Non equilibrium steady states: fluctuations and large deviations of the density and of the current

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.

35 pages, 9 figures

References in corpus (12)

• Phase Coexistence in Driven One Dimensional Transport
• Bethe Ansatz Solution of the Asymmetric Exclusion Process with Open Boundaries
• Fluctuation properties of the TASEP with periodic initial configuration
• A momentum conserving model with anomalous thermal conductivity in low dimension
• The asymmetric simple exclusion process: an integrable model for non-equilibrium statistical mechanics
• Fourier's Law for a Harmonic Crystal with Self-consistent Stochastic Reservoirs
• Work fluctuations for a Brownian particle between two thermostats
• Cumulants and large deviations of the current through non-equilibrium steady states
• Current Distribution and random matrix ensembles for an integrable asymmetric fragmentation process
• Determinant solution for the Totally Asymmetric Exclusion Process with parallel update
• An exactly solvable model for driven dissipative systems
• Exact Large Deviation Functional of a Stationary Open Driven Diffusive System: The Asymmetric Exclusion Process

Cited by in corpus (63)

• The large deviation approach to statistical mechanics
• Nonequilibrium Steady States of Matrix Product Form: A Solver's Guide
• Fluctuation theorems for stochastic dynamics
• Large Deviations in Single File Diffusion
• The Exclusion Process: A paradigm for non-equilibrium behaviour
• Mapping out of equilibrium into equilibrium in one-dimensional transport models
• Logarithmic current fluctuations in non-equilibrium quantum spin chains
• Solvable quantum nonequilibrium model exhibiting a phase transition and a matrix product representation
• Symmetries in Fluctuations Far from Equilibrium
• Fluctuation relations and coarse-graining
• Large deviation function for entropy production in driven one-dimensional systems
• DMRG-study of current and activity fluctuations near non-equilibrium phase transitions
• Mapping Nonequilibrium onto Equilibrium: The Macroscopic Fluctuations of Simple Transport Models
• Current Fluctuations and Statistics During a Large Deviation Event in an Exactly-Solvable Transport Model
• A matrix product solution for a nonequilibrium steady state of an XX chain
• Current fluctuations in stochastic systems with long-range memory
• Facilitated Asymmetric Exclusion
• Non-equilibrium steady states : maximization of the Shannon entropy associated to the distribution of dynamical trajectories in the presence of constraints
• Large Fluctuations of the Macroscopic Current in Diffusive Systems: A Confirmation of the Additivity Principle
• Survival of a static target in a gas of diffusing particles with exclusion
• Long range order in non-equilibrium interacting quantum spin chains
• Invariant quantities in shear flow
• Cumulants of the current in the weakly asymmetric exclusion process
• Asymmetric exclusion processes with constrained dynamics
• Condensation of Fluctuations in and out of Equilibrium
• Generalized Exclusion Processes: Transport Coefficients
• Parallel Coupling of Symmetric and Asymmetric Exclusion Processes
• Defect-induced phase transition in the asymmetric simple exclusion process
• A unified approach to the derivation of work theorems for equilibrium and steady-state, classical and quantum Hamiltonian systems
• Work fluctuations for a Brownian particle driven by a correlated external random force
• On and beyond entropy production: the case of Markov jump processes
• Spectrum of the totally asymmetric simple exclusion process on a periodic lattice -- first excited states
• Exclusive Queueing Processes and their Application to Traffic Systems
• Strong asymmetric limit of the quasi-potential of the boundary driven weakly asymmetric exclusion process
• Extreme Fluctuations of Current in the Symmetric Simple Exclusion Process: a Non-Stationary Setting
• Balance network of asymmetric simple exclusion process
• Dynamic Singularities in Cooperative Exclusion
• Dynamic fluctuations in unfrustrated systems: random walks, scalar fields and the Kosterlitz-Thouless phase
• Large fluctuations in diffusion-controlled absorption
• Entropy production and fluctuation relations for a KPZ interface
• Static large deviations of boundary driven exclusion processes
• Heterogeneity-induced large deviations in activity and (in some cases) entropy production
• Effects of junctional correlations in the totally asymmetric simple exclusion process on random regular networks
• Local Detailed Balance : A Microscopic Derivation
• Energy and Heat Fluctuations in a Temperature Quench
• On the symmetry of current probability distributions in jump processes
• Melting of an Ising Quadrant
• From Vicious Walkers to TASEP
• Entropy of stationary nonequilibrium measures of boundary driven symmetric simple exclusion processes
• Small noise asymptotic of the Gallavotti-Cohen functional for diffusion processes
• Current fluctuations in boundary driven diffusive systems in different dimensions: a numerical study
• Lattice gases with a point source
• Large deviation generating function for energy transport in the Pauli-Fierz model
• Exclusion Processes with Avalanches
• Experimental study of energy transport between two granular gas thermostats
• Structures of nonequilibrium fluctuations: dissipation and activity
• Long-range correlations in a simple stochastic model of coupled transport
• Realization of the open-boundary totally asymmetric simple exclusion process on a ring
• Hydrostatics and dynamical large deviations of boundary driven gradient symmetric exclusion
• A combinatorial solution for the current fluctuations in the exclusion process
• Soft bounds on diffusion produce skewed distributions and Gompertz growth
• Lagrangian view of time irreversibility of fluid turbulence
• From combinatorics to large deviations for the invariant measures of some multiclass particle systems