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.

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

References in corpus (26)

• Theoretical perspective on the glass transition and amorphous materials
• The large deviation approach to statistical mechanics
• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• Supercooled Liquids for Pedestrians
• Dynamic first-order phase transition in kinetically constrained models of glasses
• First-order dynamical phase transition in models of glasses: an approach based on ensembles of histories
• Computation of extreme heat waves in climate models using a large deviation algorithm
• Large Deviations in Single File Diffusion
• Hyperuniformity and phase separation in biased ensembles of trajectories for diffusive systems
• Entropy production of active particles and for particles in active baths
• Probing rare physical trajectories with Lyapunov weighted dynamics
• 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
• 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
• Large fluctuations and dynamic phase transition in a system of self-propelled particles
• Inactive dynamical phase of a symmetric exclusion process on a ring
• Dissipation controls transport and phase transitions in active fluids: Mobility, diffusion and biased ensembles
• Current fluctuations in stochastic systems with long-range memory
• Variational control forces for enhanced sampling of nonequilibrium molecular dynamics simulations
• Properties of a non-equilibrium heat bath
• Finite size effects in a mean-field kinetically constrained model: dynamical glassiness and quantum criticality
• 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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• Slow dynamics and large deviations in classical stochastic Fredkin chains
• Understanding the fragile-to-strong transition in silica from microscopic dynamics
• Large deviations for the Skew-Detailed-Balance Lifted-Markov processes to sample the equilibrium distribution of the Curie-Weiss model
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• Anomalous dynamical large deviations of local empirical densities and activities in the pure and in the random kinetically-constrained East Model
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• Modeling of biomolecular machines in non-equilibrium steady states
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• Large deviations of the stochastic area for linear diffusions
• Varied phenomenology of models displaying dynamical large-deviation singularities
• Large Deviations Theory of Increasing Returns
• Anomalous scalings of fluctuations of the area swept by a Brownian particle trapped in a $|x|$ potential
• Microcanonical conditioning of Markov processes on time-additive observables
• Large deviations for the Pearson family of ergodic diffusion processes involving a quadratic diffusion coefficient and a linear force
• Symmetry-induced fluctuation relations for dynamical observables irrespective of their behaviour under time-reversal
• Microscopic biasing of discrete-time quantum trajectories
• Dynamical heterogeneity and large deviations in the open quantum East glass model from tensor networks
• Finding the effective dynamics to make rare events typical in chaotic maps
• Delocalization-localization dynamical phase transition of random walks on graphs
• Revisiting boundary-driven non-equilibrium Markov dynamics in arbitrary potentials via supersymmetric quantum mechanics and explicit large deviations at various levels
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• Large deviations at level 2.5 and for trajectories observables of diffusion processes : the missing parts with respect to their random-walks counterparts
• 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 for trajectory observables of diffusion processes in dimension $d>1$ in the double limit of large time and small diffusion coefficient
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• Markov generators as non-hermitian supersymmetric quantum Hamiltonians: spectral properties via bi-orthogonal basis and Singular Value Decompositions
• K-core attack, equilibrium K-core, and kinetically constrained spin system
• Noise equals control
• Thermodynamic bound on current fluctuations in coherent conductors
• Work fluctuations for a confined Brownian particle: the role of initial conditions
• Markov spin models for image generation : explicit large deviations with respect to the number of pixels
• Dynamical Phase Transitions, Caustics, and Quantum Dark Bands
• Inverse problem in the conditioning of Markov processes on trajectory observables : what canonical conditionings can connect two given Markov generators ?
• Large-deviations approach to thermalization: the case of harmonic chains with conservative noise
• Mechanical analysis of a dynamical phase transition for particles in a channel
• Dimensionality-induced dynamical phase transition in the large deviation of local time density for Brownian motion
• Dynamical phase transition in the activity-biased fully-connected random field Ising model: connection with glass-forming systems
• Exact large deviations and emergent long-range correlations in sequential quantum East circuits
• Steering Active-Colloid Assembly by Biasing Dissipation
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