Conditioning moments of singular measures for entropy optimization. I

In order to process a potential moment sequence by the entropy optimization method one has to be assured that the original measure is absolutely continuous with respect to Lebesgue measure. We propose a non-linear exponential transform of the moment sequence of any measure, including singular ones, so that the entropy optimization method can still be used in the reconstruction or approximation of the original. The Cauchy transform in one variable, used for this very purpose in a classical context by A.\ A.\ Markov and followers, is replaced in higher dimensions by the Fantappi\`{e} transform. Several algorithms for reconstruction from moments are sketched, while we intend to provide the numerical experiments and computational aspects in a subsequent article. The essentials of complex analysis, harmonic analysis, and entropy optimization are recalled in some detail, with the goal of making the main results more accessible to non-expert readers. Keywords: Fantappi\`e transform; entropy optimization; moment problem; tube domain; exponential transformComment: Submitted to Indagnationes Mathematicae, I. Gohberg Memorial issu

Localization and Coherence in Nonintegrable Systems

We study the irreversible dynamics of nonlinear, nonintegrable Hamiltonian oscillator chains approaching their statistical asympotic states. In systems constrained by more than one conserved quantity, the partitioning of the conserved quantities leads naturally to localized and coherent structures. If the phase space is compact, the final equilibrium state is governed by entropy maximization and the final coherent structures are stable lumps. In systems where the phase space is not compact, the coherent structures can be collapses represented in phase space by a heteroclinic connection to infinity.Comment: 41 pages, 15 figure

Fast-Roll Inflation

We show that in the simplest theories of spontaneous symmetry breaking one can have a stage of a fast-roll inflation. In this regime the standard slow-roll condition |m^2| << H^2 is violated. Nevertheless, this stage can be rather long if |m| is sufficiently small. Fast-roll inflation can be useful for generating proper initial conditions for the subsequent stage of slow-roll inflation in the very early universe. It may also be responsible for the present stage of accelerated expansion of the universe. We also make two observations of a more general nature. First of all, the universe after a long stage of inflation (either slow-roll or fast-roll) cannot reach anti-de Sitter regime even if the cosmological constant is negative. Secondly, the theories with the potentials with a "stable" minimum at V(\phi)<0 in the cosmological background exhibit the same instability as the theories with potentials unbounded from below. This instability leads to the development of singularity with the properties practically independent of V(\phi). However, the development of the instability in some cases may be so slow that the theories with the potentials unbounded from below can describe the present stage of cosmic acceleration even if this acceleration occurs due to the fast-roll inflation.Comment: 21 pages, 4 figures, JHEP3, a discussion of initial conditions for the fast-roll inflation is extende

Statistical mechanics of two-dimensional point vortices: relaxation equations and strong mixing limit

We complete the literature on the statistical mechanics of point vortices in two-dimensional hydrodynamics. Using a maximum entropy principle, we determine the multi-species Boltzmann-Poisson equation and establish a form of virial theorem. Using a maximum entropy production principle (MEPP), we derive a set of relaxation equations towards statistical equilibrium. These relaxation equations can be used as a numerical algorithm to compute the maximum entropy state. We mention the analogies with the Fokker-Planck equations derived by Debye and H\"uckel for electrolytes. We then consider the limit of strong mixing (or low energy). To leading order, the relationship between the vorticity and the stream function at equilibrium is linear and the maximization of the entropy becomes equivalent to the minimization of the enstrophy. This expansion is similar to the Debye-H\"uckel approximation for electrolytes, except that the temperature is negative instead of positive so that the effective interaction between like-sign vortices is attractive instead of repulsive. This leads to an organization at large scales presenting geometry-induced phase transitions, instead of Debye shielding. We compare the results obtained with point vortices to those obtained in the context of the statistical mechanics of continuous vorticity fields described by the Miller-Robert-Sommeria (MRS) theory. At linear order, we get the same results but differences appear at the next order. In particular, the MRS theory predicts a transition between sinh and tanh-like \omega-\psi relationships depending on the sign of Ku-3 (where Ku is the Kurtosis) while there is no such transition for point vortices which always show a sinh-like \omega-\psi relationship. We derive the form of the relaxation equations in the strong mixing limit and show that the enstrophy plays the role of a Lyapunov functional

On the well-posedness of multivariate spectrum approximation and convergence of high-resolution spectral estimators

In this paper, we establish the well-posedness of the generalized moment problems recently studied by Byrnes-Georgiou-Lindquist and coworkers, and by Ferrante-Pavon-Ramponi. We then apply these continuity results to prove almost sure convergence of a sequence of high-resolution spectral estimators indexed by the sample size