This course is complementary to OIST courses B12 Statistical Physics and A225 Statistical Mechanics, Critical Phenomena and Renormalization Group. The course covers a few important topics from Statistical Physics, particularly, description of systems away from (but close to) the thermal equilibrium.

First, we discuss fluctuations of statistical quantities in a system, mostly at thermal equilibrium, and derive the very important relation (FDT) between fluctuations and dissipation in a dynamic system coupled to a noisy environment. Next, we attempt to provide a description for systems driven out of equilibrium by external forces and derive methods to account for transport of various physical quantities, such as particle’s number, momentum and thermal energy. Finally, we try to extend some of the above ideas to quantum systems, in particular those interacting with an environment, aiming to give a very basic introduction to the theory of dissipative (“open”) quantum systems.

During the course, we aim at developing a good (sometimes intuitive) understating of the physical picture rather than pursuing a rigorous mathematical description of the phenomena. Numerous examples and model problems from solid state and condensed matter physics, atomic physics, quantum optics, etc., will be discussed to illustrate the methods.

1. Statistical Fluctuations and Stochastic Processes (4-5 weeks):

- fluctuations of thermodynamic variables, statistics of fluctuations and probability distributions, correlation and spectral characteristics of a noise function, the Wiener-Khinchin theorem;

- classical system in a noisy environment, the Langevin equation, Brownian motion and diffusion, the Einstein relation, thermal noise, the Nyquist formula, shot noise, response function and Kramers-Kronig relations, the Fluctuation-Dissipation Theorem;

- equations for the probability distribution function, overdamped limit and the Einstein-Smoluchowski equation, the Boltzmann distribution, the Fokker-Plank equation, the Maxwell distribution, the Kramers problem.

2. Elements of Kinetic Theory (4-5 weeks):

- the Liouville theorem, nonequilibrium distribution function, the Boltzmann equation and relaxation-time approximation;

- particle transport and the Drude formula, electrochemical potential, the drift-diffusion equation;

- energy transport and thermal conductivity, the Wiedemann-Franz relation, thermoelectric transport, the Seebeck and Peltier effects, the reciprocal Onsager relations.

3. Introduction to Open Quantum Systems (3-4 weeks):

- density matrix formalism, reduced density matrix, open system dynamics and dephasing;

- quantum system in a noisy environment, the spin-boson model, the Heisenberg-Langevin equation, quantum noise;

- the master equation and Markov approximation, energy relaxation, the optical Bloch equations.

Statistical Physics (B12) or Statistical Mechanics, Critical Phenomena and Renormalization Group (A225); anything equivalent to a basic course on Nonrelativistic Quantum Mechanics