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1. Large Deviation Estimates and Hölder Regularity of the Lyapunov Exponents for Quasi-periodic Schrödinger Cocycles

   https://doi.org/10.1093/imrn/rnz319  

   Han, Rui; Zhang, Shiwen (May 2020, International Mathematics Research Notices)

   Abstract We consider one-dimensional quasi-periodic Schrödinger operators with analytic potentials. In the positive Lyapunov exponent regime, we prove large deviation estimates, which lead to refined Hölder continuity of the Lyapunov exponents and the integrated density of states, in both small Lyapunov exponent and large coupling regimes. Our results cover all the Diophantine frequencies and some Liouville frequencies. 

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2. Quantum chaos in a weakly-coupled field theory with nonlocality

   https://doi.org/10.1007/JHEP09(2022)097  

   Fischler, Willy; Guglielmo, Tyler; Nguyen, Phuc (September 2022, Journal of High Energy Physics)

   A bstract In order to study the chaotic behavior of a system with non-local interactions, we will consider weakly coupled non-commutative field theories. We compute the Lyapunov exponent of this exponential growth in the large Moyal-scale limit to leading order in the t’Hooft coupling and 1/ N . We found that in this limit, the Lyapunov exponent remains comparable in magnitude to (and somewhat smaller than) the exponent in the commutative case. This can possibly be explained by the infrared sensitivity of the Lyapunov exponent. Another possible explanation is that in examples of weakly coupled non-commutative field theories, non-local contributions to various thermodynamic quantities are sub-dominant. 

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3. Effective multi-scale approach to the Schrödinger cocycle over a skew-shift base

   https://doi.org/10.1017/etds.2019.19  

   HAN, R.; LEMM, M.; SCHLAG, W. (October 2019, Ergodic Theory and Dynamical Systems)

   We prove a conditional theorem on the positivity of the Lyapunov exponent for a Schrödinger cocycle over a skew-shift base with a cosine potential and the golden ratio as frequency. For coupling below $$1$$ , which is the threshold for Herman’s subharmonicity trick, we formulate three conditions on the Lyapunov exponent in a finite but large volume and on the associated large-deviation estimates at that scale. Our main results demonstrate that these finite-size conditions imply the positivity of the infinite-volume Lyapunov exponent. This paper shows that it is possible to make the techniques developed for the study of Schrödinger operators with deterministic potentials, based on large-deviation estimates and the avalanche principle, effective. 

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4. On the Norm Equivalence of Lyapunov Exponents for Regularizing Linear Evolution Equations

   https://doi.org/10.1007/s00205-023-01928-y  

   Blumenthal, Alex; Punshon-Smith, Sam (October 2023, Archive for Rational Mechanics and Analysis)

   We consider the top Lyapunov exponent associated to a dissipative linear evolution equation posed on a separable Hilbert or Banach space. In many applications in partial differential equations, such equations are often posed on a scale of nonequivalent spaces mitigating, e.g., integrability (L^p) or differentiability (W^{s,p}). In contrast to finite dimensions, the Lyapunov exponent could a priori depend on the choice of norm used. In this paper we show that under quite general conditions, the Lyapunov exponent of a cocycle of compact linear operators is independent of the norm used. We apply this result to two important problems from fluid mechanics: the enhanced dissipation rate for the advection diffusion equation with ergodic velocity field; and the Lyapunov exponent for the 2d Navier–Stokes equations with stochastic or periodic forcing. 

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5. Aquila-LCS: GPU/CPU-Accelerated Particle Advection Schemes for Large-Scale Simulations

   Christian Lagares and Guillermo Araya (March 2024, Software X)

   We introduce Aquila-LCS, GPU and CPU optimized object-oriented, in-house codes for volumetric particle advection and 3D Finite-Time Lyapunov Exponent (FTLE) and Finite-Size Lyapunov Exponent (FSLE) computations. The purpose is to analyze 3D Lagrangian Coherent Structures (LCS) in large Direct Numerical Simulation (DNS) data. Our technique uses advanced search strategies for quick cell identification and efficient storage techniques. This solver scales effectively on both GPUs (up to 62 Nvidia V100 GPUs) and multi-core CPUs (up to 32,768 CPU cores), tracking up to 8-billion particles. We apply our approach to four turbulent boundary layers at different flow regimes and Reynolds numbers. 

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