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

Annealed and quenched limit theorems for random expanding dynamical systems

In this paper, we investigate annealed and quenched limit theorems for random expanding dynamical systems. Making use of functional analytic techniques and more probabilistic arguments with martingales, we prove annealed versions of a central limit theorem, a large deviation principle, a local limit theorem, and an almost sure invariance principle. We also discuss the quenched central limit theorem, dynamical Borel-Cantelli lemmas, Erd\"os-R\'enyi laws and concentration inequalities.Comment: Appeared online in Probability Theory and Related Field

Topological Charge and the Spectrum of the Fermion Matrix in Lattice-QED_2

We investigate the interplay between topological charge and the spectrum of the fermion matrix in lattice-QED_2 using analytic methods and Monte Carlo simulations with dynamical fermions. A new theorem on the spectral decomposition of the fermion matrix establishes that its real eigenvalues (and corresponding eigenvectors) play a role similar to the zero eigenvalues (zero modes) of the Dirac operator in continuous background fields. Using numerical techniques we concentrate on studying the real part of the spectrum. These results provide new insights into the behaviour of physical quantities as a function of the topological charge. In particular we discuss fermion determinant, effective action and pseudoscalar densities.Comment: 33 pages, 10 eps-figures; reference adde

Non-Equilibrium Statistical Physics of Currents in Queuing Networks

We consider a stable open queuing network as a steady non-equilibrium system of interacting particles. The network is completely specified by its underlying graphical structure, type of interaction at each node, and the Markovian transition rates between nodes. For such systems, we ask the question ``What is the most likely way for large currents to accumulate over time in a network ?'', where time is large compared to the system correlation time scale. We identify two interesting regimes. In the first regime, in which the accumulation of currents over time exceeds the expected value by a small to moderate amount (moderate large deviation), we find that the large-deviation distribution of currents is universal (independent of the interaction details), and there is no long-time and averaged over time accumulation of particles (condensation) at any nodes. In the second regime, in which the accumulation of currents over time exceeds the expected value by a large amount (severe large deviation), we find that the large-deviation current distribution is sensitive to interaction details, and there is a long-time accumulation of particles (condensation) at some nodes. The transition between the two regimes can be described as a dynamical second order phase transition. We illustrate these ideas using the simple, yet non-trivial, example of a single node with feedback.Comment: 26 pages, 5 figure

On Energy Conditions and Stability in Effective Loop Quantum Cosmology

In isotropic loop quantum cosmology, non-perturbatively modified dynamics of a minimally coupled scalar field violates weak, strong and dominant energy conditions when they are stated in terms of equation of state parameter. The violation of strong energy condition helps to have non-singular evolution by evading singularity theorems thus leading to a generic inflationary phase. However, the violation of weak and dominant energy conditions raises concern, as in general relativity these conditions ensure causality of the system and stability of vacuum via Hawking-Ellis conservation theorem. It is shown here that the non-perturbatively modified kinetic term contributes negative pressure but positive energy density. This crucial feature leads to violation of energy conditions but ensures positivity of energy density, as scalar matter Hamiltonian remains bounded from below. It is also shown that the modified dynamics restricts group velocity for inhomogeneous modes to remain sub-luminal thus ensuring causal propagation across spatial distances.Comment: 29 pages, revtex4; few clarifications, references added, to appear in CQ