School on
Non-Equilibrium
Statistical Physics 

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 Andrea Crisanti: Stochastic processes and BEC

Lecture I: What is a stochastic process?
Markov process. Wiener process. Langevin equations. Stochastic integration: Ito and Stratonovich

Lecture II: Path Integral formulation for additive white noise Langevin equations.
Langevin equations. The Martin-Siggia-Rose-Janssen-de Domicis (MSRJD) Action. The Onsager-Machlup-Graham-Bausch-Wegner (OMGBW) Action. Response functions and response fields. Quadratic MSRJD action. Fluctuation-Dissipation Theorem for multiplicative white noise processes.

Lecture III: Quenched disorder average in statics: the replica trick.
Quenched disorder average in dynamics: the Sompolinsky-Zippelius auxiliary fields.

Lecture IV: Bose-Einstein condensation and ensemble equivalence. 

 Leticia Cugliandolo : Relaxation of quantum many-body systems 

Lecture I:   Closed and open systems.
Classical and quantum environments.

Reminder of the classical Langevin and Flokker-Planck equations. A few words about the quantum Lindblad formalism.

Lecture II:  Reduced systems. Correlations and Entanglement. Noise induced phase

transitions.

Lecture III:  Dynamics of closed integrable systems and the GGE hypothesis. A

solvable classical example.

Lecture IV:  Relaxation of glassy systems: classical and quantum aging. The role of

the effective temperature. 

 Jorge Kurchan: Out-of-equilibrium quantum dynamics

Lecture I: Time-evolution of observables in quantum systems.

General review of how quantum evolution works.

Observables, correlations, initial conditions.

Lecture II: Quantum fluctuation-dissipation theorem, KMS formalism, Quantum

Lyapunov exponents and bounds. 

Quantum and classical Fluctuation-Dissipation theorems.

Quantum and classical Lyapunov exponents as a measure of chaos.

A note on the quantum bound

Lecture III: Eigenstate thermalization hypothesis (ETH)

The consequences of chaoticity: the Eigenstate Thermalization Hypothesis.
A plausible `derivation'.

Lecture IV: Full eigenstate thermalization hypothesis and free probabilities.

ETH and applied to Lyapunov exponent: what is the problem?

Introducing the `Full' ETH. A notion of Free Cumulant expansions.

 Luca Peliti:  Stochastic Thermodynamics and Thermodynamics of Information

Lecture I: Motivation, basics

Stochastic Thermodynamics (ST) is introduced and its relation with Statistical Mechanics on the one side and Thermodynamics on the other is summarized. The stochastic dynamics describing the behavior of mesoscopic systems is introduced and the requirements allowing for its compatibility with thermodynamics are described.

Lecture II: Fluctuation relations and their uses

The fundamental relation between on the one side the ratio of the probability of a trajectory and that of its inverse and on the other the entropy production along the same trajectory is derived. We show how this relation can be put to advantage to obtain equilibrium properties in out-of-equilibrium experiments in mesoscopic systems.

Lecture III: Thermodynamics of Information

Some classical arguments highlighting the role of information in thermodynamics (like Maxwell's and Szilard's "demons") are recalled. By applying the fluctuation relation to systems with feedback these arguments are recast within ST. Some applications and experiments are discussed.

Lecture IV: ST in biological systems

The general relevance of ST for the biological machinery is discussed, in particular in the context of molecular motors and of the handling of biological information, e.g., in protein synthesis and in biopolymer copying.

 Angelo Vulpiani: Chaos and entropy production

Lecture I: Introduction to chaos:

Sensitive dependence from the initial condition, Lyapunov exponents, invariant measure, ergodicity, mixing.

Lecture II: Standard and anomalous diffusion:

Transport phenomena, asymptotic behaviour, effective diffusion coefficients,

diffusion in shear flows (Taylor, Zeldovich), anomalous diffusion.

Lecture III: Some applications of the information theory:

The uniqueness theorem for the entropy, Shannon-McMillan theorem, ϵ-entropy, connection with chaotic systems.

Lecture IV: Irreversibility and entropy production:

Mixing, irreversibility and typicality (Ehrenfest model), entropy production.

Organisers