Characterizing in a constructive way the set of real functions whose Fourier transforms are positive appears to be yet an open problem. Some sufficient conditions are known but they are far from being exhaustive. We propose two constructive sets of n
ecessary conditions for positivity of the Fourier transforms and test their ability of constraining the positivity domain. One uses analytic continuation and Jensen inequalities and the other deals with Toeplitz determinants and the Bochner theorem. Applications are discussed, including the extension to the two-dimensional Fourier-Bessel transform and the problem of positive reciprocity, i.e. positive functions with positive transforms.
We discuss dense states of QCD matter formed in high-energy hadronic and heavy-ion collisions from the point of view of statistical physics of non-equilibrium processes. For this sake, we first propose a formulation of the dynamical entropy of dense
QCD states in the saturation regime leading to a color glass condensate (CGC). The statistical physics description amounts to describe the modification of the color correlation length with energy as a compression process for which non equilibrium thermodynamic properties are applicable. We derive an expression of the dynamical entropy in terms of the rapidity evolution of the unintegrated gluon distributions in the colliding nuclei, verifying suitable positivity and irreversibility properties. We extend this approach to the initial pre-equilibrium (glasma) state of an heavy-ion collision. It allows for a definition of the initial entropy before the evolution towards the hydrodynamic regime as a function of the glasma correlation length and an overlap parameter characterizing the low-momentum spectrum of the glasma state. This initial entropy, by extension to the N=4 SYM theory, is then matched as the key input parameter to the strong coupling evaluation of thermalization towards the hydrodynamic regime based on the AdS/CFT correspondence. It thus allows to cast a bridge between the weak and strong coupling phases of an heavy-ion reaction.
The possibility that particle production in high-energy collisions is a result of two asymmetric hydrodynamic flows is investigated, using the Khalatnikov form of the 1+1-dimensional approximation of hydrodynamic equations. The general solution is di
scussed and applied to the physically appealing generalized in-out cascade where the space-time and energy-momentum rapidities are equal at initial temperature but boost-invariance is not imposed. It is demonstrated that the two-bump structure of the entropy density, characteristic of the asymmetric input, changes easily into a single broad maximum compatible with data on particle production in symmetric processes. A possible microscopic QCD interpretation of asymmetric hydrodynamics is proposed.
We define a mapping of the QCD Balitsky-Kovchegov equation in the diffusive approximation with noise and a generalized coupling allowing a common treatment of the fixed and running QCD couplings. It corresponds to the extension of the stochastic Fish
er and Kolmogorov-Petrovsky-Piscounov equation to the radial wave propagation in a medium with negative-gradient absorption responsible for anomalous diffusion,non-integer dimension and damped noise fluctuations. We obtain its analytic traveling wave solutions with a new scaling curve and in particular for running coupling a new scaling variable allowing to extend the range and validity of the geometric-scaling QCD prediction beyond the previously known domain.
The contribution presents a brief summary of the Gauge/Gravity approach to the study of hydrodynamic flow of the quark-gluon plasma formed in heavy-ion collisions, in a boost-invariant setting (Bjorken flow). Considering the ideal case of a supersymm
etric Yang-Mills theory for which the AdS/CFT correspondence gives a precise form of the Gauge/Gravity duality, the properties of the strongly coupled expanding plasma are put in one-to-one correspondence with the metric of a 5-dimensional black hole with the horizon moving away in the 5th dimension and its deformations consistent with the relevant Einstein equations. Several recently studied aspects of this framework are recalled and put in perspective. New results in collaboration with G.Beuf and M.Heller on the early time expansion towards the hydrodynamical regime are provided giving a new insight on the far-from-equilibrium behavior of the fluid at strong coupling and the thermalization and isotropization problems.
We identify the nonlinear evolution equation in impact-parameter space for the Supercritical Pomeron in Reggeon Field Theory as a 2-dimensional stochastic Fisher and Kolmogorov-Petrovski-Piscounov equation. It exactly preserves unitarity and leads in
its radial form to an high energy traveling wave solution corresponding to an universal behavior of the impact-parameter front profile of the elastic amplitude; Its rapidity dependence and form depend only on one parameter, the noise strength, independently of the initial conditions and of the non-linear terms restoring unitarity. Theoretical predictions are presented for the three typical different regimes corresponding to zero, weak and strong noise, respectively. They have phenomenological implications for total and differential hadronic cross-sections at colliders.
