Vanishing of Vacuum States and Blow-up Phenomena of the Compressible Navier-Stokes Equations

The Navier-Stokes systems for compressible fluids with density-dependent viscosities are considered in the present paper. These equations, in particular, include the ones which are rigorously derived recently as the Saint-Venant system for the motion of shallow water, from the Navier-Stokes system for incompressible flows with a moving free surface [14]. These compressible systems are degenerate when vacuum state appears. We study initial-boundary-value problems for such systems for both bounded spatial domains or periodic domains. The dynamics of weak solutions and vacuum states are investigated rigorously. First, it is proved that the entropy weak solutions for general large initial data satisfying finite initial entropy exist globally in time. Next, for more regular initial data, there is a global entropy weak solution which is unique and regular with well-defined velocity field for short time, and the interface of initial vacuum propagates along the particle path during this time period. Then, it is shown that for any global entropy weak solution, any (possibly existing) vacuum state must vanish within finite time. The velocity (even if regular enough and well-defined) blows up in finite time as the vacuum states vanish. Furthermore, after the vanishing of vacuum states, the global entropy weak solution becomes a strong solution and tends to the non-vacuum equilibrium state exponentially in time.     

Blowup of Smooth Solutions for Relativistic Euler Equations

We study the singularity formation of smooth solutions of the relativistic Euler equations in (3 + 1)-dimensional spacetime for both finite initial energy and infinite initial energy. For the finite initial energy case, we prove that any smooth solution, with compactly supported non-trivial initial data, blows up in finite time. For the case of infinite initial energy, we first prove the existence, uniqueness and stability of a smooth solution if the initial data is in the subluminal region away from the vacuum. By further assuming the initial data is a smooth compactly supported perturbation around a non-vacuum constant background, we prove the property of finite propagation speed of such a perturbation. The smooth solution is shown to blow up in finite time provided that the radial component of the initial ``generalized' momentum is sufficiently large.     

The fixed boundary value problems for the equations of ideal Magneto-Hydrodynamics with a perfectly conducting wall condition

The equations of ideal Magneto-Hydrodynamics are investigated concerning initial boundary value problems with a perfectly conducting wall condition. The local in time solution is proved to exist uniquely, provided that the normal component of the initial magnetic field vanishes everywhere or nowhere on the boundary.     

A solution for a family of perfect fluid cylinders in general relativity

A static solution of Einstein's equations representing perfect fluid cylinders of finite radius is presented. The density and pressurep are finite and monotonic decreasing from the axis to the boundary. The ratiop/ (< 1) also decreases steadily. The regularity condition at the axis is satisfied and the solution is free of singularity. The equation of state is found and special cases considered. It is shown that the solution can be joined to the vacuum cylindrical metric at the boundary. There is then at least one free parameter left in the family of solutions.     

Dynamically Broken Supergauge Symmetries

The dynamical breaking of the supergauge symmetries in the massless supergauge Wess Zumino model isdiscussed without adding the Fayet-Iliopoulos term to the Lagrangian. It is shown, in terms of the Nambu-Jona-Lasiniomechanism, that the supersymmetry breaking and the gauge symmetry breaking can be realized dynamically. It is alsoshown that the dynamical breaking moves the vacuum expectation values of two scalar fields away from zero. In order torestore the symmetry of the vacuum, one of the two scalar fields is translated and at the same time the mass spectrumis changed too.     

A regularity theorem for solutions of the spherically symmetric Vlasov-Einstein system

We show that if a solution of the spherically symmetric Vlasov-Einstein system develops a singularity at all then the first singularity has to appear at the center of symmetry. The main tool is an estimate which shows that a solution is global if all the matter remains away from the center of symmetry.Research supported in part by NSF DMS 9101517     

Boundary conditions on current carrying states and the implications for observations of bloch oscillations

The problem of the precise form (together with the associated boundary conditions) of a one-dimensional hamiltonian describing a particle of variable mass is addressed. It is shown that although hermiticity may be a necessary condition, it is not sufficient to specify uniquely the hamiltonian. In particular one form of the latter that is widely employed in the literature is shown to lead to a violation of the Heisenberg Uncertainty Principle. However imposition of the additional demand that any singular terms arising from the kinetic energy can be transformed away does lead to a unique specification of the hamiltonian. Finally the implications of the latter with regard to the standard probability interpretation of the wavefunction and for the observation of Bloch oscillations in quantum well structures is described.     

Nonlocality and local causality in the Schrödinger equation with time-dependent boundary conditions

We investigate the nonlocal dynamics of a single particle placed in an infinite well with moving walls. It is shown that in this situation, the Schrödinger equation (SE) violates local causality by causing instantaneous changes in the probability current everywhere inside the well. This violation is formalized by designing a gedanken faster-than-light communication device which uses an ensemble of long narrow cavities and weak measurements to resolve the weak value of the momentum far away from the movable wall. Our system is free from the usual features causing nonphysical violations of local causality when using the (nonrelativistic) SE, such as instantaneous changes in potentials or states involving arbitrarily high energies or velocities. We explore in detail several possible artifacts that could account for the failure of the SE to respect local causality for systems involving time-dependent boundary conditions.     

Representation of the propagator of a massive spinning particle as a BFV-BRST path integral

In the index-spinor approach, the transition amplitude for a free massive particle of arbitrary spin is obtained by calculating the relevant path integral in the BFV-BRST formalism. The calculation is performed without any renormalization of the measure in the path integral. The result coincides with the Weinberg propagator in the index-free representation. It is shown that the type of representation for the particle spin—a holomorphic or an antiholomorphic one—is determined by the choice of boundary conditions for the index spinor.     

Interface free energies in p-spin glass models

The replica method is used to calculate the interface free energy associated with the change from periodic to antiperiodic boundary conditions in finite-dimensional p-spin glass models in the phase which at mean-field level has one-step replica symmetry breaking (1RSB). In any finite dimension the interface free energy is exponentially small for a large system. This result implies that, in finite dimensions, the 1RSB state does not exist, as it is destroyed by thermal excitation of arbitrarily large droplets. The implications of this for the theory of structural glasses are discussed.     

Confinement of atoms with Robin’s condition: Spontaneous symmetry breaking

The properties of the ground state of the hydrogen atom in a spherical vacuum cavity with general boundary “not going out” conditions (i.e., when the probability current through the boundary vanishes) are studied. It is shown that in contrast to the confinement of an atom by a potential barrier, in this case depending on the parameters of the cavity, the atom could be in stable equilibrium at the center of the cavity or shift towards its periphery: spontaneous breaking of spherical symmetry occurs. The phase diagram of the shift and the dependence of the shift value and the binding energy of the ground state of the atom on the cavity parameters are presented. At the same time, the deformation properties of the electron wave function (WF) for an asymmetric distortion are so nontrivial that a non-zero shift occurs even when an electron is repulsed from the cavity boundary.     

Kinetic effects in an optically excited gas

A resonance transition in a one-dimensional layer of gas contained between transparent parallel plates and optically excited by external radiation has been treated using kinetic theory. A perturbation method has been used to obtain the “first scattering” results for the velocity distribution and number density of excited atoms and the intensity of radiation at any point in the gas. Two special cases are discussed in detail: broad band excitation with inhomogeneous broadening and monochromatic excitation with homogeneous broadening. The effects of particle streaming and wall quenching are shown to produce boundary layer behavior in the excited level density which scales with the particle mean free path. In addition, line reversal of the radiation reemited from the gas is shown to occur and to be a direct result of particle streaming. Numerical and asymptotic results are presented which show these effects. These results should be pertinent to many laboratory and industrial devices in which the particle and photon mean free paths are comparable and to diagnostic techniques which use resonance fluorescence to infer excited level densities.     

Dynamical Model of QCD Vacuum and Color Thaw at Finite Temperatures

In terms of the Nambu Jona-Lasinio (NJL) mechanism, the dynamical symmetry breaking of a simple local gauge model is investigated. An important relation between the vacuum expectation value of gauge fields and scalar fields is derived by solving the Euler equation for the gauge fields. Based on this relation the SU(3) gauge potential is given which can be used to explain the asymptotic freedom and confinement of quarks in a hadron. The confinement behavior at finite temperatures is also investigated and it is shown that color confinement at zero temperature can be melted away under high temperatures.     

Abelian duality, confinement, and chiral-symmetry breaking in a SU(2) QCD-like theory

We analyze the vacuum structure of SU(2) QCD with multiple massless adjoint representation fermions formulated on a small spatial S(1) x R(3). The absence of thermal fluctuations, and the fact that quantum fluctuations favor the vacuum with unbroken center symmetry in a weakly coupled regime, renders the interesting dynamics of these theories analytically calculable. Confinement and the generation of the mass gap in the gluonic sector are shown analytically. In this regime, theory exhibits confinement without continuous chiral-symmetry breaking. However, a flavor singlet chiral condensate (which breaks a discrete chiral symmetry) persists at arbitrarily small S(1). Under certain reasonable assumptions, we show that the theory exhibits a zero temperature chiral phase transition in the absence of any change in spatial center symmetry realizations.     

Random Tilings of High Symmetry: II. Boundary Conditions and Numerical Studies

We perform numerical studies including Monte Carlo simulations of high rotational symmetry random tilings. For computational convenience, our tilings obey fixed boundary conditions in regular polygons. Such tilings are put in correspondence with algorithms for sorting lists in computer science. We obtain statistics on path counting and vertex coordination which compare well with predictions of mean-field theory and allow estimation of the configurational entropy, which tends to the value 0.568 per vertex in the limit of continuous symmetry. Tilings with phason strain appear to share the same entropy as unstrained tilings, as predicted by mean-field theory. We consider the thermodynamic limit and argue that the limiting fixed boundary entropy equals the limiting free boundary entropy, although these differ for finite rotational symmetry.     

On the thermodynamic origin of the quantum potential

In a new thermodynamic interpretation, the quantum potential is shown to result from the presence of a subtle thermal vacuum energy distributed across the whole domain of an experimental setup. Explicitly, its form is demonstrated to be exactly identical to the heat distribution derived from the defining equation for classical diffusion wave fields. For a single free particle path, this thermal energy does not significantly affect particle motion. However, in between different paths, or at interfaces, the accumulation-depletion law for diffusion waves provides an immediate new understanding of the quantum potential’s main features.     

The kinetic boundary layer around an absorbing sphere and the growth of small droplets

Deviations from the classical Smoluchowski expression for the growth rate of a droplet in a supersaturated vapor can be expected when the droplet radius is not large compared to the mean free path of a vapor molecule. The growth rate then depends significantly on the structure of the kinetic boundary layer around a sphere. We consider this kinetic boundary layer for a dilute system of Brownian particles. For this system a large class of boundary layer problems for a planar wall have been solved. We show how the spherical boundary layer can be treated by a perturbation expansion in the reciprocal droplet radius. In each order one has to solve a finite number ofplanar boundary layer problems. The first two corrections to the planar problem are calculated explicitly. For radii down to about two velocity persistence lengths (the analog of the mean free path for a Brownian particle) the successive approximations for the growth rate agree to within a few percent. A reasonable estimate of the growth rate for all radii can be obtained by extrapolating toward the exactly known value at zero radius. Kinetic boundary layer effects increase the time needed for growth from 0 to 10 (or 2 1/2) velocity persistence lengths by roughly 35% (or 175%).     

The Asymmetric Collapse of Bubbles in Compressible Liquid

The analytical solution ora bubble collapse close to a solid boundary in a compressible water is investigatedby means of a perturbation method to first order in the bubble wall Mach number. It is shown, in this paper, that itis the Rayleigh-Plesset equation for incompressible liquid to zero order solution or similar to the Gilmore equation forcompressible water to first order solution when the effect of solid boundary is negligibly small enough, i.e., sufficientlyfar away from the bubble center.     

Complex Classical Mechanics of a QES Potential

We study a combined parity(P) and time reversal(T) invariant non-Hermitian quasi-exactly solvable(QES) potential, which exhibits PT phase transition, in the complex plane classically to demonstrate different quantum effects. The particle with real energy makes closed orbits around one of the periodic wells of the complex potential depending on the initial condition. However interestingly the particle escapes to an open orbits even with real energy if it is placed beyond a certain distance from the center of the well. On the other hand when the particle energy is complex the trajectory is open and the particle tunnels back and forth between two wells which are separated by a classically forbidden path. The tunneling time is calculated for different pair of wells and is shown to vary inversely with the imaginary component of energy. Our study reveals that spontaneous PT symmetry breaking does not affect the qualitative features of the particle trajectories in the analogous complex classical model.