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Fabian Brau

Publications and source records attributed to Fabian Brau.

At least 37 records · Page 2Linked to original sources
arXiv2009

A moving boundary model motivated by electric breakdown: II. Initial value problem

An interfacial approximation of the streamer stage in the evolution of sparks and lightning can be formulated as a Laplacian growth model regularized by a 'kinetic undercooling' boundary condition. Using this model we study both the linearized and the full nonlinear evolution of small perturbations of a uniformly translating circle. Within the linear approximation analytical and numerical results show that perturbations are advected to the back of the circle, where they decay. An initially analytic interface stays analytic for all finite times, but singularities from outside the physical region approach the interface for $t\to\infty$, which results in some anomalous relaxation at the back of the circle. For the nonlinear evolution numerical results indicate that the circle is the asymptotic attractor for small perturbations, but larger perturbations may lead to branching. We also present results for more general initial shapes, which demonstrate that regularization by kinetic undercooling cannot guarantee smooth interfaces globally in time.

nlin.PS↗
arXiv2009

Glueballs and statistical mechanics of the gluon plasma

We study a pure gluon plasma in the context of quasiparticle models, where the plasma is considered as an ideal gas of massive bosons. In order to reproduce SU(3) gauge field lattice data within such a framework, we review briefly the necessity to use a temperature-dependent gluon mass which accounts for color interactions between the gluons near $T_{\text{c}}$ and agrees with perturbative QCD at large temperatures. Consequently, we discuss the thermodynamics of systems with temperature-dependent Hamiltonians and clarify the situation about the possible solutions proposed in the literature to treat consistently those systems. We then focus our attention to two possible formulations which are thermodynamically consistent and we extract the gluon mass from the equation of state obtained in SU(3) lattice QCD. We find that the thermal gluon mass is similar in both statistical formalisms. Finally, an interpretation of the gluon plasma as an ideal gas made of glueballs and gluons is also presented. The glueball mass is consistently computed within a relativistic formalism using a potential obtained from lattice QCD. We find that the gluon plasma might be a glueball-rich medium for $T\lesssim 1.13 T_{\text{c}}$ and suggest that glueballs could be detected in future experiments dedicated to the quark-gluon plasma.

hep-ph↗
arXiv2009

A moving boundary problem motivated by electric breakdown: I. Spectrum of linear perturbations

An interfacial approximation of the streamer stage in the evolution of sparks and lightning can be written as a Laplacian growth model regularized by a `kinetic undercooling' boundary condition. We study the linear stability of uniformly translating circles that solve the problem in two dimensions. In a space of smooth perturbations of the circular shape, the stability operator is found to have a pure point spectrum. Except for the zero eigenvalue for infinitesimal translations, all eigenvalues are shown to have negative real part. Therefore perturbations decay exponentially in time. We calculate the spectrum through a combination of asymptotic and series evaluation. In the limit of vanishing regularization parameter, all eigenvalues are found to approach zero in a singular fashion, and this asymptotic behavior is worked out in detail. A consideration of the eigenfunctions indicates that a strong intermediate growth may occur for generic initial perturbations. Both the linear and the nonlinear initial value problem are considered in a second paper.

nlin.PS↗
arXiv2009

Moving boundary approximation for curved streamer ionization fronts: Numerical tests

Recently a moving boundary approximation for the minimal model for negative streamer ionization fronts was extended with effects due to front curvature; this was done through a systematic solvability analysis. A central prediction of this analysis is the existence of a nonvanishing electric field in the streamer interior, whose value is proportional to the front curvature. In this paper we compare this result and other predictions of the solvability analysis with numerical simulations of the minimal model.

physics.plasm-ph↗
arXiv2008

Moving boundary approximation for curved streamer ionization fronts: Solvability analysis

The minimal density model for negative streamer ionization fronts is investigated. An earlier moving boundary approximation for this model consisted of a "kinetic undercooling" type boundary condition in a Laplacian growth problem of Hele-Shaw type. Here we derive a curvature correction to the moving boundary approximation that resembles surface tension. The calculation is based on solvability analysis with unconventional features, namely, there are three relevant zero modes of the adjoint operator, one of them diverging; furthermore, the inner/outer matching ahead of the front has to be performed on a line rather than on an extended region; and the whole calculation can be performed analytically. The analysis reveals a relation between the fields ahead and behind a slowly evolving curved front, the curvature and the generated conductivity. This relation forces us to give up the ideal conductivity approximation, and we suggest to replace it by a constant conductivity approximation. This implies that the electric potential in the streamer interior is no longer constant but solves a Laplace equation; this leads to a Muskat-type problem.

physics.plasm-ph↗
arXiv2008

Saffman-Taylor streamers: mutual finger interaction in an electric breakdown

Bunches of streamers form the early stages of sparks and lightning but theory presently concentrates on single streamers or on coarse approximations of whole breakdown trees. Here a periodic array of interacting streamer discharges in a strong homogeneous electric field is studied in PDE approximation in two dimensions. If the period of the streamer array is small enough, the streamers do not branch, but approach uniform translation. When the streamers are close to the branching regime, the enhanced field at the tip of the streamer is close to $2 E_{\infty}$, where $E_{\infty}$ is the homogeneous field applied between the electrodes. We discuss a moving boundary approximation to the set of PDEs. This moving boundary model turns out to be essentially the same as the one for two-fluid Hele-Shaw flows. In two dimensions, this model possesses a known analytical solution. The shape of the 2D interacting streamers in uniform motion obtained from the PDE simulations is actually well fitted by the analytically known ``selected Saffman-Taylor finger''. This finding helps to understand streamer interactions and raises new questions on the general theory of finger selection in moving boundary problems.

physics.plasm-ph↗
arXiv2007

Construction and test of a moving boundary model for negative streamer discharges

Starting from the minimal model for the electrically interacting particle densities in negative streamer discharges, we derive a moving boundary approximation for the ionization fronts. The boundary condition on the moving front is found to be of 'kinetic undercooling' type. The boundary approximation, whose first results have been published in [Meulenbroek et al., PRL 95, 195004 (2005)], is then tested against 2-dimensional simulations of the density model. The results suggest that our moving boundary approximation adequately represents the essential dynamics of negative streamer fronts.

physics.plasm-ph↗
arXiv2005

Casimir scaling, glueballs and hybrid gluelumps

Assuming that the Casimir scaling hypothesis is well verified in QCD, masses of glueballs and hybrid gluelumps (gluon with a point-like $c\bar c$ pair) are computed within the rotating string formalism. In our model, two gluons are attached by an adjoint string in a glueball while the gluon and the colour octet $c\bar c$ pair are attached by two fundamental strings in a hybrid gluelump. Masses for such exotic hadrons are computed with very few free parameters. These predictions can serve as a guide for experimental searches. In particular, the ground state glueballs lie on a Regge trajectory and the lightest $2^{++}$ state has a mass compatible with some experimental candidates.

hep-ph↗
arXiv2004

Upper limit on the critical strength of central potentials in relativistic quantum mechanics

In the context of relativistic quantum mechanics, where the Schrödinger equation is replaced by the spinless Salpeter equation, we show how to construct a large class of upper limits on the critical value, $g_{\rm{c}}^{(\ell)}$, of the coupling constant, $g$, of the central potential, $V(r)=-g v(r)$. This critical value is the value of $g$ for which a first $\ell$-wave bound state appears.

math-ph↗
arXiv2004

Renormalization of the singular attractive $1/r^4$ potential

We study the radial Schrödinger equation for a particle of mass $m$ in the field of a singular attractive $g^2/{r^4}$ potential with particular emphasis on the bound states problem. Using the regularization method of Beane \textit{et al.}, we solve analytically the corresponding ``renormalization group flow" equation. We find in agreement with previous studies that its solution exhibits a limit cycle behavior and has infinitely many branches. We show that a continuous choice for the solution corresponds to a given fixed number of bound states and to low energy phase shifts that vary continuously with energy. We study in detail the connection between this regularization method and a conventional method modifying the short range part of the potential with an infinitely repulsive hard core. We show that both methods yield bound states results in close agreement even though the regularization method of Beane \textit{et al.} does not include explicitly any new scale in the problem. We further illustrate the use of the regularization method in the computation of electron bound states in the field of neutral polarizable molecules without dipole moment. We find the binding energy of s-wave polarization bound electrons in the field of C$_{60}$ molecules to be 17 meV for a scattering length corresponding to a hard core radius of the size of the molecule radius ($\sim 3.37$ Å). This result can be further compared with recent two-parameter fits using the Lennard-Jones potential yielding binding energies ranging from 3 to 25 meV.

quant-ph↗
arXiv2004

Light Meson Spectra and Instanton Induced Forces

The spinless Salpeter equation supplemented by an instanton induced force is used to describe the spectra of light mesons, including the pseudoscalar ones. The coupling constants of the instanton induced potential, as well as the quark constituent masses are not treated as simple free parameters but are calculated from the underlying instanton theory. Quite good results are obtained provided the quark are considered as effective degrees of freedom with a finite size. A further test of the model is performed by calculating the electromagnetic mass differences between S-wave mesons.

hep-ph↗
arXiv2004

Universality of Regge and vibrational trajectories in a semiclassical model

The orbital and radial excitations of light-light mesons are studied in the framework of the dominantly orbital state description. The equation of motion is characterized by a relativistic kinematics supplemented by the usual funnel potential with a mixed scalar and vector confinement. The influence of finite quark masses and potential parameters on Regge and vibrational trajectories is discussed. The case of heavy-light mesons is also presented.

hep-ph↗
arXiv2004

Bohr-Sommerfeld quantization and meson spectroscopy

We use the Bohr-Sommerfeld quantization approach in the context of constituent quark models. This method provides, for the Cornell potential, analytical formulae for the energy spectra which closely approximate numerical exact calculations performed with the Schrodinger or the spinless Salpeter equations. The Bohr-Sommerfeld quantization procedure can also be used to calculate other observables such as r.m.s. radius or wave function at the origin. Asymptotic dependence of these observables on quantum numbers are also obtained in the case of potentials which behave asymptotically as a power-law. We discuss the constraints imposed by these formulae on the dynamics of the quark-antiquark interaction.

hep-ph↗
arXiv2004

Scattering solutions of the spinless Salpeter equation

A method to compute the scattering solutions of a spinless Salpeter equation (or a Schrodinger equation) with a central interaction is presented. This method relies on the 3-dimensional Fourier grid Hamiltonian method used to compute bound states. It requires only the evaluation of the potential at equally spaced grid points and yields the radial part of the scattering solution at the same grid points. It can be easily extended to the case of coupled channel equations and to the case of non-local interactions.

hep-ph↗
arXiv2004

Sufficient conditions for the existence of bound states in a central potential

We show how a large class of sufficient conditions for the existence of bound states, in non-positive central potentials, can be constructed. These sufficient conditions yield upper limits on the critical value, $g_{\rm{c}}^{(\ell)}$, of the coupling constant (strength), $g$, of the potential, $V(r)=-g v(r)$, for which a first $\ell$-wave bound state appears. These upper limits are significantly more stringent than hitherto known results.

math-ph↗
arXiv2004

Critical strength of attractive central potentials

We obtain several sequences of necessary and sufficient conditions for the existence of bound states applicable to attractive (purely negative) central potentials. These conditions yields several sequences of upper and lower limits on the critical value, $g_{\rm{c}}^{(\ell)}$, of the coupling constant (strength), $g$, of the potential, $V(r)=-g v(r)$, for which a first $\ell$-wave bound state appears, which converges to the exact critical value.

math-ph↗
arXiv2004

Lower bounds for the spinless Salpeter equation

We obtain lower bounds on the ground state energy, in one and three dimensions, for the spinless Salpeter equation (Schrödinger equation with a relativistic kinetic energy operator) applicable to potentials for which the attractive parts are in $L^p(\R^n)$ for some $p>n$ ($n=1$ or 3). An extension to confining potentials, which are not in $L^p(\R^n)$, is also presented.

math-ph↗