Abstract:
These lecture notes give a short review of methods such as the matrix ansatz, the additivity principle or the macroscopic fluctuation theory, developed recently in the theory of non-equilibrium phenomena. They show how these methods allow to calculate the fluctuations and large deviations of the density and of the current in non-equilibrium steady states of systems like exclusion processes. The properties of these fluctuations and large deviation functions in non-equilibrium steady states (for example non-Gaussian fluctuations of density or non-convexity of the large deviation function which generalizes the notion of free energy) are compared with those of systems at equilibrium.

Abstract:
We show how certain properties of the Anderson model on a tree are related to the solutions of a non-linear integral equation. Whether the wave function is extended or localized, for example, corresponds to whether or not the equation has a complex solution. We show how the equation can be solved in a weak disorder expansion. We find that, for small disorder strength $\lambda$, there is an energy $E_c(\lambda )$ above which the density of states and the conducting properties vanish to all orders in perturbation theory. We compute perturbatively the position of the line $E_c(\lambda )$ which begins, in the limit of zero disorder, at the band edge of the pure system. Inside the band of the pure system the density of states and conducting properties can be computed perturbatively. This expansion breaks down near $E_c(\lambda )$ because of small denominators. We show how it can be resummed by choosing the appropriate scaling of the energy. For energies greater than $E_c(\lambda )$ we show that non-perturbative effects contribute to the density of states but have been unable tell whether they also contribute to the conducting properties.

Abstract:
We show how to apply the macroscopic fluctuation theory (MFT) of Bertini, De Sole, Gabrielli, Jona-Lasinio, and Landim to study the current fluctuations of diffusive systems with a step initial condition. We argue that one has to distinguish between two ways of averaging (the annealed and the quenched cases) depending on whether we let the initial condition fluctuate or not. Although the initial condition is not a steady state, the distribution of the current satisfies a symmetry very reminiscent of the fluctuation theorem. We show how the equations of the MFT can be solved in the case of non-interacting particles. The symmetry of these equations can be used to deduce the distribution of the current for several other models, from its knowledge for the symmetric simple exclusion process. In the range where the integrated current $Q_t \sim \sqrt{t}$, we show that the non-Gaussian decay $\exp [- Q_t^3/t]$ of the distribution of $Q_t$ is generic.

Abstract:
The ABC model is a simple diffusive one-dimensional non-equilibrium system which exhibits a phase transition. Here we show that the cumulants of the currents of particles through the system become singular near the phase transition. At the transition, they exhibit an anomalous dependence on the system size (an anomalous Fourier's law). An effective theory for the dynamics of the single mode which becomes unstable at the transition allows one to predict this anomalous scaling.

Abstract:
We give numerical evidence that the location of the first order phase transition between the low and the high density phases of the one dimensional asymmetric simple exclusion process with open boundaries becomes sample dependent when quenched disorder is introduced for the hopping rates.

Abstract:
We formulate a simple additivity principle allowing to calculate the whole distribution of current fluctuations through a large one dimensional system in contact with two reservoirs at unequal densities from the knowledge of its first two cumulants. This distribution (which in general is non-Gaussian) satisfies the Gallavotti-Cohen symmetry and generalizes the one predicted recently for the symmetric simple exclusion process. The additivity principle can be used to study more complex diffusive networks including loops.

Abstract:
We consider diffusive lattice gases on a ring and analyze the stability of their density profiles conditionally to a current deviation. Depending on the current, one observes a phase transition between a regime where the density remains constant and another regime where the density becomes time dependent. Numerical data confirm this phase transition. This time dependent profile persists in the large drift limit and allows one to understand on physical grounds the results obtained earlier for the totally asymmetric exclusion process on a ring.

Abstract:
Using a generalisation of the detailed balance for systems maintained out of equilibrium by contact with 2 reservoirs at unequal temperatures or at unequal densities, we recover the fluctuation theorem for the large deviation funtion of the current. For large diffusive systems, we show how the large deviation funtion of the current can be computed using a simple additivity principle. The validity of this additivity principle and the occurence of phase transitions are discussed in the framework of the macroscopic fluctuation theory.

Abstract:
The symmetric simple exclusion process is one of the simplest out-of-equilibrium systems for which the steady state is known. Its large deviation functional of the density has been computed in the past both by microscopic and macroscopic approaches. Here we obtain the leading finite size correction to this large deviation functional. The result is compared to the similar corrections for equilibrium systems.

Abstract:
A branching random walk in presence of an absorbing wall moving at a constant velocity v undergoes a phase transition as v varies. The problem can be analyzed using the properties of the Fisher-Kolmogorov-Petrovsky-Piscounov (F-KPP) equation. We find that the survival probability of the branching random walk vanishes at a critical velocity v_c of the wall with an essential singularity and we characterize the divergences of the relaxation times for vv_c. At v=v_c the survival probability decays like a stretched exponential. Using the F-KPP equation, one can also calculate the distribution of the population size at time t conditionned by the survival of one individual at a later time T>t. Our numerical results indicate that the size of the population diverges like the exponential of (v_c-v)^{-1/2} in the quasi-stationary regime below v_c. Moreover for v>v_c, our data indicate that there is no quasi-stationary regime.