Slow and glassy out of equilibrium dynamics I

In these two lectures I will present and discuss two different and paradigmatic physical mechanisms inducing slow and glassy out of equilibrium dynamics. Both are due to a very rugged energy landscape. The first corresponds to a never-ending descent within the energy landscape towards regions of the configuration space that are flatter and flatter. The second corresponds to activated dynamics across increasingly higher energy barriers. Both mechanisms will be first introduced from the energy landscape perspective but then also discussed in terms of collective motion taking place in real space.

Some references for the lectures
A crash course in ageing, G Biroli J. Stat. Mech. (2005) P05014 Aging in the random energy model, Ben Arous Bovier Gayrard, PRL 88 087201 Overview of different characterisation of dynamical heterogeneity, Berthier Biroli Bouchaud Jack http://arxiv.org/abs/1009.4765

Current fluctuations and symmetries I

Driven non-equilibrium steady states are characterized by fluxes of particles or energy. Here I will focus on currents in interacting particle systems and explain how their fluctuations can generically be expressed within a large deviation framework. I will demonstrate the formalism by calculating rate functions for toy examples and reviewing known results for various models. In particular, I will discuss the applicability of \"fluctuation relations\", well-known symmetries determining the ratio of the probabilities of positive fluctuations to negative ones. Finally, I hope to touch on some recent developments concerning fluctuations in higher dimensions and systems with memory.

Energy-exchange stochastic models in non-equilibrium

After an introduction to the problem of anomalous transport in non-equilibrium steady states, we will review a large class of interacting particle systems modeling energy-exchange. This class, that has the potential to unveil new properties of non-equilibrium, is amenable to a rigorous analysis of the stationary measure, especially correlation functions, using duality theory. We will discuss both the standard setting of density reservoirs, as well as the setting of current reservoirs, where a connection to free-boundary problems appear in the hydrodynamic limit and a new multi-scale phenomena occur on a super-hydrodynamic time scale

Extreme value statistics of correlated random variables I

Extreme value statistics (EVS) concerns the study of the statistics of the maximum or the minimum of a set of random variables. This is an important problem for any time-series and has applications in climate, finance, sports, all the way to physics of disordered systems where one is interested in the statistics of the ground state energy. While the EVS of uncorrelated\' variables are well understood, little is known for strongly correlated random variables. Only recently this subject has gained much importance both in statistical physics and in probability theory. In this tutorial, I will first review the classical EVS for uncorrelated variables and discuss few examples of correlated variables where analytical progress can be made.

Current fluctuations and symmetries II

Driven non-equilibrium steady states are characterized by fluxes of particles or energy. Here I will focus on currents in interacting particle systems and explain how their fluctuations can generically be expressed within a large deviation framework. I will demonstrate the formalism by calculating rate functions for toy examples and reviewing known results for various models. In particular, I will discuss the applicability of "fluctuation relations", well-known symmetries determining the ratio of the probabilities of positive fluctuations to negative ones. Finally, I hope to touch on some recent developments concerning fluctuations in higher dimensions and systems with memory

Energy-exchange stochastic models in non-equilibrium II

After an introduction to the problem of anomalous transport in non-equilibrium steady states, we will review a large class of interacting particle systems modeling energy-exchange. This class, that has the potential to unveil new properties of non-equilibrium, is amenable to a rigorous analysis of the stationary measure, especially correlation functions, using duality theory. We will discuss both the standard setting of density reservoirs, as well as the setting of current reservoirs, where a connection to free-boundary problems appear in the hydrodynamic limit and a new multi-scale phenomena occur on a super-hydrodynamic time scale.

Slow and glassy out of equilibrium dynamics II

In these two lectures I will present and discuss two different and paradigmatic physical mechanisms inducing slow and glassy out of equilibrium dynamics. Both are due to a very rugged energy landscape. The first corresponds to a never-ending descent within the energy landscape towards regions of the configuration space that are flatter and flatter. The second corresponds to activated dynamics across increasingly higher energy barriers. Both mechanisms will be first introduced from the energy landscape perspective but then also discussed in terms of collective motion taking place in real space

Long-Range correlations in driven, non-equilibrium systems I

Systems driven out of thermal equilibrium often reach a steady state which under generic conditions exhibits long-range correlations. As a result these systems sometimes share some common features with equilibrium systems with long-range interactions, such as the existence of long range-order and spontaneous symmetry breaking in one dimension, ensemble inequivalence and other properties. Some models of driven systems will be presented, and features resulting from the existence of long-range correlations will be discussed.

Introduction to Synchronization I

1) Synchronization: Phenomenology and introduction to the theory First, I describe different phenomena of synchronization with examples from physical and life sciences. Next, elements of the theory for periodic, chaotic, and noisy oscillators are presented. 2) T heory of synchronization in oscillator ensembles I present theoretical tools describing synchronization in populations of oscillators as a nonequilibrium phase transition. The main emphasis is on the dynamics of global variables. Approaches of Watanabe-Strogatz and Ott-Antonsen will be described in details.

Extreme value statistics of correlated random variables II

Extreme value statistics (EVS) concerns the study of the statistics of the maximum or the minimum of a set of random variables. This is an important problem for any time-series and has applications in climate, finance, sports, all the way to physics of disordered systems where one is interested in the statistics of the ground state energy. While the EVS of uncorrelated' variables are well understood, little is known for strongly correlated random variables. Only recently this subject has gained much importance both in statistical physics and in probability theory. In this tutorial, I will first review the classical EVS for uncorrelated variables and discuss few examples of correlated variables where analytical progress can be made.

A sandpile model for proportionate growth I

An interesting feature of growth in animals is that different parts of the body grow at approximately the same rate. This property is called proportionate growth. I will discuss my recent work with T. Sadhu on patterns formed by adding N grains at a single site in the abelian sandpile model. These simple models show very intricate patterns, show proportionate growth, and sometimes having a striking resemblance to natural forms. I will give several examples of such patterns, and discuss the special cases where the asymptotic pattern can be determined exactly. Proportionate growth persists even in the presence of noise in the background.

Long-Range correlations in driven, non-equilibrium systems II

Systems driven out of thermal equilibrium often reach a steady state which under generic conditions exhibits long-range correlations. As a result these systems sometimes share some common features with equilibrium systems with long-range interactions, such as the existence of long range-order and spontaneous symmetry breaking in one dimension, ensemble inequivalence and other properties. Some models of driven systems will be presented, and features resulting from the existence of long-range correlations will be discussed.

Introduction to Synchronization II

1) Synchronization: Phenomenology and introduction to the theory First, I describe different phenomena of synchronization with examples from physical and life sciences. Next, elements of the theory for periodic, chaotic, and noisy oscillators are presented. 2) T heory of synchronization in oscillator ensembles I present theoretical tools describing synchronization in populations of oscillators as a nonequilibrium phase transition. The main emphasis is on the dynamics of global variables. Approaches of Watanabe-Strogatz and Ott-Antonsen will be described in details.

A sandpile model for proportionate growth II

An interesting feature of growth in animals is that different parts of the body grow at approximately the same rate. This property is called proportionate growth. I will discuss my recent work with T. Sadhu on patterns formed by adding N grains at a single site in the abelian sandpile model. These simple models show very intricate patterns, show proportionate growth, and sometimes having a striking resemblance to natural forms. I will give several examples of such patterns, and discuss the special cases where the asymptotic pattern can be determined exactly. Proportionate growth persists even in the presence of noise in the background.