Heat and Fluctuations from Order to Chaos

The Heat theorem reveals the second law of equilibrium Thermodynamics (i.e.existence of Entropy) as a manifestation of a general property of Hamiltonian Mechanics and of the Ergodic Hypothesis, valid for 1 as well as $10^{23}$ degrees of freedom systems, {\it i.e.} for simple as well as very complex systems, and reflecting the Hamiltonian nature of the microscopic motion. In Nonequilibrium Thermodynamics theorems of comparable generality do not seem to be available. Yet it is possible to find general, model independent, properties valid even for simple chaotic systems ({\it i.e.} the hyperbolic ones), which acquire special interest for large systems: the Chaotic Hypothesis leads to the Fluctuation Theorem which provides general properties of certain very large fluctuations and reflects the time-reversal symmetry. Implications on Fluids and Quantum systems are briefly hinted. The physical meaning of the Chaotic Hypothesis, of SRB distributions and of the Fluctuation Theorem is discussed in the context of their interpretation and relevance in terms of Coarse Grained Partitions of phase space. This review is written taking some care that each section and appendix is readable either independently of the rest or with only few cross references.Comment: 1) added comment at the end of Sec. 1 to explain the meaning of the title (referee request) 2) added comment at the end of Sec. 17 (i.e. appendix A4) to refer to papers related to the ones already quoted (referee request

From order to chaos in Earth satellite orbits

We consider Earth satellite orbits in the range of semi-major axes where the perturbing effects of Earth's oblateness and lunisolar gravity are of comparable order. This range covers the medium-Earth orbits (MEO) of the Global Navigation Satellite Systems and the geosynchronous orbits (GEO) of the communication satellites. We recall a secular and quadrupolar model, based on the Milankovitch vector formulation of perturbation theory, which governs the long-term orbital evolution subject to the predominant gravitational interactions. We study the global dynamics of this two-and-a-half degrees-of-freedom Hamiltonian system by means of the fast Lyapunov indicator (FLI), used in a statistical sense. Specifically, we characterize the degree of chaoticity of the action space using angle-averaged normalized FLI maps, thereby overcoming the angle dependencies of the conventional stability maps. Emphasis is placed upon the phase-space structures near secular resonances, which are of first importance to the space debris community. We confirm and quantify the transition from order to chaos in MEO, stemming from the critical inclinations, and find that highly inclined GEO orbits are particularly unstable. Despite their reputed normality, Earth satellite orbits can possess an extraordinarily rich spectrum of dynamical behaviors, and, from a mathematical perspective, have all the complications that make them very interesting candidates for testing the modern tools of chaos theory.Comment: 30 pages, 9 figures. Accepted for publication in the Astronomical Journa

The Control of Dynamical Systems - Recovering Order from Chaos -

Following a brief historical introduction of the notions of chaos in dynamical systems, we will present recent developments that attempt to profit from the rich structure and complexity of the chaotic dynamics. In particular, we will demonstrate the ability to control chaos in realistic complex environments. Several applications will serve to illustrate the theory and to highlight its advantages and weaknesses. The presentation will end with a survey of possible generalizations and extensions of the basic formalism as well as a discussion of applications outside the field of the physical sciences. Future research avenues in this rapidly growing field will also be addressed.Comment: 18 pages, 9 figures. Invited Talk at the XXIth International Conference on the Physics of Electronic and Atomic Collisions (ICPEAC), July 22-27, 1999 (Sendai, Japan

Low-dimensional chaos induced by frustration in a non-monotonic system

We report a novel mechanism for the occurrence of chaos at the macroscopic level induced by the frustration of interaction, namely frustration-induced chaos, in a non-monotonic sequential associative memory model. We succeed in deriving exact macroscopic dynamical equations from the microscopic dynamics in the case of the thermodynamic limit and prove that two order parameters dominate this large-degree-of-freedom system. Two-parameter bifurcation diagrams are obtained from the order-parameter equations. Then we analytically show that the chaos is low-dimensional at the macroscopic level when the system has some degree of frustration, but that the chaos definitely does not occur without the frustration.Comment: 2 figure

On Birthing Dancing Stars: The Need for Bounded Chaos in Information Interaction

While computers causing chaos is acommon social trope, nearly the entirety of the history of computing is dedicated to generating order. Typical interactive information retrieval tasks ask computers to support the traversal and exploration of large, complex information spaces. The implicit assumption is that they are to support users in simplifying the complexity (i.e. in creating order from chaos). But for some types of task, particularly those that involve the creative application or synthesis of knowledge or the creation of new knowledge, this assumption may be incorrect. It is increasingly evident that perfect order—and the systems we create with it—support highly-structured information tasks well, but provide poor support for less-structured tasks.We need digital information environments that help create a little more chaos from order to spark creative thinking and knowledge creation. This paper argues for the need for information systems that offerwhat we term ‘bounded chaos’, and offers research directions that may support the creation of such interface

Wigner chaos and the fourth moment

We prove that a normalized sequence of multiple Wigner integrals (in a fixed order of free Wigner chaos) converges in law to the standard semicircular distribution if and only if the corresponding sequence of fourth moments converges to 2, the fourth moment of the semicircular law. This extends to the free probabilistic, setting some recent results by Nualart and Peccati on characterizations of central limit theorems in a fixed order of Gaussian Wiener chaos. Our proof is combinatorial, analyzing the relevant noncrossing partitions that control the moments of the integrals. We can also use these techniques to distinguish the first order of chaos from all others in terms of distributions; we then use tools from the free Malliavin calculus to give quantitative bounds on a distance between different orders of chaos. When applied to highly symmetric kernels, our results yield a new transfer principle, connecting central limit theorems in free Wigner chaos to those in Gaussian Wiener chaos. We use this to prove a new free version of an important classical theorem, the Breuer-Major theorem.Comment: Published in at http://dx.doi.org/10.1214/11-AOP657 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Holographic Order from Modular Chaos

We argue for an exponential bound characterizing the chaotic properties of modular Hamiltonian flow of QFT subsystems. In holographic theories, maximal modular chaos is reflected in the local Poincare symmetry about a Ryu-Takayanagi surface. Generators of null deformations of the bulk extremal surface map to modular scrambling modes -positive CFT operators saturating the bound- and their algebra probes the bulk Riemann curvature, clarifying the modular Berry curvature proposal of arXiv:1903.04493.Comment: 22 pages + Appendi

Transition from order to chaos in the wake of an airfoil

An experimental effort is presented here that examines the nonlinear interaction of multiple frequencies in the forced wake of an airfoil. Wakes with one or two distinct frequencies behave in an ordered manner – being either locked or quasi-periodic. When a third incommensurate frequency is added to the system, the flow demonstrates chaotic behaviour. Previously, the existence of the three-frequency route to chaos has been reported only for closed system flows. It is important to note that this chaotic state is obtained at a low Reynolds number. However, the chaotic flow shows localized characteristics similar to those of high Reynolds number turbulent flows. The degree of chaotic behaviour is verified by applying ideas from nonlinear dynamics (such as Lyapunov exponents and Poincaré sections) to the experimental data, thus relating the basic physics of the system to the concepts of mode interaction and chaos. Significant changes to the vortex configuration in the wake and to the r.m.s. velocity profile occur during the transition from order to chaos