibrium conditions and their theoretical treatment is challenging. This course will give an introduction to the theoretical concepts and mathematical tools needed to describe time-dependent irreversible phenomena in systems away from equilibrium. The course will start with a quick reminder of the basic notions of equilibrium thermodynamics and statistical mechanics, although it is assumed that the participants have already taken a class on these topics at the level of T4. We will then introduce the fundamentals of phenomenological irreversible thermodynamics and discuss the macroscopic equations governing the transport of mass, momentum and energy. The main part of the course deals with statistical-mechanical treatment of time-dependent processes on the microscopic and mesoscopic scale. We will learn how linear response theory can be used to understand the behavior of systems close to equilibrium in terms is their equilibrium fluctuations. In subsequent chapters of the course, we will learn about how to describe non-equilibrium phenomena as stochastic processes, both in terms of stochastic differential equations (Langevin equation) and stochastic partial differential equations (Fokker-Planck equation). Recent exact results obtained for systems driven arbitrarily far from equilibrium, such as the Jarzynski equality and the Crooks fluctuation theorem, will be discussed in the chapter on stochastic thermodynamics. Finally, we will examine the basic ideas and results of kinetic theory including the Boltzmann equation and the H-theorem following from it.

The lectures will be complemented by weekly exercise sessions, in which the concepts discussed in class will be applied to solve specific problems either analytically or computationally.

After taking this course, students
- have an overview of the basic ideas and methods of non-equilibrium thermodynamics and statistical mechanics
- understand their range of applicability and know their limitations
- are able to apply the concepts and tools discussed in the course to solve concrete problems
- are prepared to read the current research literature in this field

Tentative table of contents:

1. Equilibrium thermodynamics and statistical mechanics in a nutshell

1.1. Thermodynamics (first and second law, equiibrium conditions)
1.2. Statistical mechanics (ensembles, fluctuations)

2. Non-equilibrium thermodynamics

2.1. Balance equations (entropy production)
2.2. Phenomenological equations and Onsager relations

3. Non-equilibrium statistical mechanics

3.1. Fluctuations and microscopic reversibility
3.2. Microscopic derivation of Onsager relations
3.3. Linear Response theory and transport phenomena (fluctuation-dissipation theorem)
3.4. Brownian motion and Langevin equations
3.5. Fokker-Planck equations
3.6. Master equations
3.7. Stochastic thermodynamics (Jarzynski and Crooks theorems)
3.8. Kinetic theory (Boltzmann equation)