Bayesian Inverse Quantum Theory

A Bayesian approach is developed to determine quantum mechanical potentials from empirical data. Bayesian methods, combining empirical measurements and "a priori" information, provide flexible tools for such empirical learning problems. The paper presents the basic theory, concentrating in particular on measurements of particle coordinates in quantum mechanical systems at finite temperature. The computational feasibility of the approach is demonstrated by numerical case studies. Finally, it is shown how the approach can be generalized to such many-body and few-body systems for which a mean field description is appropriate. This is done by means of a Bayesian inverse Hartree-Fock approximation.Comment: LaTex, 32 pages, 19 figure

Bayesian Reconstruction of Approximately Periodic Potentials at Finite Temperature

The paper discusses the reconstruction of potentials for quantum systems at finite temperatures from observational data. A nonparametric approach is developed, based on the framework of Bayesian statistics, to solve such inverse problems. Besides the specific model of quantum statistics giving the probability of observational data, a Bayesian approach is essentially based on "a priori" information available for the potential. Different possibilities to implement "a priori" information are discussed in detail, including hyperparameters, hyperfields, and non--Gaussian auxiliary fields. Special emphasis is put on the reconstruction of potentials with approximate periodicity. The feasibility of the approach is demonstrated for a numerical model.Comment: 18 pages, 17 figures, LaTe

Mean Field Methods for Atomic and Nuclear Reactions: The Link between Time--Dependent and Time--Independent Approaches

Three variants of mean field methods for atomic and nuclear reactions are compared with respect to both conception and applicability: The time--dependent Hartree--Fock method solves the equation of motion for a Hermitian density operator as initial value problem, with the colliding fragments in a continuum state of relative motion. With no specification of the final state, the method is restricted to inclusive reactions. The time--dependent mean field method, as developed by Kerman, Levit and Negele as well as by Reinhardt, calculates the density for specific transitions and thus applies to exclusive reactions. It uses the Hubbard--Stratonovich transformation to express the full time--development operator with two--body interactions as functional integral over one--body densities. In stationary phase approximation and with Slater determinants as initial and final states, it defines non--Hermitian, time--dependent mean field equations to be solved self--consistently as boundary value problem in time. The time--independent mean field method of Giraud and Nagarajan is based on a Schwinger--type variational principle for the resolvent. It leads to a set of inhomogeneous, non--Hermitian equations of Hartree--Fock type to be solved for given total energy. All information about initial and final channels is contained in the inhomogeneities, hence the method is designed for exclusive reactions. A direct link is established between the time--dependent and time--independent versions. Their relation is non--trivial due to the non--linear nature of mean field methods.Comment: 21 pages, to be published in European Physical Journal

Singularities in cascade models of the Euler equation

The formation of singularities in the three-dimensional Euler equation is investigated. This is done by restricting the number of Fourier modes to a set which allows only for local interactions in wave number space. Starting from an initial large-scale energy distribution, the energy rushes towards smaller scales, forming a universal front independent of initial conditions. The front results in a singularity of the vorticity in finite time, and has scaling form as function of the time difference from the singularity. Using a simplified model, we compute the values of the exponents and the shape of the front analytically. The results are in good agreement with numerical simulations.Comment: 33 pages (REVTeX) including eps-figures, Stylefile here.st

On the chemical equilibration of strangeness-exchange reaction in heavy-ion collisions

The strangeness-exchange reaction pi + Y -> K- + N is shown to be the dynamical origin of chemical equilibration for K- production in heavy-ion collisions up to beam energies of 10 A GeV. The hyperons occurring in this process are produced associately with K+ in baryon-baryon and meson-baryon interactions. This connection is demonstrated by the ratio K-/K+ which does not vary with centrality and shows a linear correlation with the yield of pions per participant. At incident energies above AGS this correlation no longer holds due to the change in the production mechanism of kaons.Comment: 9 pages, 4 figure