# Theory and numerical simulation of mass, charge, and heat transport in condensed matter

### Advanced course given by Stefano Baroni, Federico Grasselli

**Term: **0

**Start:** 20-04-2020
**End:** 15-05-2020
**Room:** zoom

**Credits: **3

**Schedule:** TBA

## Program

**Lectures 1-2: Introduction**

- What diffusion is all about;
- Extensive variables, intensive variables, thermal equilibrium;
- Conserved densities and currents: continuity equation;
- Osmotic forces, Fick's law, and Einstein's relation between friction and diffusion coefficients;
- A mathematical intermezzo: stochastic processes and related concepts;
- Brownian motion.

**Lecture 2-3: Theoretical foundations**

- Hydrodynamic variables and Onsager's transport equations;
- Hamilton's and Liouville's equations of classical mechanics;
- Mixing, ergodicity, and the spectral properties of the Liouvillian
- Linear-response theory and the Green-Kubo and Einstein-Helfand expressions for transport coefficients.

**Lecture 3-4: Heat transport**

- Thermal
*vs.*mechanical perturbations: heat transport; - The classical formula for the heat current;
- Shear viscosity;
- Gauge invariance of transport coefficients;
- Quasi-harmonic theory of heat transport in crystals and glasses;
- Density-functional theory of adiabatic heat transport.

**Lecture 4-5: Adiabatic charge transport**

- Adiabatic charge current and Born's effective charges;
- Thouless' and Resta's topological quantization of adiabatic charge transport;
- Atomic oxidation numbers as topological charges;
- Gauge invariance and topological quantization of adiabatic charge transport.

**Lectures 5-6: Computer simulations**

- Computer simulations: classical and
*ab initio*molecular dynamics; - Data analysis for equilibrium properties: the correlation time;
- Computer simulation of transport phenomena;
- Spectral theory of time series: the Wiener-Khintchine theorem;
- Cepstral analysis for uni- and multi-variate time series.