Matrix Quantization of Turbulence

Based on our recent work on Quantum Nambu Mechanics \cite{af2}, we provide an explicit quantization of the Lorenz chaotic attractor through the introduction of Non-commutative phase space coordinates as Hermitian $N \times N$ matrices in $R^{3}$. For the volume preserving part, they satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction it violates the quantum commutation relations. We demonstrate that the Heisenberg-Nambu evolution equations for the Matrix Lorenz system develop fast decoherence to N independent Lorenz attractors. On the other hand there is a weak dissipation regime, where the quantum mechanical properties of the volume preserving non-dissipative sector survive for long times.Comment: 14 pages, Based on invited talks delivered at: Fifth Aegean Summer School, "From Gravity to Thermal Gauge theories and the AdS/CFT Correspondance", September 2009, Milos, Greece; the Intern. Conference on Dynamics and Complexity, Thessaloniki, Greece, 12 July 2010; Workshop on "AdS4/CFT3 and the Holographic States of Matter", Galileo Galilei Institute, Firenze, Italy, 30 October 201

Phase Space Geometry and Chaotic Attractors in Dissipative Nambu Mechanics

Following the Nambu mechanics framework we demonstrate that the non-dissipative part of the Lorenz system can be generated by the intersection of two quadratic surfaces that form a doublet under the group SL(2,R). All manifolds are classified into four dinstict classes; parabolic, elliptical, cylindrical and hyperbolic. The Lorenz attractor is localized by a specific infinite set of one parameter family of these surfaces. The different classes correspond to different physical systems. The Lorenz system is identified as a charged rigid body in a uniform magnetic field with external torque and this system is generalized to give new strange attractors.Comment: 22 pages, 13 figure

Strange Attractors in Dissipative Nambu Mechanics : Classical and Quantum Aspects

We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation in $R^{3}$ phase space. We demonstrate that it accommodates the phase space dynamics of low dimensional dissipative systems such as the much studied Lorenz and R\"{o}ssler Strange attractors, as well as the more recent constructions of Chen and Leipnik-Newton. The rotational, volume preserving part of the flow preserves in time a family of two intersecting surfaces, the so called {\em Nambu Hamiltonians}. They foliate the entire phase space and are, in turn, deformed in time by Dissipation which represents their irrotational part of the flow. It is given by the gradient of a scalar function and is responsible for the emergence of the Strange Attractors. Based on our recent work on Quantum Nambu Mechanics, we provide an explicit quantization of the Lorenz attractor through the introduction of Non-commutative phase space coordinates as Hermitian $N \times N$ matrices in $R^{3}$. They satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction it violates the quantum commutation relations. We demonstrate that the Heisenberg-Nambu evolution equations for the Quantum Lorenz system give rise to an attracting ellipsoid in the $3 N^{2}$ dimensional phase space.Comment: 35 pages, 4 figures, LaTe

Study of the 3D Euler equations using Clebsch potentials: dual mechanisms for geometric depletion (vol 31, pg R25, 2018)

After the publication of [1], it has come to the author’s attention that a class of Clebsch potentials for the Kida-Pelz flow, similar to what was derived in Appendix B of [1], has been studied in detail in [2]. We also note that there are typos in the formulas for one such example in [3], and these are corrected in [1]

Observational evidence for exponential tornado intensity distributions over specific kinetic energy

Observational evidence supports the recent analytical prediction that tornado intensities are exponentially distributed over peak wind speed squared (nu(2)), or equivalently, Rayleigh-distributed over v. For large USA data samples, exponential tails are found in the tornado intensity distributions over nu(2) from about F2 intensity on. Similar results follow for smaller worldwide data samples. For the 1990s data from the USA and Oklahoma, deviations from the Rayleigh distribution for weak tornadoes can be explained by the emergence of a separate, likely non-mesocyclonic tornado mode. These bimodal datasets can be modeled by superposition of two Rayleigh distributions. The change in modal dominance occurs at about the F2 threshold (nu approximate to 50 m s(-1)). In France, likely mainly the mesocyclonic tornado mode has been recorded, while in the UK, only a non-mesocyclonic mode seems to be present

A group action principle for Nambu dynamics of spin degrees of freedom

International audienceWe describe a formulation of the group action principle, for linear Nambu flows, that explicitly takes into account all the defining properties of Nambu mechanics and illustrate its relevance by showing how it can be used to describe the off–shell states and superpositions thereof that define the transition amplitudes for the quantization of Larmor precession of a magnetic moment. It highlights the relation between the fluctuations of the longitudinal and transverse components of the magnetization. This formulation has been shown to be consistent with the approach that has been developed in the framework of the non commutative geometry of the 3–torus. In this way the latter can be used as a consistent discretization of the former