### 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?