Leticia F. Cugliandolo (University of Paris VI)

Slow dynamics of quantum and classical systems (15 h)

1. Introduction

Falling out of equilibrium in classical systems: simple examples showing the role played by flat directions in the energy landscape (the oscillator and the scalar field); coarsening: the phenomenon, dynamic scaling and the solution to the O(N) model; nucleation and growth; relaxation in glasses; driven dynamics (rheology, two thermal baths).
Quantum glasses: relaxation, driven dynamics (two thermal baths).
Annealing and the Kibble-Zurek mechanism.

2. Formalism

Classical stochastic Brownian dynamics: Langevin equation derived for an isolated system's with an oscillators thermal bath; Martin-Siggia-Rose path integral.
Quantum mechanics: Green functions for Bosons and Fermions (definitions, conventions); linear response and Kubo formula; Feynman path integral (time-ordered); Schwinger-Keldysh path integrals (symmetric correlations).

3. Environments

Oscillators Feynman-Vernon model and the Caldeira-Leggett localisation; electronic environments; spin baths (short and quick reminder of all of this since it will be covered in more detail by other lecturers).

4. Equilibrium quantum statistical mechanics

Matsubara imaginary time formalism; quenched disorder: replicas for equilibrium dynamics; mean-field models [rotors, p-spin and O(N) models]; second and first order phase transitions, the role of the bath.

5. Equilibrium dynamics, symmetries and fluctuation-dissipation relations

Classical stochastic processes: in Fokker-Planck formalism with operators; in Martin-Siggia-Rose with symmetries.
Quantum mechanical systems: KMS condition, canonical and gran-canonical fluctuation-dissipation theorems; Schwinger-Keldysh field theory; initial conditions and factorisation of the density matrix.

6. Out of equilibrium dynamics

Mean-field models [rotors, p-spin and O(N) models]: how to deal with SU(2) spins?; how to deal with the bath?
Tricks to solve the equations numerically and analytically.
Second and first order dynamic phase transitions.
The role of the bath.
Driven problems.
Effective temperatures; quantum mechanical formalism?