Non Hamiltonian Chaos from Nambu Dynamics of Surfaces

We discuss recent work with E.Floratos (JHEP 1004:036,2010) on Nambu Dynamics of Intersecting Surfaces underlying Dissipative Chaos in $R^{3}$. We present our argument for the well studied Lorenz and R\"{o}ssler strange attractors. We implement a flow decomposition to their equations of motion. Their volume preserving part preserves in time a family of two intersecting surfaces, the so called {\em Nambu Hamiltonians}. For dynamical systems with linear dissipative sector such as the Lorenz system, they are specified in terms of Intersecting Quadratic Surfaces. For the case of the R\"{o}ssler system, with nonlinear dissipative part, they are given in terms of a Helicoid intersected by a Cylinder. In each case they foliate the entire phase space and get deformed by Dissipation, the irrotational component to their flow. It is given by the gradient of a surface in $R^{3}$ specified in terms of a scalar function. All three intersecting surfaces reproduce completely the dynamics of each strange attractor.Comment: 10 pages, Invited Talks at the International Conferences on: Nonlinear Dynamics and Complexity; Theory, Methods and Applications, 12-16 July 2010, Thessaloniki, Greece; Second Greek-Turkish Conference on Statistical Mechanics and Dynamical Systems, Turunc-Symi, 5-12 September 201

Charged Cosmic String Nucleation in de Sitter Space

We investigate the quantum nucleation of pairs of charged circular cosmic strings in de Sitter space. By including self-gravity we obtain the classical potential energy barrier and compute the quantum mechanical tunneling probability in the semiclassical approximation. We also discuss the classical evolution of charged circular strings after their nucleation.Comment: 12 pages Latex + 3 figures (not included), Nordita 94/38

The Omega-Infinity Limit of Single Spikes

A new infinite-size limit of strings in RxS2 is presented. The limit is obtained from single spike strings by letting by letting the angular velocity parameter omega become infinite. We derive the energy-momenta relation of omega-infinity single spikes as their linear velocity v-->1 and their angular momentum J-->1. Generally, the v-->1, J-->1 limit of single spikes is singular and has to be excluded from the spectrum and be studied separately. We discover that the dispersion relation of omega-infinity single spikes contains logarithms in the limit J-->1. This result is somewhat surprising, since the logarithmic behavior in the string spectra is typically associated with their motion in non-compact spaces such as AdS. Omega-infinity single spikes seem to completely cover the surface of the 2-sphere they occupy, so that they may essentially be viewed as some sort of "brany strings". A proof of the sphere-filling property of omega-infinity single spikes is given in the appendix.Comment: 35 pages, 14 figures. Matches published version; Contains equation (4.21) that gives the first few finite-size corrections to the energy of omega-infinity single spike

Chaotic Information Processing by Extremal Black Holes

We review an explicit regularization of the AdS$_2$/CFT$_1$ correspondence, that preserves all isometries of bulk and boundary degrees of freedom. This scheme is useful to characterize the space of the unitary evolution operators that describe the dynamics of the microstates of extremal black holes in four spacetime dimensions. Using techniques from algebraic number theory to evaluate the transition amplitudes, we remark that the regularization scheme expresses the fast quantum computation capability of black holes as well as its chaotic nature.Comment: 8 pages, 2 JPEG figues. Contribution to the VII Black Holes Workshop, Aveiro PT, Decemeber 201

The quantum cat map on the modular discretization of extremal black hole horizons

Based on our recent work on the discretization of the radial AdS$_2$ geometry of extremal BH horizons,we present a toy model for the chaotic unitary evolution of infalling single particle wave packets. We construct explicitly the eigenstates and eigenvalues for the single particle dynamics for an observer falling into the BH horizon, with time evolution operator the quantum Arnol'd cat map (QACM). Using these results we investigate the validity of the eigenstate thermalization hypothesis (ETH), as well as that of the fast scrambling time bound (STB). We find that the QACM, while possessing a linear spectrum, has eigenstates, which are random and satisfy the assumptions of the ETH. We also find that the thermalization of infalling wave packets in this particular model is exponentially fast, thereby saturating the STB, under the constraint that the finite dimension of the single--particle Hilbert space takes values in the set of Fibonacci integers.Comment: 28 pages LaTeX2e, 8 jpeg figures. Clarified certain issues pertaining to the relation between mixing time and scrambling time; enhanced discussion of the Eigenstate Thermalization Hypothesis; revised figures and updated references. Typos correcte

M2-brane Dynamics in the Classical Limit of the BMN Matrix Model

We investigate the large-N limit of the BMN matrix model by analyzing the dynamics of ellipsoidal M2-branes that spin in the 11-dimensional maximally supersymmetric SO(3)xSO(6) plane-wave background. We identify finite-energy solutions by specifying the local minima of the corresponding energy functional. These configurations are static in SO(3) due to the Myers effect and rotate in SO(6) with an angular momentum that is bounded from above. As a first step towards studying their chaotic properties, we evaluate the Lyapunov exponents of their radial fluctuations.Comment: 7 pages, 8 figure

Exact Multiparticle Amplitudes at Threshold in ϕ4\phi^4 Theories with Softly Broken O(∞)O(\infty) Symmetry

We consider the problem of multiparticle production at threshold in a $\phi^4$-theory with an $O(N_1$$+$$N_2)$ symmetry softly broken down to $O(N_1)\times O(N_2)$ by nonequal masses. We derive the set of recurrence relations between the multiparticle amplitudes which sums all relevant diagrams with arbitrary number of loops in the large-$N$ limit with fixed number of produced particles. We transform it into a quantum mechanical problem and show how it can be obtained directly from the operator equations of motion by applying the factorization at large $N$. We find the exact solutions to the problem by using the Gelfand--Diki\u{\i} representation of the diagonal resolvent of the Schr\"{o}dinger operator. The result coincides with the tree amplitudes while the effect of loops is the renormalization of the coupling constant and masses. The form of the solution is due to the fact that the exact amplitude of the process $2$\ra$n$ vanishes at $n$$>$$2$ on mass shell when averaged over the $O(N_{1,2})$-indices of incoming particles. We discuss what dynamical symmetry is behind this property. We also give an exact solution in the large-$N$ limit for the model of the $O(N)$$+$$singlet$ scalar particle with the spontaneous breaking of a reflection symmetry.Comment: Latex, 33 pages, NBI-HE-94-3