An introduction to large deviations with applications in physics

Abstract

These notes are based on the lectures that one of us (HT) gave at the Summer School on the "Theory of Large Deviations and Applications", held in July 2024 at Les Houches in France. They present the basic definitions and mathematical results that form the theory of large deviations, as well as many simple motivating examples of applications in statistical physics, which serve as a basis for the many other lectures given at the school that covered more specific applications in biophysics, random matrix theory, nonequilibrium systems, geophysics, and the simulation of rare events, among other topics. These notes extend the lectures, which can be accessed online, by presenting exercises and pointer references for further reading.

48 pages, 12 figures, Lecture notes for Les Houches summer school 2024, typos corrected, minor revisions

References in corpus (19)

• The large deviation approach to statistical mechanics
• Nonequilibrium Markov processes conditioned on large deviations
• Large deviations and ensembles of trajectories in stochastic models
• Nonequilibrium microcanonical and canonical ensembles and their equivalence
• Introduction to dynamical large deviations of Markov processes
• Aspects of non-equilibrium in classical and quantum systems: slow relaxation and glasses, dynamical large deviations, quantum non-ergodicity, and open quantum dynamics
• Variational and optimal control representations of conditioned and driven processes
• Effective interactions and large deviations in stochastic processes
• A formal view on 2.5 large deviations and fluctuation relations
• Rules for transition rates in nonequilibrium steady states
• Current large deviations for driven periodic diffusions
• Detailed balance has a counterpart in non-equilibrium steady states
• Diffusions conditioned on occupation measures
• A basic introduction to large deviations: Theory, applications, simulations
• Dynamical large deviations of reflected diffusions
• Learning nonequilibrium control forces to characterize dynamical phase transitions
• Dynamical large deviations of linear diffusions
• Large deviations of currents in diffusions with reflective boundaries
• Large deviations

Cited by in corpus (3)

• Remarkable similarities in distributions of dynamical observables in chaotic systems
• A first passage problem for a Poisson counting process with a linear moving boundary
• Classification of diffusion processes in dimension $d$ via the Carleman approach with applications to models involving additive, multiplicative or square-root noises