Asymptotic Degrees Research Articles

2D holography beyond the Jackiw-Teitelboim model

Having in mind extensions of 2D holography beyond the Jackiw-Teitelboim model we propose holographic counterterms and asymptotic conditions for a family of asymptotically AdS2 dilaton gravity models leading to a consistent variational problem and a finite on-shell action. We show the presence of asymptotic Virasoro symmetries in all these models. The Schwarzian action generates (a part) of the equations of motion governing the asymptotic degrees of freedom. We also analyse the applicability of various entropy formulae. By a dilaton-dependent conformal transformation our results are extended to an even larger class of models having exotic asymptotic behavior. We also analyse asymptotic symmetries for some other classes of dilaton gravities without, however, constructing holographic counterterms.

Measure-theoretic degrees and topological pressure for non-expanding transformations

We consider invariant sets Λ of saddle type, for non-invertible smooth maps f, and equilibrium measures μϕ associated to Hölder potentials ϕ on Λ. We define a notion of measure-theoretic asymptotic degree off|Λ:Λ→Λ, with respect to the measure μϕ on the fractal set Λ. In our case, the equilibrium measure μϕ is the unique linear functional in C(Λ)⁎ tangent to the pressure function P(⋅):C(Λ)→R at ϕ. In particular, for the measure of maximal entropy μ0 of f|Λ, we obtain the asymptotic degree of f|Λ, which represents the average rate of growth of the number of n-preimages of x that remain in Λ when n→∞; notice that, in general, Λ is not totally invariant for f. To this end, we will obtain first a formula for the Jacobians of the probability μϕ, with respect to arbitrary iterates fm, m≥2. We then show that a formula for the topological pressure P(ϕ) that holds in the expanding case, is no longer true on saddle sets. In the saddle case we find a new formula for the pressure, involving weighted sums on preimage sets. We also apply the asymptotic degrees, together with various pressure functionals, in order to obtain estimates for the Hausdorff dimension of stable slices through certain sets of full μϕ-measure in the fractal Λ. In the end, we give also some concrete examples on saddle folded sets.

Chiral hadronic mean field model including quark degrees of freedom

In an approach inspired by Polyakov loop extended Nambu–Jona-Lasinio models, we present a nonlinear hadronic SU(3) σ–ω mean field model augmented by quark degrees of freedom. By introducing the effective Polyakov loop related scalar field Φ and an associated effective potential, the model includes all known hadronic degrees of freedom at low temperatures and densities as well as a quark phase at high temperatures and densities. Hadrons in the model exhibit a finite volume in order to suppress baryons at high T and μ. This ensures that the right asymptotic degrees of freedom are attained for the description of strongly interacting matter and allows one to study the QCD phase diagram in a wide range of temperatures and chemical potentials. Therefore, with this model it is possible to study the phase transition of chiral restoration and deconfinement. In this paper, the impact of quarks on the resulting phase diagram is shown. The results from the chiral model are compared with recent data from lattice QCD.

An effective chiral hadron–quark equation of state

We construct an effective model for the QCD equation of state, taking into account chiral symmetry restoration as well as the deconfinement phase transition. The correct asymptotic degrees of freedom at the high and low temperature limits are included (quarks ↔ hadrons). The model shows a rapid crossover for both order parameters, as expected from lattice calculations. We then compare the thermodynamic properties of the model at μB = 0 which turn out to be in qualitative agreement with lattice data, while apparent quantitative differences can be attributed to hadronic contributions and excluded-volume corrections. Furthermore we discuss the effects of a repulsive vector-type quark interaction at finite baryon number densities on the resulting phase diagram of the model. Our current model is able to reproduce a first-order liquid–gas phase transition as expected, but does not show any signs of a first-order deconfinement or chiral phase transition. Both transitions rather appear as a very wide crossover in which heavily medium modified hadron coexists with free quarks.

Finite Dimensional Behaviors of the Primitive Equations Under Small Depth Assumption

In this article, we study the asymptotic degrees of freedom for solutions to the primitive equation (PEs for brevity). More precisely, we will prove that the long-time behavior of solutions to PEs is determined by the set of either finite Fourier modes, line elements, or volume elements. Our results show that the long-time behavior of the PEs is determined by the baro-tropic flows that are independent of the vertical direction in ℝ3. This study builds upon the previous article by the author concerning the existence and uniqueness of strong solutions to the PEs in thin domains.

Analytic study of shell models of turbulence

In this paper we study analytically the viscous “sabra” shell model of the energy turbulent cascade. We prove the global regularity of solutions and show that the shell model has finitely many asymptotic degrees of freedom, specifically: a finite dimensional global attractor and globally invariant inertial manifolds. Moreover, we establish the existence of an exponentially decaying energy dissipation range for sufficiently smooth forcing.

Sharp Estimates for the Number of Degrees of Freedom for the Damped-Driven 2-D Navier-Stokes Equations

We derive upper bounds for the number of asymptotic degrees (determining modes and nodes) of freedom for the two-dimensional Navier-Stokes system and Navier-Stokes system with damping. In the first case we obtain the previously known estimates in an explicit form, which are larger than the fractal dimension of the global attractor. However, for the Navier-Stokes system with damping, our estimates for the number of the determining modes and nodes are comparable to the sharp estimates for the fractal dimension of the global attractor. Our investigation of the damped-driven 2-D Navier-Stokes system is inspired by the Stommel-Charney barotropic model of ocean circulation where the damping represents the Rayleigh friction. We remark that our results equally apply to the viscous Stommel-Charney model.

Evaporation of the gluon condensate: a model for pure gauge SU(3) c phase transition

We interpret lattice data for the equation of state of pure gauge SU(3) c by an evaporation model. At low temperatures gluons are frozen inside the gluon condensate, whose dynamics is described in terms of a dilaton Lagrangian. Above the critical temperature quasi-free gluons evaporate from the condensate: a first order transition is obtained by minimizing the thermodynamical potential of the system. Within the model it is possible to reproduce lattice QCD results at finite temperature for thermodynamical quantities such as pressure and energy. The gluonic longitudinal mass can also be evaluated; it vanishes below the critical temperature, where it shows a discontinuity. At very large temperatures we recover the perturbative scenario and gluons are the only asymptotic degrees of freedom.

Fast spline smoothing via spectral factorization concepts

When tuning the smoothness parameter of nonparametric regression splines, the evaluation of the so-called degrees of freedom is one of the most computer-intensive tasks. In the paper, a closed-form expression of the degrees of freedom is obtained for the case of cubic splines and equally spaced data when the number of data tends to infinity. State-space methods, Kalman filtering and spectral factorization techniques are used to prove that the asymptotic degrees of freedom are equal to the variance of a suitably defined stationary process. The closed-form expression opens the way to fast spline smoothing algorithms whose computational complexity is about one-half of standard methods (or even one-fourth under further approximations).

Estimates of asymptotic degrees of freedom for solutions to the Navier-Stokes equations

We derive estimates on the upper bounds of the asymptotic degrees of freedom for strong solutions of the two-dimensional Navier-Stokes equations with no-slip and non-homogeneous boundary conditions. The analysis is also extended to an abstract equation modelling some other equations in fluid mechanics. Similar results are obtained.

The Navier-Stokes equations on the rotating 2-D sphere: Gevrey regularity and asymptotic degrees of freedom

In this article we prove a Gevrey class global regularity to the Navier-Stokes equations on the rotating two dimensional sphere, S 2 - a fundamental model that arises naturally in large scale atmospheric dynamics. As a result one concludes the exponential convergence of the spectral Galerkin numerical method, based on spherical harmonic functions. Moreover, we provide an upper bound for the number of asymptotic degrees of freedom for this system.

Anti–de Sitter–CFT correspondence in three-dimensional supergravity

Anti--de Sitter supergravity models are considered in three dimensions. Precise asymptotic conditions involving a chiral projection are given on the Rarita-Schwinger fields. Together with the known boundary conditions on the bosonic fields, these ensure that the asymptotic symmetry algebra is the superconformal algebra. The classical central charge is computed and found to be equal to the one of pure gravity. It is also indicated that the asymptotic degrees of freedom are described by 2D ``induced supergravity'' and that the boundary conditions ``transmute'' the non-vanishing components of the WZW supercurrent into the supercharges.

Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems

We show that the long-time behavior of the projection of the exact solutions to the Navier-Stokes equations and other dissipative evolution equations on the finite-dimensional space of interpolant polynomials determines the long-time behavior of the solution itself provided that the spatial mesh is fine enough. We also provide an explicit estimate on the size of the mesh. Moreover, we show that if the evolution equation has an inertial manifold, then the dynamics of the evolution equation is equivalent to the dynamics of the projection of the solutions on the finite-dimensional space spanned by the approximating polynomials. Our results suggest that certain numerical schemes may capture the essential dynamics of the underlying evolution equation.

Mass and dual mass: A gravitational analogue of the Dirac quantization rule and its physical implications

It is shown that (asymptotically multi-NUT) gravitational magnetic monopoles, which can be described by anS 3/Z N principal Hopf-bundle structure at conformal null infinity (Z N is a cyclic subgroup of orderN ofZ). provide a gravitational analogue of the Dirac quantization rule, which involves the total magnetic (dual) mass of the space-time-a measurement of the first Chern class of the bundle-and the mass of a test particle located in the rest frame defined at infinity by the Bondi (or dual Bondi) 4-momentum. It is shown thatSU 2/U 1 preserves the asymptotic structure. A definition of the angular momentum operator which extends that available for test electric charges in the field of a (Maxwellian) Yang-Wu magnetic monopole is presented. The commutation relations are dictated by the quantization rule. Various physical consequences are mentioned. SinceSU 2 is a double covering ofSO 3, gravitational magnetic monopoles provide a topological explanation for the existence of particles with half-integer spin. Abelian (U 1), non-AbelianSU 1 asymptotic degrees of freedom of the gravitational field could be related to suitable nontrivial cohomology classes; Penrose's nonlinear graviton modes could emerge as self (antiself) adjoint (Yang-Mills) gauge connections.

The Schwarzschild H space

The usual construction of anH space from asymptotic data onI+ uses only the radiative component of the asymptotic gravitational field and ignores the longitudinal part of the field [2]. While Penrose and Sparling [1] have given the construction of a deformed twistor space encoding the Schwarzschild structure by exponentiating the linearized spin 2 monopole field, one would like to have a directH space construction as a step toward constructingH spaces containing all the asymptotic degrees of freedom. The first step of giving a directH space construction for the Schwarzschild asymptotic data is contained in the following. Several interesting new structures arise in this process.