Anomalous scaling and first-order dynamical phase transition in large deviations of the Ornstein-Uhlenbeck process

Abstract

We study the full distribution of $A={\int }_0^T{x}^n\left(t\right)dt$, $n=1,2,\dots$, where $x\left(t\right)$ is an Ornstein-Uhlenbeck process. We find that for $n>2$ the long-time ($T\to \infty$) scaling form of the distribution is of the anomalous form $P\left(A;T\right)\sim e^{-T^{\mu }{f}_n\left(\Delta A/T^{\nu }\right)}$ where $\Delta A$ is the difference between $A$ and its mean value, and the anomalous exponents are $\mu =2/\left(2n-2\right)$, and $\nu =n/\left(2n-2\right)$. The rate function $f_n\left(y\right)$, that we calculate exactly, exhibits a first-order dynamical phase transition which separates between a homogeneous phase that describes the Gaussian distribution of typical fluctuations, and a "condensed" phase that describes the tails of the distribution. We also calculate the most likely realizations of $\mathcal{A}(t)={\int }_0^t{x}^n\left(s\right)ds$ and the distribution of $x(t)$ at an intermediate time $t$ conditioned on a given value of $A$. Extensions and implications to other continuous-time systems are discussed.

arXiv comment9 pages, 2 figures

References in corpus (43)

• The large deviation approach to statistical mechanics
• Nonequilibrium Statistical Mechanics of the Zero-Range Process and Related Models
• Stationary and Transient Work-Fluctuation Theorems for a Dragged Brownian Particle
• Statistical mechanics of a nonlinear discrete system
• Condensation in the zero range process: stationary and dynamical properties
• Introduction to dynamical large deviations of Markov processes
• Power and heat fluctuation theorems for electric circuits
• WKB theory of large deviations in stochastic populations
• The Nature of the Condensate in Mass Transport Models
• Canonical Analysis of Condensation in Factorised Steady State
• Generalised extreme value statistics and sum of correlated variables
• Large deviations for random walks under subexponentiality: the big-jump domain
• Wealth Condensation in Pareto Macro-Economies
• Mortality, Redundancy, and Diversity in Stochastic Search
• The Single Big Jump Principle in Physical Modelling
• Microscopic versus macroscopic approaches to non-equilibrium systems
• Sampling rare events: statistics of local sequence alignments
• A First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle
• Interaction driven real-space condensation
• Lévy walks and scaling in quenched disordered media
• Constraint driven condensation in large fluctuations of linear statistics
• Anomalous scaling of dynamical large deviations
• Condensation and Extreme Value Statistics
• Density profiles, dynamics, and condensation in the ZRP conditioned on an atypical current
• Large-Deviation Functions for Nonlinear Functionals of a Gaussian Stationary Markov Process
• Condensation transition in the late-time position of a Run-and-Tumble particle
• Localization transition in the Discrete Non-Linear Schrödinger Equation: ensembles inequivalence and negative temperatures
• Rare events in generalized Lévy Walks and the Big Jump principle
• Transport in disordered systems: the single big jump approach
• Large deviations of the maximum of independent and identically distributed random variables
• Rare events and scaling properties in field-induced anomalous dynamics
• Condensation transition in joint large deviations of linear statistics
• Condensation transition and ensemble inequivalence in the Discrete Nonlinear Schrödinger Equation
• Anomalous scaling of dynamical large deviations of stationary Gaussian processes
• Giant leaps and long excursions: fluctuation mechanisms in systems with long-range memory
• Superdiffusion and Transport in 2d-systems with Lévy Like Quenched Disorder
• Relaxation and coarsening of weakly-interacting breathers in a simplified DNLS chain
• Condensation Transition in Polydisperse Hard Rods
• Finite-size localization scenarios in condensation transitions
• Master equation for She-Leveque scaling and its classification in terms of other Markov models of developed turbulence
• Participation ratio for constraint-driven condensation with superextensive mass
• Conditioned random walks and interaction-driven condensation
• Large-scale lognormality in turbulence modeled by Ornstein-Uhlenbeck process

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