Symmetry vector fields and similarity solutions of a nonlinear field equation describing the relaxation to a maxwell distribution

In the study of the formulation of Maxwellian tails the nonlinear partial differential equation ² u/x +u/x+u ²=0 arises. We determine the Lie point symmetry vector fields and calculate the similarity ansätze. Then we discuss the resulting ordinary differential equations. Finally, the existence of Lie Bäcklund vector fields is studied and a Painlevé analysis is performed.     

Quantization of the Maxwell field in linear conformal invariant gauges

It is shown that the generalized Lorentz gauges provide all linear conformal invariant gauges, i.e. gauges such that A =0.     

From Time Inversion to Nonlinear QED

In Minkowski flat space-time, it is perceived that time inversion is unitary rather than antiunitary, with energy being a time vector changing sign under time inversion. The Dirac equation, in the case of electromagnetic interaction, is not invariant under unitary time inversion, giving rise to a Klein paradox. To render unitary time inversion invariance, a nonlinear wave equation is constructed, in which the Klein paradox disappears. In the case of Coulomb interaction, the revised nonlinear equation can be linearized to give energy solutions for Hydrogen-like ions without singularity when nuclear number Z>137, showing a reversed energy order pending for experimental tests such as Zeeman effects. In non-relativistic limit, this nonlinear equation reduces to nonlinear Schrödinger equation with soliton-like solutions. Moreover, particle conjugation and electron-proton scattering with a nonsingular current-potential interaction are discussed. Finally the explicit form of gauge function is found, the uniqueness of Lorentz gauge is proven and the Lagrangian density of quantum electrodynamics (QED) is revised as well. The implementation of unitary time inversion leads to the ultimate derivation of nonlinear QED.     

Nonlocality in (1/N)-expanded nonrenormalizable interactions

Nonrenormalizable interactions are studied on the example of the (1/N) ₆ ⁴ model with interaction via the collective = ² mode. In 4<d<6 dimensions, the model is nonrenormalizable by power counting but it does not require more subtractions than in the four-dimensional case. In six dimensions, nonrenormalizability is essential: There are two induced couplings ³ and . The ³ coupling strength can be determined from the requirement that the effective potential has a minimum at ² = = 0. The nonlocal formafactor produced by an infinite number of insertions cannot be determined uniquely, however the ambiguity concentrates in the leading order only. We construct this formfactor using the method of Efimov and Mogilevsky. The resulting theory satisfies unitarity, microcausality and correct spectral properties.     

Weyl quantization and metaplectic representation

The formal expansion defining the twisted exponential of an element of the Lie algebra _n ( ⁿ Sp(2, )) can be summed and this result is used to explicitly obtain the classical function u _t corresponding to an evolution operator associated to a quantum Hamiltonian belonging to the above mentioned Lie algebra.Then, by applying the Weyl quantization procedure to u _t we get a representation of the group W _n ( ⁿ Sp(2, )) in terms of integral operators, the kernels of which are expressed by means of the classical action. The family u _t being only locally defined, it must be considered as a distribution on the classical phase space in order to get the metaplectic representation.Membres du Centre National de la Recherche Scientifique.     

H2-induced changes in electrical conductance of β-Ga2O3 thin-film systems

H₂-induced changes of electrical conductivity in polycrystalline, undoped -Ga₂O₃ thin films in the temperature range of 400–650° C are described. The sheet conductance of these films depends reversibly, according to a power law p ^1/3, on the partial pressure of hydrogen in the ambient atmosphere of the Ga₂O₃ film. A bulk vacancy mechanism is excluded by experiments and it is shown that the interaction is based on a surface effect. Changes in conductance are discussed to result from the formation of an accumulation layer due to chemisorption on the grain surfaces. Typical coverages are determined to be approximately 10^–4 ML for pH₂=0.05 bar and T=600° C. A possible explanation of the p ^1/3 power law is provided.     

Bianchi and Weyl equations as sufficient equations for Einstein spaces

When the Bianchi equation and the wave equation for the Weyl spinor are written in the form which they take for Einstein spaces, but with the symmetric 4-spinor _ABCD considered arbitrary and with the background space unspecified, ^EA _EBCD=0; (+12) _ABCD –6(AB ^EF ( _CD )EF =0 it is shown that — in general — for this pair of equations to be consistent, the background space has to be an Einstein space, and the symmetric 4-spinor _ABCD has to be the Weyl spinor of this space.     

Long range scattering for nonlinear Schrödinger equations in one space dimension

We consider the scattering problem for the nonlinear Schrödinger equation in 1+1 dimensions: where = /x,R{0},R,p>3. We show that modified wave operators for (*) exist on a dense set of a neighborhood of zero in the Lebesgue spaceL ²(R) or in the Sobolev spaceH ¹(R)., The modified wave operators are introduced in order to control the long range nonlinearity |u|² u.Laboratoire associé au Centre National de la Recherche Scientifique     

Implementation of comparative probability by normal states. Infinite dimensional case

Let be an infinite dimensional Hilbert space and () the set of all (orthogonal) projections on . A comparative probability on () is a linear preorder on () such thatOP1,1O and such that ifP┴R,Q┴R, thenPQP+RQ+R for allP, Q, R in (). We give a sufficient and necessary condition for to be implemented in a canonical way by a normal state onB(), the bounded linear operators on .     

Quantum supports and modal logic

LetA be a quasi-manual with finite operations. Associate to each E = {e ₁ ,..., e_n} A the set _E of modal formulas: (e ₁ ··· e_n), e_i (e ₁ ··· e_i–1 e_i+1 ··· e_n), i=1,..., n. Set _A = {_E|E A}. We show that supports ofA are in one-to-one correspondence with certain Kripke models of _A where the supports are given by {x |A x is true}.On leave from the Department of Mathematics, Pontificia Universidade Catolica, Rio de Janeiro, Brazil.     

Unitarizable scalar multipoles

We show that the scalar multipole ( +m ²) ⁿ = 0 forn odd and all massesm0 allows a Gupta-Bleuler-type quantization with a unitary space of physical states.Feodor-Lynen fellow of the Humboldt foundation. On leave from Institut für Theoretische Physik, TU Clausthal, Leibnizstr. 10, D3392 Clausthal-Zellerfeld, W. Germany.     

Kinetics of small perturbations in an isotropic world

Long-wavelength gravitational perturbations are studied in an isotropic expanding universe filled with an ultrarelativistic gas. A kinetic study in the collisionless approximation shows that scalar and vector perturbations which appear at a time ₀ 1/n, where N is the wave vector and is the time coordinate x⁴, grow if the perturbation of the macroscopic momentum density of the gas at time ₀ is nonvanishing. The growth continues until the time ₁=27₀, at which the perturbation of the macroscopic momentum density of the gas vanishes. A solution is also derived for tensor perturbations in the limit n 1.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 37–45, April, 1978.     

Long-range correlations at depinning transitions

Semi-infinite systems are considered with long-range surface fields B _z ^–(1+r) for large distancesz from the surface. The influence of such fields on the global phase diagram and on the critical singularities of depinning transitions is studied within Landau theory. For |B|0, the correlation length diverges as b ^–1/2 withb=|Bln|B^–(1+r). For finiteB, t ^v withv =(2+r)/(2+2r) wheret measures the distance from bulk coexistence. In the latter case, a Ginzburg criterion leads to the upper critical dimensiond ^*=(2+3r)/(2+r).     

Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits

We present and discuss the derivation of a nonlinear nonlocal integrodifferential equation for the macroscopic time evolution of the conserved order parameter (r, t) of a binary alloy undergoing phase segregation. Our model is ad-dimensional lattice gas evolving via Kawasaki exchange with respect to the Gibbs measure for a Hamiltonian which includes both short-range (local) and long-range (nonlocal) interactions. The nonlocal part is given by a pair potential ^dJ(|x–y|), >0 x and y in ^d, in the limit 0. The macroscopic evolution is observed on the spatial scale ^–1 and time scale ^–2, i.e., the density (r, t) is the empirical average of the occupation numbers over a small macroscopic volume element centered atr=x. A rigorous derivation is presented in the case in which there is no local interaction. In a subsequent paper (Part II) we discuss the phase segregation phenomena in the model. In particular we argue that the phase boundary evolutions, arising as sharp interface limits of the family of equations derived in this paper, are the same as the ones obtained from the corresponding limits for the Cahn-Hilliard equation.     

High frequency gravitational radiation in Kerr-Schild space-times

Vaidya has obtained general solutions of the Einstein equationsR _ab= _{a b} by means of the Kerr-Schild metricsg _ab= _ab +H _{a b} . The vector field _a generates a shear free null geodetic congruence both in Minkowski space and in the Kerr-Schild space-time. If in addition it is hypersurface orthogonal, the Kerr-Schild metric may be interpreted as the background metric in a space-time perturbed by a high frequency gravitational wave. It is shown that Vaidya's solutions satisfying this additional condition are of only two types: (1) Kinnersley's accelerating point mass solution and (2) a similar solution where a space-like curve plays the role of the time-like curve describing the world line of the accelerating mass. The solution named by Vaidya as the radiating Kerr metric does not satisfy the hypersurface orthogonal condition.Supported in part by National Science Foundation Grant MPS 741029.     

Letter: The Quantum Theory of the Quadratic Gravity Action for Heterotic Strings

The wave function for the quadratic gravity theory derived from the heterotic string effective action is deduced to first order in by solving a perturbed second-order Wheeler-DeWitt equation, assuming that the potential is slowly varying with respect to . Predictions for inflation based on the solutionto the second-order Wheeler-DeWitt equation continue to hold for this higher-order theory. It is shown how formal expressions for the average paths in minisuperspace {a(t), (t)} for this theory can be used to determine the shifts from the classical solutions a _cl (t) and _cl (t), which occur only at third order in the expansion of the functional integrals representing the expectation values.     

Analyticity of scattering for the ϕ4 theory

We consider scattering for the equation (+m ²)+³=0 on four-dimensional Minkowski space. Form>0, one-to-one and onto wave operatorsW _± :HH are known to exist for all 0, whereH denotes the Hilbert space of finite-energy Cauchy data. We prove that the maps (,u)W _± (u) and (,u)(W _± )^–1 (u) are continuous from [0, )×H toH, and extend to real-analytic functions from an open neighborhood of {0}×H×{0}×H to the Hilbert spaceH _–1 of Cauchy data with Poincaré-invariant norm. Form=0, wave operatorsW _± are known to exist as diffeomorphisms ofH for all 0, where hereH denotes the Hilbert space of finite Einstein energy Cauchy data. In this case we prove that the maps (,u)(W _± ) (u) and (,u)(W _± )^–1 (u) extend to real-analytic functions from a neighborhood of [0, )×H×H toH.     

Excitation of low-frequency waves in the topside ionosphere in heating experiments at the leading edge of accelerated suprathermal electron beams

We consider the excitation of low-frequency waves in the upper ionosphere at the leading edge of suprathermal electron beams produced in the reflection region of a powerful radio wave. The integral growth rate of electron-beam instability is found for a wave packet intersecting an instability domain. The growth rate is strongly dependent on the ratio R=(/ ) /(/ )of the group velocity of the wave packet along the geomagnetic field to its phase velocity, which is equal to the velocity of the resonant particles. It is shown that in the upper ionosphere (the electron gyrofrequency f _Be 1MHz and the ratio of ion to electron masses M/m=18364the growth rate of longitudinal lower-hybrid oscillations is maximal at the frequency f 200kHz when R 1and the phase velocity / begins to grow due to the influence of ions. The excitation of longitudinal oscillations with f < f _Bi increases with decreasing f, since the ratio R approaches unity in this case. The enhancement of ion cyclotron oscillations is most effective in the vicinity of the second ion cyclotron harmonic f<2f _Bi 1MHz. Magnetohydrodynamic (MHD) waves interact with higher-energy particles having velocities on the order of the Alfvén velocity V _A ~ 10⁸ cm/sec. This interaction is damped in the small parameter (m/M)=1/1836.However, the integral growth rate of low-frequency Alfvén waves becomes considerable in the case of their quasiorthogonal (with respect to the magnetic field) propagation ( )in a strongly magnetized ionospheric plasma when the Alfvén velocity is much greater than the thermal velocity of the electrons. The interaction of a fast magnetosound (FMS) wave with resonant electrons increases considerably for f f _Bi when R=1and the wave packet moves along the magnetic field with resonant particle velocity (this condition is similar to the well-known R. Gendrin condition obtained earlier for high-frequency whistlers excited by higher-energy electrons). The magnitude and direction of the vector of the FMS wave as a function of its frequency f are found for R=1. The possibility of using these processes for explanation of the same features of low-frequency waves excited in heating experiments is discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 39, No. 2, pp. 163–178, February, 1996.This work was supported in part by the Russian Foundation for Fundamental Research.     

A generalized fluctuation-dissipation theorem for the one-dimensional diffusion process

The [, , ]-Langevin equation describes the time evolution of a real stationary process withT-positivity (reflection positivity) originating in the axiomatic quantum field theory. For this [, , ]-Langevin equation a generalized fluctuation-dissipation theorem is proved. We shall obtain, as its application, a generalized fluctuation-dissipation theorem for the one-dimensional non-linear diffusion process, which presents one solution of Ryogo Kubo's problem in physics.     

Rate equations in the hopping transport

The usual kinetic equations for the site occupation probabilities in an external field are solved exactly in a simple one-dimensional periodic model with two kinds of atoms using a) free boundary conditions and order of limitsN, 0 needed for a proper treatment of the dc conductivity here b) boundary conditions with metallic contacts and order of limitsN, 0 and c) the same boundary conditions but reversed order of limiting processes 0,N typical of e.g. numerical and percolation treatments. (N and are the number of sites and frequency.) It is demonstrated that though the bulk dc conductivity is the same in all three cases, local bulk properties of the material are strongly dependent on the régime used. The role of the order of all three limiting processes 0,N+ andn+ (N–n+) for local shifts of the chemical potential _n in the dc limit is examined (n is the number of the relevant site calculated from a boundary of the chain). It is shown especially that the rate equation treatment (régime a) on the one hand and numerical or percolation treatments (régime c) on the other hand never yield the same bulk values of _r.