Introduction to dynamical large deviations of Markov processes

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.Comment: 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. I dedicate this paper to the memory of E. G. D. Cohen (1923-2017

Asymptotic equivalence of probability measures and stochastic processes

Let $P_n$ and $Q_n$ be two probability measures representing two different probabilistic models of some system (e.g., an $n$-particle equilibrium system, a set of random graphs with $n$ vertices, or a stochastic process evolving over a time $n$) and let $M_n$ be a random variable representing a 'macrostate' or 'global observable' of that system. We provide sufficient conditions, based on the Radon-Nikodym derivative of $P_n$ and $Q_n$, for the set of typical values of $M_n$ obtained relative to $P_n$ to be the same as the set of typical values obtained relative to $Q_n$ in the limit $n\rightarrow\infty$. This extends to general probability measures and stochastic processes the well-known thermodynamic-limit equivalence of the microcanonical and canonical ensembles, related mathematically to the asymptotic equivalence of conditional and exponentially-tilted measures. In this more general sense, two probability measures that are asymptotically equivalent predict the same typical or macroscopic properties of the system they are meant to model.Comment: v1: 16 pages. v2: 17 pages, precisions, examples and references added. v3: Minor typos corrected. Close to published versio

Comment on "Towards a large deviation theory for strongly correlated systems"

I comment on a recent paper by Ruiz and Tsallis [Phys. Lett. A 376, 2451 (2012)] claiming to have found a '$q$-exponential' generalization of the large deviation principle for strongly correlated random variables. I show that the basic scaling results that they find numerically can be reproduced with a simple example involving independent random variables, and are not specifically related to the $q$-exponential function. In fact, identical scaling results can be obtained with any other power-law deformations of the exponential. Thus their results do not conclusively support their claim of a $q$-exponential generalization of the large deviation principle.Comment: Comment, 3 pages, 2 figure