Advanced Statistical Mechanics

Classical statistical physics deals with systems at equilibrium or, in some of its ramifications, with steady-states or short-time processes for which memory effects are not important. The majority of systems of interest for modern physics, however, evolve as a function of time far away from equilibrium or any other steady state. Our universe, but also any biological living system, are just some well known examples of systems driven out of equilibrium. Also, the time-dependence of any solid or liquid material which adjusts, relaxes or transforms under applied external influences, is a problem of central importance in condensed matter and materials science. The standard tools of statistical mechanics and condensed matter theory provide no answer to the many questions arising from the observation of these systems.

This course will provide an introduction to modern theoretical tools which allow one to describe the time-dependent relaxation of strongly-interacting many-body systems on their way to equilibrium. First the language of correlation functions, and especially of time-correlation functions will be introduced. The basic tools and concepts of liquid dynamics across the length and time scales, from molecular to hydrodynamics, will then be presented, together with a modern account of linear response theory. A key problem in this context is to account for the non-Markovian time-propagation of earlier disturbances affecting the later evolution of the system, a phenomoneon with profound implications in biophysics, chemical physics, and consensed matter. To this aim, the formalism of memory functions and Zwanzig-Mori projector operators will be explained which is the only theoretical tool available in order to tackle these problems. Applications of this formalism will be discussed, starting with the simple example of Brownian motion, and progressing towards more advanced topics such as the mode-coupling theory of relaxation in liquids and supercooled liquids (glasses). Finally, the long standing paradox of irreversibility and the arrow of time will be addressed from the perspective of the Zwanzig formalism.

The successful students will be able to understand and use theoretical techniques for the analysis and description of dynamics in many-body systems across different fields. They will be able to set up, solve and analyse theoretical models of relaxation dynamics in chemical and biophysical system. They will be able to develop mathematical descriptions of materials evolving as a function of time under applied external disturbances. They will be able to understand and analyse scattering techniques based on the interaction between radiation and a strongly-interacting many-body system, as well as the vibrational relaxation spectrum of liquids.

Knoledge of equilibrium statistical mechanics and quantum mechanics.

There will be a series of excercises and problems involving calculations to illustrate case studies and applications of the methods illustrated in the lectures.

Blackboard

Chaikin & Lubensky "Principles of Condensed Matter Physics"

Hansen & MacDonald "Theory of Simple Liquids"

Hansen & Barrat "Basic concepts of simple and complex fluids"

Goetze "Complex Dynamics of Glass-forming Liquids"

In a written exam the learning outcome is tested using comprehension questions and sample problems.

In accordance with §12 (8) APSO the exam can be done as an oral exam. In this case the time duration is 25 minutes.