Introduction to dynamical large deviations of Markov processes

Abstract

These notes give a summary of techniques used in large deviation theory to study the fluctuations of time-additive quantities, called dynamical observables, defined in the context of Langevin-type equations, which model equilibrium and nonequilibrium processes driven by external forces and noise sources. These fluctuations are described by large deviation functions, obtained by solving a dominant eigenvalue problem similar to the problem of finding the ground state energy of quantum systems. This analogy is used to explain the differences that exist between the fluctuations of equilibrium and nonequilibrium processes. An example involving the Ornstein-Uhlenbeck process is worked out in detail to illustrate these methods. Exercises, at the end of the notes, also complement the theory.

19 pages. Lecture notes for the 2017 Summer School on Fundamental Problems in Statistical Physics XIV, 16-29 July 2017, Bruneck (Brunico), Italy. v2: Typos corrected, exercises added. v3: Typos corrected. v4: More typos corrected, footnote and references added. v5: Close to published version. v6: typo corrected. I dedicate this paper to the memory of E. G. D. Cohen (1923-2017)

References in corpus (8)

• The large deviation approach to statistical mechanics
• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• Fluctuation theorems for stochastic dynamics
• Stationary and Transient Work-Fluctuation Theorems for a Dragged Brownian Particle
• Extended Heat-Fluctuation Theorems for a System with Deterministic and Stochastic Forces
• The Exclusion Process: A paradigm for non-equilibrium behaviour
• A minimal model of dynamical phase transition
• Thermodynamic formalism and large deviation functions in continuous time Markov dynamics

Cited by in corpus (131)

• Universal trade-off between power, efficiency, and constancy in steady-state heat engines
• Aspects of non-equilibrium in classical and quantum systems: slow relaxation and glasses, dynamical large deviations, quantum non-ergodicity, and open quantum dynamics
• Ergodicity and large deviations in physical systems with stochastic dynamics
• Learning force fields from stochastic trajectories
• Properties of additive functionals of Brownian motion with resetting
• Dynamical phase transition in drifted Brownian motion
• Anomalous scaling of dynamical large deviations
• Large deviations conditioned on large deviations I: Markov chain and Langevin equation
• Effect of a magnetic field on the thermodynamic uncertainty relation
• Direct Route to Thermodynamic Uncertainty Relations and Their Saturation
• Entropy production fluctuations encode collective behavior in active matter
• Large Deviation Function for a Driven Underdamped Particle in a Periodic Potential
• Effective driven dynamics for one-dimensional conditioned Langevin processes in the weak-noise limit
• Large deviations conditioned on large deviations II: Fluctuating hydrodynamics
• Large-deviation theory for a Brownian particle on a ring: a WKB approach
• Spectral theory of fluctuations in time-average statistical mechanics of reversible and driven systems
• Collective motion in large deviations of active particles
• Thermodynamic uncertainty relation for underdamped Langevin systems driven by a velocity-dependent force
• Dynamical large deviations of reflected diffusions
• Large deviations for continuous time random walks
• Learning nonequilibrium control forces to characterize dynamical phase transitions
• The Airy distribution: experiment, large deviations and additional statistics
• Anomalous scaling and first-order dynamical phase transition in large deviations of the Ornstein-Uhlenbeck process
• Microscopic Fluctuation Theory (mFT) for interacting Poisson processes
• Dynamical large deviations of linear diffusions
• Large deviations for Markov processes with stochastic resetting : analysis via the empirical density and flows or via excursions between resets
• Process interpretation of current entropic bounds
• Large deviations of the Lyapunov exponent in 2D matrix Langevin dynamics with applications to one-dimensional Anderson Localization models
• Large Deviations for the density and the current in Non-Equilibrium-Steady-States on disordered rings
• Nonequilibirum steady state for harmonically-confined active particles
• Anomalous scaling of dynamical large deviations of stationary Gaussian processes
• Condensation transition in large deviations of self-similar Gaussian processes with stochastic resetting
• Large deviations of currents in diffusions with reflective boundaries
• Mathematical, Thermodynamical, and Experimental Necessity for Coarse Graining Empirical Densities and Currents in Continuous Space
• Inertial effects on the Brownian gyrator
• Giant leaps and long excursions: fluctuation mechanisms in systems with long-range memory
• Large Deviations for dynamical fluctuations of Open Markov processes, with application to random cascades on trees
• Nonequilibrium diffusion processes via non-Hermitian electromagnetic quantum mechanics with application to the statistics of entropy production in the Brownian gyrator
• Exact quench dynamics of the Floquet quantum East model at the deterministic point
• Nonequilibrium thermodynamics of the asymmetric Sherrington-Kirkpatrick model
• Role of current fluctuations in nonreversible samplers
• Revisiting the Ruelle thermodynamic formalism for Markov trajectories with application to the glassy phase of random trap models
• Exponential volume dependence of entropy-current fluctuations at first-order phase transitions in chemical reaction networks
• On the molecular correlations that result in field-dependent conductivities in electrolyte solutions
• Optimal sampling of dynamical large deviations in two dimensions via tensor networks
• Coarse Graining Empirical Densities and Currents in Continuous-Space Steady States
• Large deviations at various levels for run-and-tumble processes with space-dependent velocities and space-dependent switching rates
• Large deviations in chaotic systems: exact results and dynamical phase transition
• Statistical physics of long dynamical trajectories for a system in contact with several thermal reservoirs
• Transport Coefficients from Large Deviation Functions
• Work fluctuations of self-propelled particles in the phase separated state
• Large deviations for the Skew-Detailed-Balance Lifted-Markov processes to sample the equilibrium distribution of the Curie-Weiss model
• A large deviation theory perspective on nanoscale transport phenomena
• Time-averaged height distribution of the Kardar-Parisi-Zhang interface
• Jump-Drift and Jump-Diffusion Processes : Large Deviations for the density, the current and the jump-flow and for the excursions between jumps
• Structural-dynamical transition in the Wahnström mixture
• Extreme events in stochastic transport on networks
• Noise correction of large deviations with anomalous scaling
• Large deviations for metastable states of Markov processes with absorbing states with applications to population models in stable or randomly switching environment
• Universal bounds on the performance of information-thermodynamic engine
• Anomalous dynamical large deviations of local empirical densities and activities in the pure and in the random kinetically-constrained East Model
• Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon
• Efficiency of one-dimensional active transport conditioned on motility
• Tricritical behavior in dynamical phase transitions
• A comparison of dynamical fluctuations of biased diffusion and run-and-tumble dynamics in one dimension
• Dynamical phase transition in the occupation fraction statistics for non-crossing Brownian particles
• Nonequilibrium steady state of trapped active particles
• Modeling of biomolecular machines in non-equilibrium steady states
• Large deviations of the stochastic area for linear diffusions
• Extinctions of coupled populations, and rare-event dynamics under non-Gaussian noise
• Large Deviations Theory of Increasing Returns
• Microcanonical conditioning of Markov processes on time-additive observables
• Anomalous scalings of fluctuations of the area swept by a Brownian particle trapped in a $|x|$ potential
• Large deviations for the Pearson family of ergodic diffusion processes involving a quadratic diffusion coefficient and a linear force
• Confined run and tumble particles with non-Markovian tumbling statistics
• Area fluctuations on a subinterval of Brownian excursion
• Large deviations in statistics of the local time and occupation time for a run and tumble particle
• Markov trajectories : Microcanonical Ensembles based on empirical observables as compared to Canonical Ensembles based on Markov generators
• Fractal catastrophes
• Delocalization-localization dynamical phase transition of random walks on graphs
• Efficient large deviation estimation based on importance sampling
• Revisiting boundary-driven non-equilibrium Markov dynamics in arbitrary potentials via supersymmetric quantum mechanics and explicit large deviations at various levels
• Large-deviation principles of switching Markov processes via Hamilton-Jacobi equations
• Convergence of the Integral Fluctuation Theorem estimator for nonequilibrium Markov systems
• Inhomogeneous asymmetric exclusion processes between two reservoirs : large deviations for the local empirical observables in the Mean-Field approximation
• Large deviations and conditioning for chaotic non-invertible deterministic maps: analysis via the forward deterministic dynamics and the backward stochastic dynamics
• An introduction to large deviations with applications in physics
• Force-dependent diffusion coefficient of molecular Brownian ratchets
• Adaptive power method for estimating large deviations in Markov chains
• Dynamical large deviations of the fractional Ornstein-Uhlenbeck process
• Large deviations in statistics of the convex hull of passive and active particles: A theoretical study
• Local time of an Ornstein-Uhlenbeck particle
• Physics-informed graph neural networks enhance scalability of variational nonequilibrium optimal control
• Partial versus total resetting for Lévy flights in d dimensions: similarities and discrepancies
• Rare Events and Single Big Jump Effects in Ornstein-Uhlenbeck Processes
• Asymptotic equivalence of probability measures and stochastic processes
• First-passage properties of the jump process with a drift. Two exactly solvable cases
• Martingale approach for first-passage problems of time-additive observables in Markov processes
• Nonequilibrium steady state of Brownian motion in an intermittent potential
• A giant disparity and a dynamical phase transition in large deviations of the time-averaged size of stochastic populations
• Conditioning diffusion processes with respect to the local time at the origin
• On correlations and fluctuations of time-averaged densities and currents with general time-dependence
• Efficient post-selection in light-cone correlations of monitored quantum circuits
• Feynman-Kac theory of time-integrated functionals: Itô versus functional calculus
• Joint distribution of two Local Times for diffusion processes with the application to the construction of various conditioned processes
• Infinite family of universal profiles for heat current statistics in Fourier's law
• On the Kemeny time for continuous-time reversible and irreversible Markov processes with applications to stochastic resetting and to conditioning towards forever-survival
• Current fluctuations of a self-interacting diffusion on a ring
• Explicit dynamical properties of the Pelikan random map in the chaotic region and at the intermittent critical point towards the non-chaotic region
• Large deviations at level 2.5 and for trajectories observables of diffusion processes : the missing parts with respect to their random-walks counterparts
• A supersymmetric quantum perspective on the explicit large deviations for reversible Markov jump processes, with applications to pure and random spin chains
• What is a chemostat? Insights from hybrid dynamics and stochastic thermodynamics
• Large deviations for trajectory observables of diffusion processes in dimension $d>1$ in the double limit of large time and small diffusion coefficient
• A pedestrian's approach to large deviations in semi-Markov processes with an application to entropy production
• Inverse problem in the conditioning of Markov processes on trajectory observables : what canonical conditionings can connect two given Markov generators ?
• Full distribution of the ground-state energy of potentials with weak disorder
• Markov spin models for image generation : explicit large deviations with respect to the number of pixels
• Subleading-order theory for condensation transitions in large deviations of sums of independent and identically distributed random variables
• Large Deviations of Irreversible Processes
• Large-deviations approach to thermalization: the case of harmonic chains with conservative noise
• Effective Langevin equations leading to large deviation function of time-averaged velocity for a nonequilibrium Rayleigh piston
• Exact large deviations and emergent long-range correlations in sequential quantum East circuits
• Microcanonical ensemble out of equilibrium
• Conditioning two diffusion processes with respect to their first-encounter properties
• A first passage problem for a Poisson counting process with a linear moving boundary
• Giant Fluctuations in Self-Propelled Particles with Age-Dependent Switching
• Density-dependent stochastic resetting: a large deviations framework for achieving target distributions over networks
• Large deviation of long time average for a stochastic process : an alternative method
• Thermodynamics of quantum-jump trajectories of open quantum systems subject to stochastic resetting
• Short-time large deviations of first-passage functionals for high-order stochastic processes
• Dimensionality-induced dynamical phase transition in the large deviation of local time density for Brownian motion