High Energy Asymptotics Research Articles

On Soot Inception in Nonpremixed Flames and the Effects of Flame Structure

A simplified three-step model of soot inception has been employed with high activation energy asymptotics to study soot inception in nonpremixed counterflow systems with emphasis on understanding the effects of hydrodynamics and transport. The resulting scheme yields three zones: (1) a fuel oxidation zone wherein the fuel and oxidizer react to form product as well as a radical R, (e.g., H), (2) a soot/precursor formation zone where the radical R reacts with fuel to form “soot/precursor” S, and (3) a soot/precursor consumption zone where S reacts with the oxidizer to form product. The kinetic scheme, although greatly simplified, allows the coupling between soot inception and flame structure to be assessed. The results yield flame temperature, flame location, and a soot/precursor index S1 as functions of Damkohler number for S formation. The soot/precursor index indicates the amount of S at the boundary of the formation region. The flame temperature indirectly indicates the total amount of S integrated over the formation region because as S is formed less heat release is available. The results show that unlike oxidation reactions, an extinction turning-point behavior does not exist for soot. Instead, the total amount of S slowly decreases with decreasing Damkohler number (increasing strain rate), which is consistent with counterflow flame experiments. When the Lewis number of the radical is decreased from unity, the total S reduces due to reduced residence time for the radical in the soot formation region. Similarly, when the Lewis number of the soot/precursor is increased from unity the amount of S increases for all Damkohler numbers. In addition to studying fuel-air (low stoichiometric mixture fraction) flames, the air-side nitrogen was substituted into the fuel, yielding diluted fuel-oxygen (high stoichiometric mixture fraction) flames with the same flame temperature as the fuel-air flames. The relative flame locations were different however, and, consistent with counterflow flame experiments, this difference was found to dramatically reduce the total amount of S generated because the change in stoichiometric mixture fraction affects residence times, temperatures and concentrations in the soot/precursor formation and consumption zones. Furthermore, while the soot/precursor consumption reaction had a negligible effect on the soot process for fuel-air flames it was very important to diluted fuel-oxygen flames.

Combustion waves with wide reaction zones

Combustion waves are traditionally studied using high activation energy asymptotics. The structure of the combustion wave involves a thin reaction zone which separates the initial reactants from the combustion product. All previous mathematical studies were focused on such combustion waves. However, even if the activation energy of the reaction is high, combustion waves exhibiting very different structures may exist. We develop a new asymptotic theory to describe experimentally observed combustion waves with wide reaction zones. They occur when the reaction rate is significantly reduced in the course of the reaction as the product is formed.

Spectral Properties and Identities for the Polar Operator with Nonsmooth Periodic Density

Spectral properties of the polar operator depending on the smoothness of the periodic coefficient are studied. The width of the far gaps in the Bloch spectrum is shown to grow for piecewise continuous coefficients, to be asymptotically constant if the coefficient derivative is piecewise continuous, and to decrease in the more smooth cases. The high energy asymptotics of the Lyapunov function, of the quasimomentum and of ‘effective masses’ are obtained. The spectral identities for the corresponding classical string equation are derived.

A stationary approach to inverse scattering for schrodinger operators with first order perturbation

In this paper, we study the inverse scattering of Schrodinger operators with short-range (resp. long-range) electric and magnetic potentials. We develop a stationary approach to determine the high energy asymptotics of the scattering operator (resp. modified scattering operator). As a corollary, we show that the electric potential and the magnetic field are uniquely determined by the first two terms of this asymptotic expansion.

One dimensional periodic Dirac Hamiltonians: Semiclassical and high-energy asymptotics for gaps

In this article we consider a one-dimensional Dirac operator with a potential of Gevrey class α and study the semiclassical and high-energy asymptotics of the spectral gaps for a region of energies that in the Schrödinger case corresponds to unbounded motion. An exponential upper bound for the gap’s widths as well as the asymptotic expansion of their positions are derived for both cases; the first two terms in the asymptotic expansions are explicitly written down for the scalar potential case.

New effective Feynman-like rules for the multi-Regge QCD asymptotics of inclusive multijet production

New effective Feynman-like rules are defined for inclusive multijet cross sections in the multi-Regge regime. The solution of the BFKL equation is used as a starting point. The resulting rules involve conformal weight and rapidity as a momentum and a coordinate respectively and are translation invariant in the coordinates. We use the effective rules to calculate ultra high energy asymptotics of inclusive multijet production. The dependence on the parton densities occurs only in the overall normalization of the asymptotic cross sections.

High-energy effective action in gravity

The high-energy effective action in the multi-Regge kinematics can be derived starting from the original gravity action. The result is a starting point for studying corrections to the eikonal and to the leading logarithmic approximation to the high energy asymptotics.

Bethe ansatz for QCD pomeron

The equivalence is found between high-energy QCD in the generalized leading logarithmic approximation and the one-dimensional Heisenberg magnet. According to Regge theory, the high-energy asymptotics of hadronic scattering amplitudes are related to singularities of partial waves in the complex angular momentum plane. In QCD, the partial waves are determined by nontrivial two-dimensional dynamics of the transverse gluonic degrees of freedom. The “bare” gluons interact with each other to form a collective excitation, the Reggeon. The partial waves of the scattering amplitude satisfy the Bethe-Salpeter equation whose solutions describe the color singlet compound states of Reggeons - Pomeron, Odderon and higher Reggeon states. We show that the QCD Hamiltonian for reggeized gluons coincides in the multi-color limit with the Hamiltonian of XXX Heisenberg magnet for spin s = 0 and spin operators being the generators of the conformal SL(2,C) group. As a result, the Schrödinger equation for the compound states of Reggeons has a sufficient number of conservation laws to be completely integrable. A generalized Bethe ansatz is developed for the diagonalization of the QCD Hamiltonian and for the calculation of hadron-hadron scattering. Using the Bethe Ansatz solution of high-energy QCD we investigate the properties of the Reggeon compound states which govern the Regge behavior of the total hadron-hadron cross sections and the small-x behavior of the structure functions deep inelastic scattering.

High-energy scattering in QCD and cross singularities of Wilson loops

We consider elastic quark-quark scattering at high energy and fixed transferred momentum. Performing factorization of soft-gluon exchanges into the Wilson-line expectation value we find that there is one-to-one correspondence between high-energy asymptotics in QCD and renormalization properties of the so-called cross singularities of the Wilson lines. Using this relation we show that the asymptotic behavior of the quark-quark scattering amplitude is controlled by a 2 × 2 matrix of the cross anomalous dimensions. We evaluate the matrix of the cross anomalous dimension to two-loop order and study the properties of the obtained expressions to higher loop order.

On the uniqueness of inverse scattering for N-body systems

We study the high-energy asymptotics of the free-channel-free-channel scattering matrix for the generalized N-body Schrodinger operator P=- Delta + Sigma a in AVa(Xa) with short-range interactions and prove that all the interactions Va are uniquely determined by the free-channel-free-channel scattering matrix at high energies. The potentials treated are more general than the sums of pair potentials, and in fact include all possible k-particle interactions with 2<or=k<or=N.

High-energy asymptotics of resonances for three-dimensional Schrödinger operator with screened Coulomb potential

Resonances for one-particle three-dimensional Schrödinger operators with Coulomb potential perturbed by a spherically symmetric compactly supported function q(r) are studied. The mth derivative of q(r) has a jump on the boundary of the support (m≥0). Resonances are defined as poles of an analytical continuation of the quadratic form of the resolvent to the second Riemann sheet through the branch cut along the continuous spectrum. It is shown that there exists an infinite set of resonance poles which splits into an infinite sequence of infinite series corresponding to different values of an angular momentum. Resonances in each series have the only point of accumulation at infinity. The main result of the work is an asymptotic formula for resonances in each series. It follows from this formula that high-energy resonances of the above operator are asymptotically close to those of the Schrödinger operator with the same potential q(r) but without Coulomb term. It was shown in one of the previous works of the author that, in contrast with the non-Coulomb case, some perturbations q(r) of the Coulomb potential can produce a sequence of resonances converging to zero. The result of this work together with the above result shows that the Coulomb part of the potential, while dramatically changing the geometry of resonances at low energies, does not destroy their asymptotic behavior at high energies.

Total cross sections inN-body problems: Finiteness and high energy asymptotics

We study the finiteness of total scattering cross sections from an arbitrary channel to a two-cluster channel and establish the high energy asymptotics for total scattering cross sections with initial two-cluster channel and those from an arbitrary channel to a two-cluster channel.

High energy asymptotics of multi-colour QCD and two-dimensional conformal field theories

In the multi-colour limit of perturbative QCD the holomorphic factorization of wave function of compound states of n reggeized gluons in the impact parameter space is shown. The conformally invariant Hamiltonian for each holomorphic factor has a nontrivial integral of motion. The odderon in QCD is the simplest example of the composite system with these properties.

Temperature Surges in Current-Limiting Circuit Devices

This paper studies the problem of heat transfer in a thermistor, which is used as a switching device in electronic circuits. The temperature field is coupled to the current flow by ohmic heating in the device, and the problem is rendered highly nonlinear by a very rapid variation of electrical conductivity with temperature. Approximate methods based on high activation energy asymptotics are developed to describe the transient heat flow, which occurs when the circuit is switched on. In particular, it is found that a transient “surge” phenomenon (akin to thermal runaway, but self-saturating) occurs, and we conjecture that this phenomenon may be associated with cracking of thermistors, which sometimes occurs during operation.

Etude d'une limite singulière d'un modèle intervenant en combustion

Glangetas, L., Etude d'une limite singuliere d'un modele intervenant en combustion, Asymptotic Analysis 5 (1992) 317-342. We consider solutions (us, u,,' c.) of a system of two nonlinear differential equations -u~ + cu: = f,,(u.)u E , -Au; + cu~ = -f(u.)u,, on IR with the boundary conditions u,,(-oo)=O, u,,(+oo) = 1, U,,(-oo) = 1, u,,(+oo)=O. We investigate the asymptotic behavior of (u,,' u,,' c.) as [5 ..... 0 and f,,(u)(1 - u) behaves as a Dirac distribution. This singular limit corresponds to some combustion models (planar flame propagations) for high activation energy asymptotics.

Four-fermi interaction in (2+1) dimensions beyond leading order in 1/N

We study the renormalizable Gross-Neveu model in (2 + 1) dimensions at next to leading order in the 1/ N expansion. The corrections to the value of the critical coupling and β-function are calculated. The nature of the finite fixed point as found in the leading order remains unchanged. We also obtain the effective potential for the scalar field interpolating the bound state meson and the high energy asymptotics of the fermionic Green's functions.

High-energy asymptotics for the scattering matrix on the line

The authors consider the Schrodinger equation on the line and prove asymptotic expansions for the S-matrix S(k) in inverse powers of k as k to infinity . The order of these expansions depends on the number of L1 derivatives of the potential. They establish and exploit a connection between S(k) and the two Titchmarsh-Weyl m-functions (m+or-) associated with this problem.

Stretch effects and solid propellant burning

The combustion of a premixed solid reactant is investigated using a two-step mechanism, i.e., an endothermal pyrolysis of the solid surface and a premixed gaseous flame located at a distance off the solid. Interaction between the solid burning rate and a nonuniform outer flow is analyzed as a result of unbalanced conduction of heat and diffusion of species. Considering a self-similar steady situation and using the technique of high activation energy asymptotics for the gaseous phase reaction zone, it is shown how a small rate of strain modifies the solid burning rate.

Negative binomial multiplicity distributions in high energy hadron collisions

This paper concerns the results recently obtained by the UA5 Collaboration on charged particle multiplicity distributions at the CERN $$p\bar p$$ collider $$(\sqrt s = 540 GeV)$$ and their comparison with lower energy data [1, 2]. Our aim is to interpret in general terms the experimental finding that the distributions are of negative binomial (NB) form both for total multiplicities and in finite pseudorapidity intervals. This can be understood in terms of partial stimulated emission of bosons, or of a simple form of cascade process, or (more artificially) with both mechanisms. The cascade interpretation appears to be the most attractive. It leads to a new concept of clustering, the empirical data imply unexpected properties for the clusters, and a novel type of high energy asymptotics seems to emerge. We briefly discuss the relation with QCD-based models for hadron production and the case ofe + e − annihilation.

Flame fronts with complex chemical networks

For laminar flame fronts with general complex chemistry networks, we present a systematic method to reduce the detailed chemistry in the limit of high activation energy asymptotics. We detail the method on a global multistep hydrocarbon model. Generic flame profile structures are discussed, and extensions to networks with zero activation energies are outlined.