# Statistical Fluctuations and Elements of Physical Kinetics

Course Description

Explore and explain key ideas of physical kinetics, for systems both at equilibrium and then driven out of equilibrium by a variety of factors. Derive the very important relation (FDT) between fluctuations and dissipation in a dynamic system coupled to a noisy environment. Describe (within certain approximations) the dynamics of classical systems driven out of equilibrium. Apply models and equations to quantify the transport properties of some idealized solid-state and condensed-matter systems. Extend some of these ideas to quantum systems, in particular those interacting with an environment, and explore the dynamics of dissipative ("open") quantum systems. Develop an intuitive understating of the physical picture rather than pursuing a rigorous mathematical description of the phenomena with numerous examples and model problems from solid state and condensed matter physics, atomic physics, and quantum optics, and reinforce these with regular problem sets.

Course Contents

Weekly schedule will be decided based on the progress in understanding of the material covered throughout the course. A (tentative) content of the proposed course is outlined below.

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.

Assessment

Lecture attendance (25%);
Homework (50%);
Midterm and final exams (25%)

Prerequisites or Prior Knowledge

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

Textbooks

Any textbook on Statistical Physics, e.g.
F. Reif , Fundamentals of Statistical and Thermal Physics, 2009 Waveland Press

Reference Books

1. Landau and Lifshits, Statistical Physics, Vol. V;
2. Landau and Lifshitz, Physical Kinetics, Vol. X;
3. Coffey and Kalmykov, The Langevin Equation, 2017 World Scientific Publishing;
4. Blum, Density Matrix: Theory and Applications, 2012 Springer