Asymptotics of entropy rate of hidden Markov chains at weak Black Holes

We generalize a result in [8] and derive an asymptotic formula for entropy rate of a hidden Markov chain around a "weak Black Hole". We also discuss applications of the asymptotic formula to certain channels. © 2008 IEEE.published_or_final_versio

Analyticity of Entropy Rate of Hidden Markov Chains With Continuous Alphabet

We first prove that under certain mild assumptions, the entropy rate of a hidden Markov chain, observed when passing a finite-state stationary Markov chain through a discrete-time continuous-output channel, is analytic with respect to the input Markov chain parameters. We then further prove, under strengthened assumptions on the chan- nel, that the entropy rate is jointly analytic as a function of both the input Markov chain parameters and the channel parameters. In particular, the main theorems estab- lish the analyticity of the entropy rate for two representative channels: Cauchy and Gaussian.published_or_final_versio

Operator Spreading in Random Unitary Circuits

Random quantum circuits yield minimally structured models for chaotic quantum dynamics, able to capture for example universal properties of entanglement growth. We provide exact results and coarse-grained models for the spreading of operators by quantum circuits made of Haar-random unitaries. We study both 1+1D and higher dimensions, and argue that the coarse-grained pictures carry over to operator spreading in generic many-body systems. In 1+1D, we demonstrate that the out-of-time-order correlator (OTOC) satisfies a biased diffusion equation, which gives exact results for the spatial profile of the OTOC, and the butterfly speed $v_{B}$. We find that in 1+1D the `front' of the OTOC broadens diffusively, with a width scaling in time as $t^{1/2}$. We address fluctuations in the OTOC between different realizations of the random circuit, arguing that they are negligible in comparison to the broadening of the front. Turning to higher D, we show that the averaged OTOC can be understood exactly via a remarkable correspondence with a classical droplet growth problem. This implies that the width of the front of the averaged OTOC scales as $t^{1/3}$ in 2+1D and $t^{0.24}$ in 3+1D (KPZ exponents). We support our analytic argument with simulations in 2+1D. We point out that, in a lattice model, the late time shape of the spreading operator is in general not spherical. However when full spatial rotational symmetry is present in 2+1D, our mapping implies an exact asymptotic form for the OTOC in terms of the Tracy-Widom distribution. For an alternative perspective on the OTOC in 1+1D, we map it to the partition function of an Ising-like model. As a result of special structure arising from unitarity, this partition function reduces to a random walk calculation which can be performed exactly. We also use this mapping to give exact results for entanglement growth in 1+1D circuits.Comment: 29 pages, 16 figures. v2: new appendix on 'mean field

Scaling Limits in Models of Statistical Mechanics

The workshop brought together researchers interested in spatial random processes and their connection to statistical mechanics. The principal subjects of interest were scaling limits and, in general, limit laws for various two-dimensional critical models, percolation, random walks in random environment, polymer models, random fields and hierarchical diffusions. The workshop fostered interactions between groups of researchers in these areas and led to interesting and fruitful exchanges of ideas

From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics

This review gives a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from quantum chaos and random matrix theory (RMT). To this end, we present a brief overview of classical and quantum chaos, as well as RMT and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the implications of quantum chaos and ETH to thermodynamics. Basic thermodynamic relations are derived, including the second law of thermodynamics, the fundamental thermodynamic relation, fluctuation theorems, the fluctuation–dissipation relation, and the Einstein and Onsager relations. In particular, it is shown that quantum chaos allows one to prove these relations for individual Hamiltonian eigenstates and thus extend them to arbitrary stationary statistical ensembles. In some cases, it is possible to extend their regimes of applicability beyond the standard thermal equilibrium domain. We then show how one can use these relations to obtain nontrivial universal energy distributions in continuously driven systems. At the end of the review, we briefly discuss the relaxation dynamics and description after relaxation of integrable quantum systems, for which ETH is violated. We present results from numerical experiments and analytical studies of quantum quenches at integrability. We introduce the concept of the generalized Gibbs ensemble and discuss its connection with ideas of prethermalization in weakly interacting systems.This work was supported by the Army Research Office [grant number W911NF1410540] (L.D., A.P, and M.R.), the U.S.-Israel Binational Science Foundation [grant number 2010318] (Y.K. and A.P.), the Israel Science Foundation [grant number 1156/13] (Y.K.), the National Science Foundation [grant numbers DMR-1506340 (A.P.)and PHY-1318303 (M.R.)], the Air Force Office of Scientific Research [grant number FA9550-13-1-0039] (A.P.), and the Office of Naval Research [grant number N000141410540] (M.R.). The computations were performed in the Institute for CyberScience at Penn State. (W911NF1410540 - Army Research Office; 2010318 - U.S.-Israel Binational Science Foundation; 1156/13 - Israel Science Foundation; DMR-1506340 - National Science Foundation; PHY-1318303 - National Science Foundation; FA9550-13-1-0039 - Air Force Office of Scientific Research; N000141410540 - Office of Naval Research)Accepted manuscrip

Cosmological Inflation, Dark Matter and Dark Energy

Various cosmological observations support not only cosmological inflation in the early universe, which is also known as exponential cosmic expansion, but also that the expansion of the late-time universe is accelerating. To explain this phenomenon, the existence of dark energy is proposed. In addition, according to the rotation curve of galaxies, the existence of dark matter, which does not shine, is also suggested. If primordial gravitational waves are detected in the future, the mechanism for realizing inflation can be revealed. Moreover, there exist two main candidates for dark matter. The first is a new particle, the existence of which is predicted in particle physics. The second is an astrophysical object which is not found by electromagnetic waves. Furthermore, there are two representative approaches to account for the accelerated expansion of the current universe. One is to assume the unknown dark energy in general relativity. The other is to extend the gravity theory to large scales. Investigation of the origins of inflation, dark matter, and dark energy is one of the most fundamental problems in modern physics and cosmology. The purpose of this book is to explore the physics and cosmology of inflation, dark matter, and dark energy