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.