Geometry of the ergodic quotient reveals coherent structures in flows

Dynamical systems that exhibit diverse behaviors can rarely be completely understood using a single approach. However, by identifying coherent structures in their state spaces, i.e., regions of uniform and simpler behavior, we could hope to study each of the structures separately and then form the understanding of the system as a whole. The method we present in this paper uses trajectory averages of scalar functions on the state space to: (a) identify invariant sets in the state space, (b) form coherent structures by aggregating invariant sets that are similar across multiple spatial scales. First, we construct the ergodic quotient, the object obtained by mapping trajectories to the space of trajectory averages of a function basis on the state space. Second, we endow the ergodic quotient with a metric structure that successfully captures how similar the invariant sets are in the state space. Finally, we parametrize the ergodic quotient using intrinsic diffusion modes on it. By segmenting the ergodic quotient based on the diffusion modes, we extract coherent features in the state space of the dynamical system. The algorithm is validated by analyzing the Arnold-Beltrami-Childress flow, which was the test-bed for alternative approaches: the Ulam's approximation of the transfer operator and the computation of Lagrangian Coherent Structures. Furthermore, we explain how the method extends the Poincar\'e map analysis for periodic flows. As a demonstration, we apply the method to a periodically-driven three-dimensional Hill's vortex flow, discovering unknown coherent structures in its state space. In the end, we discuss differences between the ergodic quotient and alternatives, propose a generalization to analysis of (quasi-)periodic structures, and lay out future research directions.Comment: Submitted to Elsevier Physica D: Nonlinear Phenomen

Gauge/string duality for QCD conformal operators

Renormalization group evolution of QCD composite light-cone operators, built from two and more quark and gluon fields, is responsible for the logarithmic scaling violations in diverse physical observables. We analyze spectra of anomalous dimensions of these operators at large conformal spins at weak and strong coupling with the emphasis on the emergence of a dual string picture. The multi-particle spectrum at weak coupling has a hidden symmetry due to integrability of the underlying dilatation operator which drives the evolution. In perturbative regime, we demonstrate the equivalence of the one-loop cusp anomaly to the disk partition function in two-dimensional Yang-Mills theory which admits a string representation. The strong coupling regime for anomalous dimensions is discussed within the two pictures addressed recently, -- minimal surfaces of open strings and rotating long closed strings in AdS background. In the latter case we find that the integrability implies the presence of extra degrees of freedom -- the string junctions. We demonstrate how the analysis of their equations of motion naturally agrees with the spectrum found at weak coupling.Comment: Latex, 59 pages, 6 figure

Collective variables between large-scale states in turbulent convection

The dynamics in a confined turbulent convection flow is dominated by multiple long-lived macroscopic circulation states, which are visited subsequently by the system in a Markov-type hopping process. In the present work, we analyze the short transition paths between these subsequent macroscopic system states by a data-driven learning algorithm that extracts the low-dimensional transition manifold and the related new coordinates, which we term collective variables, in the state space of the complex turbulent flow. We therefore transfer and extend concepts for conformation transitions in stochastic microscopic systems, such as in the dynamics of macromolecules, to a deterministic macroscopic flow. Our analysis is based on long-term direct numerical simulation trajectories of turbulent convection in a closed cubic cell at a Prandtl number $Pr = 0.7$ and Rayleigh numbers $Ra = 10^6$ and $10^7$ for a time lag of $10^5$ convective free-fall time units. The simulations resolve vortices and plumes of all physically relevant scales resulting in a state space spanned by more than 3.5 million degrees of freedom. The transition dynamics between the large-scale circulation states can be captured by the transition manifold analysis with only two collective variables which implies a reduction of the data dimension by a factor of more than a million. Our method demonstrates that cessations and subsequent reversals of the large-scale flow are unlikely in the present setup and thus paves the way to the development of efficient reduced-order models of the macroscopic complex nonlinear dynamical system.Comment: 24 pages, 12 Figures, 1 tabl

c-map as c=1 string

We show the existence of a duality between the c-map space describing the universal hypermultiplet at tree level and the matrix model description of two-dimensional string theory compactified at a self-dual radius and perturbed by a sine-Liouville potential. It appears as a particular case of a general relation between the twistor description of four-dimensional quaternionic geometries and the Lax formalism for Toda hierarchy. Furthermore, we give an evidence that the instanton corrections to the c-map metric coming from NS5-branes can be encoded into the Baker-Akhiezer function of the integrable hierarchy.Comment: 19 pages, 2 figure