* Francois Petrelis (ENS Paris)*

Many natural phenomena are far from thermodynamic equilibrium and keep
on exchanging matter, energy or information with their surroundings,
producing currents that break time-reversal invariance. Such systems
lie beyond the realm of traditional thermodynamics: the principles of
equilibrium statistical mechanics do not apply to them.

In these lectures, we will study such non-equilibrium systems in
which the competition between fluctuations and deterministic dynamics
leads to original and non-trivial phenomena. For example, we shall
investigate the effect of noise in the vicinity of the onset of an
instability and in particular probe the impact of stochasticity on the
stability and the dynamics of the system.

We will start with a reminder on the tools required to study such
systems: normal form, dynamical systems, Langevin equation.

We will then study the following topics:

• Effect of noise on an instability : the additive noise case. Random jumps between different solutions.

• Effect of noise on an instability : the multiplicative noise case. Parametric stabilization. On-off intermittency.

• Fluctuation driven dynamics. Exit from a potential well Excitability.
Models of Earth magnetic field reversals. Stochastic resonance. Stochastic coherence.

• Anomalous diffusion and fractional Brownian motion. Wiener-Khintchin theorem. 1/f noise.

** Prerequisites:**

Basic statistical physics

**Bibliography:**

1. N. Van Kampen, Stochastic Processes in Physics and Chemistry , (North-Holland, Amsterdam, 2007).

2. P. L. Krapivsky, S. Redner and E. Ben Naim, A Kinetic View of Statistical Physics, (Cambridge University Press, 2010).

**Timing:**
The Course is offered in the first part (september-november) of the M2 year.

It consists of 8 Lectures on Wednesday from 1.45pm to 5.45pm at Sorbonne University, **room 24.34.310**

First lecture on **Wednesday 18th September**

**ECTS Credits:**3

**Hours:** 32 hours.