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 consider high order current cumulants in disordered systems out of equilibrium. They are interesting and reveal information which is not easily exposed by the traditional shot noise. Despite the fact that the dynamics of the electrons is classical, the standard kinetic theory of fluctuations needs to be modified to account for those cumulants. We perform a quantum-mechanical calculation using the Keldysh technique and analyze its relation to the quasi classical Boltzmann-Langevin scheme. We also consider the effect of inelastic scattering. Strong electron-phonon scattering renders the current fluctuations Gaussian, completely suppressing the $n>2$ cumulants. Under strong electron-electron scattering the current fluctuations remain non-Gaussian.

Abstract:
We evaluate the exact energy current and scaled cumulant generating function (related to the large-deviation function) in non-equilibrium steady states with energy flow, in any integrable model of relativistic quantum field theory (IQFT) with diagonal scattering. Our derivations are based on various recent results of D. Bernard and B. Doyon. The steady states are built by connecting homogeneously two infinite halves of the system thermalized at different temperatures $T_l$, $T_r$, and waiting for a long time. We evaluate the current $J(T_l,T_r)$ using the exact QFT density matrix describing these non-equilibrium steady states and using Al.B. Zamolodchikov's method of the thermodynamic Bethe ansatz (TBA). The scaled cumulant generating function is obtained from the extended fluctuation relations which hold in integrable models. We verify our formula in particular by showing that the conformal field theory (CFT) result is obtained in the high-temperature limit. We analyze numerically our non-equilibrium steady-state TBA equations for three models: the sinh-Gordon model, the roaming trajectories model, and the sine-Gordon model at a particular reflectionless point. Based on the numerics, we conjecture that an infinite family of non-equilibrium $c$-functions, associated to the scaled cumulants, can be defined, which we interpret physically. We study the full scaled distribution function and find that it can be described by a set of independent Poisson processes. Finally, we show that the "additivity" property of the current, which is known to hold in CFT and was proposed to hold more generally, does not hold in general IQFT, that is $J(T_l,T_r)$ is not of the form $f(T_l)-f(T_r)$.

Abstract:
Recently a remarkable connection has been proposed between the fluctuating hydrodynamic equations of a one-dimensional fluid and the Kardar-Parizi-Zhang (KPZ) equation for interface growth. This connection has been used to relate equilibrium correlation functions of the fluid to KPZ correlation functions. Here we use this connection to compute the exact cumulant generating function for energy current in the fluid system. This leads to exact expressions for all cumulants and in particular to universal results for certain combinations of the cumulants. As examples, we consider two different systems which are expected to be in different universality classes, namely a hard particle gas with Hamiltonian dynamics and a harmonic chain with momentum conserving stochastic dynamics. Simulations provide excellent confirmation of our theory.

Abstract:
The purpose of the present paper is to establish moderate deviation principles for a rather general class of random variables fulfilling certain bounds of the cumulants. We apply a celebrated lemma of the theory of large deviations probabilities due to Rudzkis, Saulis and Statulevicius. The examples of random objects we treat include dependency graphs, subgraph-counting statistics in Erd\H{o}s-R\'enyi random graphs and $U$-statistics. Moreover, we prove moderate deviation principles for certain statistics appearing in random matrix theory, namely characteristic polynomials of random unitary matrices as well as the number of particles in a growing box of random determinantal point processes like the number of eigenvalues in the GUE or the number of points in Airy, Bessel, and $\sin$ random point fields.

Abstract:
We calculate steady-state properties of a strongly correlated quantum dot under voltage bias by means of non-equilibrium Cluster Perturbation Theory and the non-equilibrium Variational Cluster Approach, respectively. Results for the steady-state current are benchmarked against data from accurate Matrix Product State based time evolution. We show that for low to medium interaction strength, non-equilibrium Cluster Perturbation Theory already yields good results, while for higher interaction strength the self-consistent feedback of the non-equilibrium Variational Cluster Approach significantly enhances the accuracy. We report the current-voltage characteristics for different interaction strengths. Furthermore we investigate the non-equilibrium local density of states of the quantum dot and illustrate that within the variational approach a linear splitting and broadening of the Kondo resonance is predicted which depends on interaction strength. Calculations with applied gate voltage, away from particle hole symmetry, reveal that the maximum current is reached at the crossover from the Kondo regime to the doubly-occupied or empty quantum dot. Obtained stability diagrams compare very well to recent experimental data [Phys. Rev. B, 84, 245316 (2011)].

Abstract:
Let an infinite, homogeneous, many-body quantum system be unitarily evolved for a long time from a state where two halves are independently thermalized. One says that a non-equilibrium steady state emerges if there are nonzero steady currents in the central region. In particular, their presence is a signature of ballistic transport. We analyze the consequences of the current observable being a conserved density; near equilibrium this is known to give rise to linear wave propagation and a nonzero Drude peak. Using the Lieb-Robinson bound, we derive, under a certain regularity condition, a lower bound for the non-equilibrium steady-state current determined by equilibrium averages. This shows and quantifies the presence of ballistic transport far from equilibrium. The inequality suggests the definition of "nonlinear sound velocities", which specialize to the sound velocity near equilibrium in non-integrable models, and "generalized sound velocities", which encode generalized Gibbs thermalization in integrable models. These are bounded by the Lieb-Robinson velocity. The inequality also gives rise to a bound on the energy current noise in the case of pure energy transport. We show that the inequality is satisfied in many models where exact results are available, and that it is saturated at one-dimensional criticality.

Abstract:
The BGK model kinetic equation is applied to spatially inhomogeneous states near steady uniform shear flow. The shear rate of the reference steady state can be large so the states considered include those very far from equilibrium. The single particle distribution function is calculated exactly to first order in the deviations of the hydrodynamic field gradients from their values in the reference state. The corresponding non-linear hydrodynamic equaitons are obtained and the set of transport coefficients are identified as explicit functions of the shear rate. The spectrum of the linear hydrodynamic equation is studied in detail and qualitative differences from the spectrum for equilibrium fluctuations are discussed. Conditions for instabilities at long wavelengths are identified and disccused.

Abstract:
We consider the problem of constructing a thermodynamic theory of non-equilibrium steady states as a formal extension of the equilibrium theory. Specifically, studying a particular system, we attempt to construct a phenomenological theory describing the interplay between heat and mechanical work that takes place during operations through which the system undergoes transitions between non-equilibrium steady states. We find that, in contrast to the case of the equilibrium theory, apparently, there exists no systematic way within a phenomenological formulation to describe the work done by the system during such operations. With this observation, we conclude that the attempt to construct a thermodynamic theory of non-equilibrium steady states in analogy to the equilibrium theory has limited prospects for success and that the pursuit of such a theory should be directed elsewhere.

Abstract:
We show that the system entropy change for the transitions between non-equilibrium steady states arbitrarily far from equilibrium for any constituting process is given by the relative entropy of the distributions of these steady states. This expression is then shown to relate to the dissipation relations of both Vaikuntanathan and Jarzynski [EPL 87, 60005 (2009)] and Kawai, Parrondo and Van den Broeck [Phys. Rev. Lett. 98, 080602 (2007)] in the case of energy-conserving driving.