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.

arXiv comment35 pages, 9 figures

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• Asymmetric Exclusion Model and Weighted Lattice Paths
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• Bethe Ansatz calculation of the spectral gap of the asymmetric exclusion process
• Entropy of Open Lattice Systems
• Diffusion algebras
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• An equilibrium state with macroscopic correlations
• Exact probability function for bulk density and current in the asymmetric exclusion process
• Large deviations in weakly interacting boundary driven lattice gases
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• Shift Equivalence of Measures and the Intrinsic Structure of Shocks in the Asymmetric Simple Exclusion Process
• The Asymmetric Exclusion Process revisited: Fluctuations and Dynamics in the Domain Wall Picture
• Dynamical Windings of Random Walks and Exclusion Models. Part I: Thermodynamic Limit

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• Jump-Drift and Jump-Diffusion Processes : Large Deviations for the density, the current and the jump-flow and for the excursions between jumps
• Time-averaged height distribution of the Kardar-Parisi-Zhang interface
• Exact solution of an integrable non-equilibrium particle system
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• Nonequilibrium thermodynamic potentials for continuous-time Markov chains
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• Large deviations in the symmetric simple exclusion process with slow boundaries
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• Dynamical Phase Transitions for Flows on Finite Graphs
• Work, work fluctuations, and the work distribution in a thermal non-equilibrium steady state
• Large deviations and conditioning for chaotic non-invertible deterministic maps: analysis via the forward deterministic dynamics and the backward stochastic dynamics
• Integrability of two-species partially asymmetric exclusion processes
• Entropy Governed by the Absorbing State of Directed Percolation
• Entropy of stationary nonequilibrium measures of boundary driven symmetric simple exclusion processes
• Large deviations of density fluctuations in the boundary driven Quantum Symmetric Simple Inclusion Process
• From Vicious Walkers to TASEP
• Active Fluctuation Symmetries
• Emerging universality classes in thermally-assisted activation of interacting diffusive systems: A perturbative hydrodynamic approach
• Large Deviations in the Symmetric Simple Exclusion Process with Slow Boundaries: A Hydrodynamic Perspective
• Multivariate analysis of short time series in terms of ensembles of correlation matrices
• Current fluctuations in boundary driven diffusive systems in different dimensions: a numerical study
• The boundary driven zero-range process
• Inverse Design of Non-Equilibrium Steady-States: A Large Deviation Approach
• Fluctuations of absorption of interacting diffusing particles by multiple absorbers
• Large deviations of a tracer position in the dense and the dilute limits of a single-file diffusion
• Long range correlations in stochastic transport with energy and momentum conservation
• Large deviation generating function for energy transport in the Pauli-Fierz model
• Current fluctuations in a partially asymmetric simple exclusion process with a defect particle
• Dynamical phase transitions in one-dimensional hard-particle systems
• Full absorption statistics of diffusing particles with exclusion
• Perturbative large deviation analysis of non-equilibrium dynamics
• Scaling laws and bulk-boundary decoupling in heat flow
• Modified saddle-point integral near singularity for the large deviation function
• Asymmetric Exclusion Process with Global Hopping
• A low temperature analysis of the boundary driven Kawasaki Process
• Interfacial and density fluctuations in a lattice model of motility-induced phase separation
• Remarks on the multi-species exclusion process with reflective boundaries
• Lattice gases with a point source
• Hydrodynamic limit for a boundary driven stochastic lattice gas model with many conserved quantities
• Small noise asymptotic of the Gallavotti-Cohen functional for diffusion processes
• Direct numerical method for counting statistics in stochastic processes
• Work probability distribution and tossing a biased coin
• Nonequilibrium steady states in coupled asymmetric and symmetric exclusion processes
• Nonequilibrium chemical potentials of steady-state lattice gas models in contact: A large-deviations approach
• Particle current fluctuations in a variant of asymmetric Glauber model
• Exclusion Processes with Avalanches
• Finite-time fluctuation theorem for diffusion-influenced surface reactions on spherical and Janus catalytic particles
• Anomalous long-range correlations at a non-equilibrium phase transition
• Tuning the distance to the equipartition by controlling the collision rate in a driven granular gas experiment
• Finite-time fluctuation theorem for diffusion-influenced surface reactions
• Dynamical phase transitions for the activity biased Ising model in a magnetic field
• Current loops and fluctuations in the zero-range process on a diamond lattice
• Conservative stochastic PDE and fluctuations of the symmetric simple exclusion process
• Experimental study of energy transport between two granular gas thermostats
• Symmetry shapes thermodynamics of macroscopic quantum systems
• Exact solution of a quantum asymmetric exclusion process with particle creation and annihilation
• Matrix Product representation of the stationary state of the open Zero Range Process
• Long-ranged correlations in large deviations of local clustering
• Critical behavior of the exclusive queueing process
• Long-range correlations in a simple stochastic model of coupled transport
• Transport-induced correlations in weakly interacting systems
• Importance Sampling for counting statistics in one-dimensional systems
• Exact asymptotics of long-range quantum correlations in a nonequilibrium steady state
• Work distribution function for a Brownian particle driven by a nonconservative force
• Thermal Contact I : Symmetries ruled by Exchange Entropy Variations
• Hydrodynamics of a hard-core active lattice gas
• Minimum Dissipation Principle in Nonlinear Transport
• Structures of nonequilibrium fluctuations: dissipation and activity
• Large deviations for out of equilibrium correlations in the symmetric simple exclusion process
• The statistical mechanics of Twitter
• Steady state large deviations for one-dimensional, symmetric exclusion processes in weak contact with reservoirs
• Feynman-Kac theory of time-integrated functionals: Itô versus functional calculus
• Exclusion in Junction Geometries
• Full Counting Statistics and Fluctuation Theorem for the Currents in the Discrete Model of Feynman's Ratchet
• Current statistics in the q-boson zero range process
• Complete Integrability of the Problem of Full Statistics of Nonstationary Mass Transfer in the Simple Inclusion Process
• Positional information trade-offs in boundary-driven reaction-diffusion systems
• Current fluctuations in the symmetric exclusion process beyond the one-dimensional geometry
• Emergence of the Gibbs ensemble as a steady state in Lindbladian dynamics
• The chemical birth-death process with Gillespie noise
• Dynamic fluctuations of current and mass in nonequilibrium mass transport processes
• A two-line representation of stationary measure for open TASEP
• Explicit dynamical properties of the Pelikan random map in the chaotic region and at the intermittent critical point towards the non-chaotic region
• Programmable time crystals from higher-order packing fields
• Trajectory phase transitions in non-interacting systems: all-to-all dynamics and the random energy model
• Large deviations at level 2.5 and for trajectories observables of diffusion processes : the missing parts with respect to their random-walks counterparts
• Tracer and current fluctuations in driven diffusive systems
• The Moving Born-Oppenheimer Approximation
• Kinetically constrained model for gravity-driven granular flow and clogging
• Cooperative transport with selective kinetic constraints
• Diffusion and entanglement in open quantum systems
• Glauber's Ising chain between two thermostats
• Realization of the open-boundary totally asymmetric simple exclusion process on a ring
• Imitating non-equilibrium steady states using time-varying equilibrium force in many-body diffusive systems
• A combinatorial solution for the current fluctuations in the exclusion process
• Large deviations of current for the symmetric simple exclusion process on a semi-infinite line, and on an infinite line with a slow bond
• On the Kemeny time for continuous-time reversible and irreversible Markov processes with applications to stochastic resetting and to conditioning towards forever-survival
• Large deviations for correlated random variables described by a matrix product ansatz
• Soft bounds on diffusion produce skewed distributions and Gompertz growth
• Totally asymmetric limit for models of heat conduction
• Hydrostatics and dynamical large deviations of boundary driven gradient symmetric exclusion
• Infinite family of universal profiles for heat current statistics in Fourier's law
• Joint distribution of two Local Times for diffusion processes with the application to the construction of various conditioned processes
• Finite size corrections to the large deviation function of the density in the one dimensional symmetric simple exclusion process
• Inferring intermediate states by leveraging the many-body Arrhenius law
• Lecture notes on large deviations in non-equilibrium diffusive systems
• Markov generators as non-hermitian supersymmetric quantum Hamiltonians: spectral properties via bi-orthogonal basis and Singular Value Decompositions
• A supersymmetric quantum perspective on the explicit large deviations for reversible Markov jump processes, with applications to pure and random spin chains
• Thermal contact through a two-temperature kinetic Ising chain
• Stochastic Dynamics of Extended Objects in Driven Systems: I. Higher-Dimensional Currents in the Continuous Setting
• Uncertainty Relations for Mesoscopic Coherent Light
• Numerical Estimation of the Current Large Deviation Function in the Asymmetric Simple Exclusion Process with Open Boundary Conditions
• Current fluctuations in the symmetric zero-range process below and at critical density
• Large-time asymptotic of heavy tailed renewal processes
• Random walk through a fertile site
• Large deviations for trajectory observables of diffusion processes in dimension $d>1$ in the double limit of large time and small diffusion coefficient
• Fluctuating Hydrodynamics Describes Transport in Cellular Aggregates
• Inhomogeneous discrete-time exclusion processes
• An exactly solvable asymmetric $K$-exclusion process
• Systematic analysis of critical exponents in continuous dynamical phase transitions of weak noise theories
• Inverse problem in the conditioning of Markov processes on trajectory observables : what canonical conditionings can connect two given Markov generators ?
• Limits to positional information in boundary-driven systems
• Universal and non-universal signatures in the scaling functions of critical variables
• Stationary Distribution of open Asymmetric Simple Exclusion Processes on an Interval as a marginal of a two-layer ensemble
• Full distribution of the ground-state energy of potentials with weak disorder
• Entropy of non-equilibrium stationary measures of boundary driven TASEP
• Markov spin models for image generation : explicit large deviations with respect to the number of pixels
• Towards neural reinforcement learning for large deviations in nonequilibrium systems with memory
• Critical behavior of a programmable time-crystal lattice gas
• Exact large-scale correlations in diffusive systems with general interactions: explicit characterisation without the Dean--Kawasaki equation
• Stationary measure for six-vertex model on a strip
• Phase coexistence phenomena in an extreme case of the misanthrope process with open boundaries
• Fluctuations of TASEP on a ring in relaxation time scale
• Integrability in out-of-equilibrium systems
• Inverse Scattering Method Solves the Problem of Full Statistics of Nonstationary Heat Transfer in the Kipnis-Marchioro-Presutti Model
• Large deviations of avalanches in the raise and peel model
• Non-additive large deviations function for the particle densities of driven systems in contact
• Effective bandwidth of non-Markovian packet traffic
• Nonequilibrium Phase Transition in a 2D Ferromagnetic Spins with Effective Interactions
• Approximating the entire spectrum of nonequilibrium steady state distributions using relative entropy: An application to thermal conduction
• Microscopic and macroscopic perspectives on stationary nonequilibrium states
• Current statistics and depinning transition for a one-dimensional Langevin process in the weak-noise limit
• Non-equilibrium stationary properties of the boundary driven zero-range process with long jumps
• Conditioning two diffusion processes with respect to their first-encounter properties
• Structure of the optimal path to a fluctuation
• Dynamical large deviations for a boundary driven stochastic lattice gas model with many conserved quantities
• Mechanical analysis of a dynamical phase transition for particles in a channel
• A stochastic model solvable without integrability
• Dynamical phase transition in the activity-biased fully-connected random field Ising model: connection with glass-forming systems
• Introduction to stochastic models of transportation flows. Part I
• System-size independence of a large deviation function for frequency of events in a one-dimensional forest-fire model with a single ignition site
• Density-dependent transport coefficients in two-dimensional cellular aggregates
• Current fluctuations in finite-sized one-dimensional non-interacting passive and active systems
• Domain-wall melting in all-to-all QSSEP from random-matrix theory
• Macroscopic fluctuation theory of interacting Brownian particles
• Symmetry Breaking of Current Response in Disordered Exclusion Processes
• Dimensionality-induced dynamical phase transition in the large deviation of local time density for Brownian motion
• From combinatorics to large deviations for the invariant measures of some multiclass particle systems
• Anomalous current fluctuations and mobility-driven clustering
• Asymmetric scaling in large deviations for rare values bigger or smaller than the typical value
• Limit fluctuations of stationary measure of totally asymmetric simple exclusion process with open boundaries on the coexistence line
• Transition state theory applied to self-diffusion of hard spheres
• Mapping current and activity fluctuations in exclusion processes: consequences and open questions
• Lagrangian view of time irreversibility of fluid turbulence
• Dispersal and organization of polarized cells: non-linear diffusion and cluster formation without adhesion
• Effective Langevin equations leading to large deviation function of time-averaged velocity for a nonequilibrium Rayleigh piston