Stability and causality in dissipative relativistic fluids

The standard theory of relativistic dissipative fluid mechanics developed by Eckart contains several undesirable features: thermal and viscous fluctuations propagate acausally; there exist generic short wavelength secular instabilities; and there is not a well posed initial value problem for rotating fluids. In this paper we examine whether the generalization of Eckart's theory developed by Israel has succeeded in eliminating these features. We first generalize Israel's theory to include the possibility of nonuniform equilibrium configurations. This generalization allows us to describe equilibrium configurations which may be rotating and self-gravitating such as neutron stars. We then evaluate the stability conditions for these fluids and compute the characteristic velocities at which perturbations propagate. Our main result is that if these fluids are stable, then the characteristic velocities are subluminal and the perturbations propagate via hyperbolic equations. Thus Israel's theory is causal for all stable fluids. In addition, there is no generic instability, and the initial value problem is well posed. In our opinion, for these reasons, Israel's theory should replace Eckart's as the standard theory of relativistic dissipative fluid mechanics.     

Field-theoretic approach to the properties of the solid state

The techniques of quantum field theory are used to investigate the thermodynamic ion displacement correlation function—or Green's function of the phonon field—in a crystal and especially in a metal. The structure of thermodynamic Green's functions is outlined and the method for solving for them at finite temperature is fully discussed.The analytic structure of the phonon Green's function is then considered. This function is shown to be bounded and invertible everywhere off the real axis; a spectral form is derived for its inverse. The symmetries imposed by the point group of the crystal are then discussed.Assuming small ionic oscillations, we find the inverse of the phonon Green's function as a linear function of the electronic contribution to the dielectric response function of the metal. This dielectric function is shown to be simply related to the longitudinal part of the conductivity tensor that gives the response of the electrons to the effective electric field in the metal. The assumption of translational invariance then leads to an explicit expression for the phonon Green's function in terms of this conductivity.The deformations in the lattice induced by an arbitrarily time varying external force are calculated in terms of the retarded phonon Green's function. In the static long wavelength limit the phonon Green's function yields the macroscopic elastic constants of the crystal. Their relation to the conductivity is exhibited, and several elastic constants are estimated. We also see that the complete phonon spectrum and the lifetimes of the phonon states may be calculated from this Green's function. A relation between the long wavelength acoustic attenuation in metals and the de conductivity is derived, which is in good agreement with recent experiments. Furthermore, the ions in a metal are shown to have a high-frequency oscillation along with the electrons, at essentially the electron plasma frequency.     

A cluster plus effective medium tight-binding study of SiOx systems

The valence band density of states of SiO_x amorphous systems is calculated using a tight-binding treatment. The Green's function method is applied to a small cluster embeddied in a suitable effective medium. An average of this Green's function is then performed assuming a statistical distribution of composition for the small clusters. The effective medium is treated as a Bethe lattice of the correct composition. The results are completely consistent with experiment if the Si O Si angle is assumed to decrease with x.     

Application of complex energy integration to selfconsistent electronic structure calculations

In electronic structure calculations the charge density is obtained by an energy integral over the one-electron Green's function, which especially for transition or rare earth metals is strongly structured. We show that a replacement of this integral by a contour integral in the complex energy plane allows a very efficient and accurate calculation of the charge density. Schrödinger's equation has to be solved only for a rather small number of complex energies along the integration path. We demonstrate the efficiency of this method for impurity calculations in Cu using the KKR-Green's function method and also discuss a possible application to band structure calculations.     

An extended car-following model considering the appearing probability of truck and driver's characteristics

In this paper, the appearing probability of truck is introduced and an extended car-following model is presented to analyze the traffic flow based on the consideration of driver's characteristics, under honk environment. The stability condition of this proposed model is obtained through linear stability analysis. In order to study the evolution properties of traffic wave near the critical point, the mKdV equation is derived by the reductive perturbation method. The results show that the traffic flow will become more disorder for the larger appearing probability of truck. Besides, the appearance of leading truck affects not only the stability of traffic flow, but also the effect of other aspects on traffic flow, such as: driver's reaction and honk effect. The effects of them on traffic flow are closely correlated with the appearing probability of truck. Finally, the numerical simulations under the periodic boundary condition are carried out to verify the proposed model. And they are consistent with the theoretical findings.     

Dual models as saddle point approximations to Polyakov's quantized string

We show that if the metric becomes singular on the boundary, then Polyakov's quantized string theory has a saddle point. This leads to an off-shell Green function, the S-matrix of which is the standard dual (Veneziano) model.     

Extended Feynman formula for time-dependent forced harmonic oscillator

Horváthy's modification of Feynman's path integral formula is generalized to the time-dependent forced harmonic oscillator. The propagator at caustics is then obtained by using its modified semi-group property. Finally, with our new formula, the propagator for a charged particle in a time-dependent electromagnetic field is evaluated exactly beyond and at caustics.     

Definite paths and upper bounds on regular regions of velocity phase space

Hamilton's principle states that the path integral of the Langrangian is stationary with respect to variations of a classical path. It does not distinguish between a local minimum, a local maximum or a saddle point in path space. A simple algorithm is devised which provides strict and useful upper bounds on the region of velocity phase space occupied by paths that are either local maxima or local minima. The technique is illustrated graphically for the standard map. It is found that the bounds provide accurate numerical upper estimates for the region of velocity phase space filled by the rotational KAM surfaces at arbitrarily chosen values of the perturbation parameter.     

An improved separability approximation for line radiative transport in nonhomogeneous media

A simple modification to the constant half-width approximation of Wilson and Greif, which is an extension of the well-known Curtis-Godson method to the treatment of temperature variations along the integration path, is introduced which permits a more accurate evaluation of line radiative transport in non- homogeneous gases. To demonstrate the method's accuracy, comparisons are made with Wilson and Greif and numerical frequency integrated results for the line equivalent width and radiative flux in a planar slab with prescribed Planck function and line half-width spatial variations are chosen to represent typical shock layer conditions. It is found that the modified procedure reduces the inaccuracies inherent in Wilson and Greif's approximation by factors ranging from 5 to 10, while retaining the latter method's ease of application.     

Towards a theory of spacetime structure at very short distances

Renormalization-group arguments are summarized which suggest that at distances shorter than the Planck length the spacetime geometry should be asymptotically scale invariant. A new locally scale-invariant extension of general relativity is then proposed based on Weyl's conformally invariant geometry. It is shown that if the theory contains a Higgs phase, then it reduces to Einstein's theory in the limit of large distances. Finally, the theory incorporates the non-linear sigma model, which suggests a new approach to the calculation of non-perturbative, short-distance effects in quantum gravity.     

The fast-time instability of diffusion flames

A diffusion flame burns with varying intensity if the pressure is varied, and the nonlinear steady-state response is typically S-shaped; that is, multiple solutions exist for some range of pressure. The physically relevant branch can only be determined by unsteady analyses, and in this paper we discuss the stability of a large class of diffusion flames when the activation energy characterizing the reaction rate is large. Specifically, we examine the evolution from the stationary solution on a time scale so short that changes are confined to the thin flame sheet where all the reaction occurs. In this region time derivatives are added to the steady state equations, which otherwise describe a balance between diffusion and chemical reaction. The stability problem is then reduced to the determination of the spectrum of Schrödinger's equation, defined on the infinite interval, with a potential that is not of one sign on this interval. In this way certain conclusions about extinction are drawn and certain past misconceptions are clarified.     

Biorthogonality and radiative transfer in finite slab atmospheres

It is shown that the exact solution of transfer problems of polarized light in finite slab atmospheres can be obtained from an eigenmode expansion, if there is a known set of adjoints defined appropriately to treat two-point, half-range boundary-value problems. The adjoints must obey a half-range biorthogonality relation.The adjoints are obtained in terms of Case's eigenvectors and the reflection or the transmission matrices. Half-range characteristic equations for the eigenvectors and their adjoints are derived, where the kernel functions of the integral operators are given by the boundary values of the source function matrix of the slab albedo problem. Spectral formulae are obtained for the surface Green's functions. A relationship is noted between the biorthogonality concept and some half-range forms of the transfer equation for the surface Green's functions and their adjoints. Linear and non-linear functional equations that are well known from an invariance approach, are derived from a new point of view. The biorthogonality concept offers the opportunity for a better understanding of mathematical structures and the nonuniqueness problem for solutions of such functional equations.     

Path integral approach to the Schrödinger equation with a complex potential

We show that Feynman's path integral method can be extended to include complex potentials by introducing a complex action function. The resulting path integral,which yields the propagator of the corresponding Schrödinger equation, involves only real paths.     

Ferromagnetic 3d-metals: On the definition of “local bands”

The basic concept of a picture for itinerant ferromagnetism is discussed. The central point is that a local exchange splitting and local moments, exist even above the transition temperature T_c. Transverse fluctuations (and not magnitude fluctuations, as in Stoner theory) are the dominant source for the phase transition to the paramagnetic state. The author's Green's function method is extended to the use of a full bandstructure including hybridization and general electron-electron interactions. Spin waves are also discussed.     

Some non-perturbative semi-classical methods in quantum field theory (a pedagogical review)

A pedagogical introduction is given to non-perturbative semi-classical methods for finding solutions to quantum field theories. Both the weak coupling method based on a time-independent classical solution, and the WKB method based on all periodic orbits are developed in detail, proceeding ffrom elementary quantum mechanics to field theory in stages. Both methods are then illustrated in model field theories. The [λø⁴]₂ theory to which the weak coupling method is applied yields a new family of “kink” states whose properties are discussed.The WKB method is illustrated by quantizing “soliton” and “doublet” solutions of the two-dimensional sine-Gordon theory. The results are compared to consequences of Coleman's equivalence proof relating this system to the massive Thirring model. The speculation that solitons may be fermions is discussed, along with indications that the WKB method may ne yielding exact mass ratios for this theory.A final section is devoted to solutions of more realistic four-dimensional models containing fermions, internal symmetry etc. Some quark-confinement models and vortex type solutions come under this category. General remarks are made on this family of solutions, and illustrated using 't Hooft's monopole solution.     

Positive energy without supersymmetry

Witten's positive energy theorem and its generalizations can be viewed as stating that supersymmetric solutions of any supergravity theory are stable. In this paper we give a criterion to test the stability of non-supersymmetric solutions of supergravity theories and solutions of theories which cannot be embedded in a supergravity theory. Previously some of these solutions might have been considered to be unstable. In particular, we show that the non-supersymmetric stationary point of the scalar potential of the gauged N = 5 supergravity theory is stable. We also give an elegant derivation of the Breitenlohner-Freedman condition for (small fluctuation) stability.     

Direct Monte Carlo simulation of scattering processes of kV electrons in aluminum; comparison of theoretical N(E) spectra with experiment

A Monte Carlo simulation of the scattering processes of kV electrons penetrating into aluminum was performed. The simulation is based on the use of different types of differential cross-sections for individual elastic and inelastic scattering: (i) Elastic scattering; the differential cross-sections derived by partial wave expansion method. (ii) Inelastic scattering; Gryzinski's excitation function for inner-shell electron excitation, Streitwolfs excitation function for conduction electron excitation, and Quinn's mean free path for plasmon excitation. For verification the energy loss spectra obtained from the Monte Carlo calculations were then compared with experiment done with commercial type Auger microprobes, JAMP-3, for angle of incidence 45° and JAMP-10 for normal incidence at primary electron energies of 1.5 and 3.0 keV, respectively. The results show satisfactory agreement between theory and experiment.     

A microscopic theory of the lambda transition,II

The previously proposed finite temperature field theory of the lambda transition based on the Schwinger functional method is investigated further. A systematic method for calculating the higher-order loop terms is presented by introducing the one-loop Green's functions, which are found to be a natural finite temperature extension of the Beliaev-Hugenholtz-Pines-Gavoret-Nozières zero-temperature Green's functions. The application of the finite temperature loop expansion to the dynamical properties is presented by calculating the retarded density correlation functions at the one-loop level. The result gives a microscopic basis for the form of the dynamical structure factor recently proposed by Woods and Svensson. From a general point of view, without using any approximations or model interactions, Goldstone's theorem for the lambda transition at finite temperature is presented.     

Solution of the path integral for the H-atom

The Green's function of the H-atom is calculated by a simple reduction of Feynman's path integral to gaussian form.     

Slavnov-Taylor identities in Weinberg's renormalization scheme

Weinberg's renormalization scheme, although more cumbersome from the computational point of view, has a more immediate physical interpretation than 't Hooft's minimal renormalization scheme. It is expected to lead to smaller higher-order coefficients in a perturbative approach to QCD. However, it a priori violates the Slavnov-Taylor identities. A complete study of this problem is performed, both theoretically and for the practitioner's sake. The ambiguities in the choice of the tensorial basis of some of the QCD vertices, as well as the dependence in the gauge parameter are used for substantiating, eventually, the Slavnov-Taylor identities in this renormalization scheme.