Ergodicity and large deviations in physical systems with stochastic dynamics

Abstract

In ergodic physical systems, time-averaged quantities converge (for large times) to their ensemble-averaged values. Large deviation theory describes rare events where these time averages differ significantly from the corresponding ensemble averages. It allows estimation of the probabilities of these events, and their mechanisms. This theory has been applied to a range of physical systems, where it has yielded new insights into entropy production, current fluctuations, metastability, transport processes, and glassy behaviour. We review some of these developments, identifying general principles. We discuss a selection of dynamical phase transitions, and we highlight some connections between large-deviation theory and optimal control theory.

arXiv commentaccepted version, small additions and modifications. 23 pages, "colloquium" article

References in corpus (108)

• Stochastic thermodynamics, fluctuation theorems, and molecular machines
• Theoretical perspective on the glass transition and amorphous materials
• The large deviation approach to statistical mechanics
• A Gallavotti-Cohen Type Symmetry in the Large Deviation Functional for Stochastic Dynamics
• Athermal Phase Separation of Self-Propelled Particles with no Alignment
• Path ensembles averages in systems driven far-from-equilibrium
• Dynamical ensembles in stationary states
• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• Supercooled Liquids for Pedestrians
• Local Density Fluctuations, Hyperuniformity, and Order Metrics
• Dissipation bounds all steady-state current fluctuations
• Macroscopic fluctuation theory
• How far from equilibrium is active matter?
• Dynamics on the Way to Forming Glass: Bubbles in Space-time
• The Fluctuation Theorem as a Gibbs Property
• Dynamic first-order phase transition in kinetically constrained models of glasses
• Macroscopic fluctuation theory for stationary non equilibrium states
• Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations
• First-order dynamical phase transition in models of glasses: an approach based on ensembles of histories
• Current fluctuations in non-equilibrium diffusive systems: an additivity principle
• Exact Large Deviation Function in the Asymmetric Exclusion Process
• Thermodynamic formalism for systems with Markov dynamics
• Direct evaluation of large-deviation functions
• Nonequilibrium Markov processes conditioned on large deviations
• Thermodynamics of Quantum Jump Trajectories
• Large deviations and ensembles of trajectories in stochastic models
• The Statistical Mechanics of Dynamic Pathways to Self-assembly
• Distribution of current in non-equilibrium diffusive systems and phase transitions
• Space-time Thermodynamics of the Glass Transition
• Nonequilibrium microcanonical and canonical ensembles and their equivalence
• Computation of extreme heat waves in climate models using a large deviation algorithm
• Canonical structure of dynamical fluctuations in mesoscopic nonequilibrium steady states
• Facets of glass physics
• Introduction to dynamical large deviations of Markov processes
• Unravelling the large deviation statistics of Markovian open quantum systems
• Entropy production and fluctuation theorems for active matter
• Chaotic Scattering Theory, Thermodynamic Formalism, and Transport Coefficients
• Universal cumulants of the current in diffusive systems on a ring
• Large Deviations in Single File Diffusion
• Hidden entropy production and work fluctuations in an ideal active gas
• Hyperuniformity and phase separation in biased ensembles of trajectories for diffusive systems
• Entropy production of active particles and for particles in active baths
• Variational and optimal control representations of conditioned and driven processes
• Probing rare physical trajectories with Lyapunov weighted dynamics
• On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion
• Current fluctuations in the zero-range process with open boundaries
• Population dynamics method with a multi-canonical feedback control
• Metastable states in glassy systems
• Effective interactions and large deviations in stochastic processes
• A formal view on 2.5 large deviations and fluctuation relations
• Construction of a Coordinate Bethe Ansatz for the asymmetric simple exclusion process with open boundaries
• Dynamical symmetry breaking and phase transitions in driven diffusive systems
• Bounds on current fluctuations in periodically driven systems
• Optimizing active work: Dynamical phase transitions, collective motion, and jamming
• A minimal model of dynamical phase transition
• Finite-temperature critical point of a glass transition
• Space-time thermodynamics and subsystem observables in a kinetically constrained model of glassy systems
• Finite-Size Scaling of a First-Order Dynamical Phase Transition: Adaptive Population Dynamics and an Effective Model
• Thermodynamics of currents in nonequilibrium diffusive systems: theory and simulation
• Test of the Additivity Principle for Current Fluctuations in a Model of Heat Conduction
• Large fluctuations and dynamic phase transition in a system of self-propelled particles
• Action Principle in Nonequilibrium Statistical Dynamics
• Using matrix product states to study the dynamical large deviations of kinetically constrained models
• Exact fluctuations of nonequilibrium steady states from approximate auxiliary dynamics
• Thermodynamic formula for the cumulant generating function of time-averaged current
• Constrained dynamics of localized excitations causes a non-equilibrium phase transition in an atomistic model of glass formers
• Inactive dynamical phase of a symmetric exclusion process on a ring
• Anomalous scaling of dynamical large deviations
• Large-Deviation Functions for Nonlinear Functionals of a Gaussian Stationary Markov Process
• Preparation and relaxation of very stable glassy states of a simulated liquid
• Long range correlations and phase transition in non-equilibrium diffusive systems
• Dynamic transition in an atomic glass former: a molecular dynamics evidence
• Dynamic phase transitions in simple driven kinetic networks
• Dissipation controls transport and phase transitions in active fluids: Mobility, diffusion and biased ensembles
• Current fluctuations in stochastic systems with long-range memory
• Importance sampling large deviations in nonequilibrium steady states. I
• Dynamical fluctuations for semi-Markov processes
• Metastable states and space-time phase transitions in a spin-glass model
• Experimental Evidence for a Structural-Dynamical Transition in Trajectory Space
• Adaptive sampling of large deviations
• Fluctuations of time averages for Langevin dynamics in a binding force field
• Finite size scaling of the dynamical free-energy in a kinetically constrained model
• Current fluctuations of interacting active Brownian particles
• Activity phase transition for constrained dynamics
• Current fluctuations in systems with diffusive dynamics, in and out of equilibrium
• Order and symmetry-breaking in the fluctuations of driven systems
• Large deviations of the dynamical activity in the East model: analysing structure in biased trajectories
• Canonical structure and orthogonality of forces and currents in irreversible Markov chains
• Sampling rare events across dynamical phase transitions
• Variational control forces for enhanced sampling of nonequilibrium molecular dynamics simulations
• Non differentiable large-deviation functionals in boundary-driven diffusive systems
• Dynamical phase transitions in supercooled liquids: interpreting measurements of dynamical activity
• Rare-event trajectory ensemble analysis reveals metastable dynamical phases in lattice proteins
• Level 2.5 large deviations for continuous time Markov chains with time periodic rates
• Properties of a non-equilibrium heat bath
• Direct evaluation of dynamical large-deviation rate functions using a variational ansatz
• Large deviations and optimal control forces for hard particles in one dimension
• Efficient characterisation of large deviations using population dynamics
• Finite size effects in a mean-field kinetically constrained model: dynamical glassiness and quantum criticality
• Large deviations for Generalized Polya Urns with arbitrary urn function
• Fluctuations in interacting particle systems with memory
• Large deviations in models of growing clusters with symmetry-breaking transitions
• Heat current fluctuations and anomalous transport in low dimensional carbon lattices
• Thermodynamics of trajectories of the one-dimensional Ising model
• Activity statistics in a colloidal glass former: experimental evidence for a dynamical transition
• Conditional reversibility in nonequilibrium stochastic systems
• Dynamical phase transitions for the activity biased Ising model in a magnetic field
• Symmetries and Geometrical Properties of Dynamical Fluctuations in Molecular Dynamics

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• Large deviations for the Pearson family of ergodic diffusion processes involving a quadratic diffusion coefficient and a linear force
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• Nonequilibrium fluctuations of chemical reaction networks at criticality: The Schlögl model as paradigmatic case
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• Conditioning diffusion processes with respect to the local time at the origin
• Long-ranged correlations in large deviations of local clustering
• Efficient post-selection in light-cone correlations of monitored quantum circuits
• Reaction-Path Statistical Mechanics of Enzymatic Kinetics
• Joint distribution of two Local Times for diffusion processes with the application to the construction of various conditioned processes
• Explicit dynamical properties of the Pelikan random map in the chaotic region and at the intermittent critical point towards the non-chaotic region
• On the Kemeny time for continuous-time reversible and irreversible Markov processes with applications to stochastic resetting and to conditioning towards forever-survival
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• Trajectory phase transitions in non-interacting systems: all-to-all dynamics and the random energy model
• A supersymmetric quantum perspective on the explicit large deviations for reversible Markov jump processes, with applications to pure and random spin chains
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• Large-deviations approach to thermalization: the case of harmonic chains with conservative noise
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• Dimensionality-induced dynamical phase transition in the large deviation of local time density for Brownian motion
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