Radiating gravitational collapse with shear viscosity revisited

A new model is proposed to a collapsing radiating star consisting of an isotropic fluid with shear viscosity undergoing radial heat flow with outgoing radiation. In a previous paper we have introduced a function time dependent into the g _rr , besides the time dependent metric functions and . The aim of this work is to generalize this previous model by introducing shear viscosity and compare it to the non-viscous collapse. The behavior of the density, pressure, mass, luminosity and the effective adiabatic index is analyzed. Our work is compared to the case of a collapsing shearing fluid of a previous model, for a star with 6 . The pressure of the star, at the beginning of the collapse, is isotropic but due to the presence of the shear the pressure becomes more and more anisotropic. The black hole is never formed because the apparent horizon formation condition is never satisfied. An observer at infinity sees a radial point source radiating exponentially until reaches the time of maximum luminosity and suddenly the star turns off. The effective adiabatic index has a very unusual behavior because we have a non-adiabatic regime in the fluid due to the heat flow.     

Regular exact models for a nonstatic gas sphere in general relativity

Some new exact models for an expanding or a contracting gaseous sphere (i.e., the density is to vanish at the outer boundary together with the pressurep) are given. The physical properties of the models are investigated, and it is found that both the pressure and the density are positive inside the outer boundary of the sphere, and their respective gradients are negative. The density is increasing for contracting spheres, and it is decreasing for expanding spheres. It is also shown that this is the case for the pressure at any moment for the layers close to the boundary of the spheres. For these layers it is further shown that the adiabatic speed of sound is less than the speed of light, and the trace of the energy-momentum tensor is positive. The rate of change of the circumference as measured by an observer riding on the boundary of the sphere is increasing for expanding spheres and it is decreasing for collapsing spheres. We also find that the physical radius is an increasing function of comoving radial coordinate. The mass function is further shown to be positive.     

Solutions for static gas spheres in general relativity

A method for constructing solutions for a relativistic static gaseous sphere is examined. Gaseous here means that the density vanishes at the outer boundary together with the pressurep. Two different classes of solutions are investigated in detail. The models of both these classes have the property that the density gradient is zero both at the center and at the surface. It is further shown that both classes yield models which are physically acceptable, i.e. both the pressure and the density are positive and finite inside the outer boundary of the sphere, and their respective gradients are negative. The trace of their energy-momentum tensors are positive, and the adiabatic sound speeds are decreasing outwards throughout the sphere. The relativistic adiabatic indices are examined, and it is found that they are decreasing functions of radial coordinate. It is shown that for the first class this index is 3/2 at the surface, while for the second class it is 4/3 at the boundary. We find that the models of the first class arestable with respect to small radial disturbances. Putting the density at the center equal to 10¹⁶g cm^–3, the maximum mass for the stable class is found to be 0.87 solar masses.     

Exact model for a gaseous regular bouncing sphere in general relativity

An exact model for a relativistic gaseous sphere (i.e., one whose density vanishes at the outer boundary of the nonstatic sphere together with the pressurep) is given. The model has a bounce: The collapsing sphere comes momentarily to rest when the boundary is still outside the Schwarzschild radius of the matter sphere, then there is a macroscopic bounce, and the matter of the expanding sphere spreads all over the universe. This bouncing solution of Einstein's field equations is physically valid at any moment, i.e., the pressure and the density are positive inside the fluid sphere, and their respective gradients are negative. The mass function is positive, and the circumference is an increasing function of radial coordinate. This solution may represent at easily surveyable model for a supernova explosion where the explosion is so violent that no remnant whatsoever is left.     

Transport properties of the Lorentz gas: Fourier's law

We investigate the stationary nonequilibrium (heat transporting) states of the Lorentz gas. This is a gas of classical point particles moving in a region gL containing also fixed (hard sphere) scatterers of radiusR. The stationary state considered is obtained by imposing stochastic boundary conditions at the top and bottom of , i.e., a particle hitting one of these walls comes off with a velocity distribution corresponding to temperaturesT ₁ andT ₂ respectively,T ₁ <T ₂. Letting be the average density of the randomly distributed scatterers we show that in the Boltzmann-Grad limit,,R 0 with the mean free path fixed, the stationary distribution of the Lorentz gas converges in theL ¹-norm to the stationary distribution of the corresponding linear Boltzmann equation with the same boundary conditions. In particular, the steady state heat flow in the Lorentz gas converges to that of the linear Boltzmann equation, which is known to behave as (T ₂-T ₁)/L for largeL, whereL is the distance from the bottom to the top wall: i.e., Fourier's law of heat conduction is valid in the limit. The heat flow converges even in probability. Generalizations of our result for scatterers with a smooth potential as well as the related diffusion problem are discussed.Research supported in part by NSF Grant no. Phy 77-22302.On leave of absence from the Fachbereich Physik der Universität, München. Work supported by a DFG fellowship.     

Spectral Boundary Conditions for Generalizations of Laplace and Dirac Operators

Spectral boundary conditions for Laplace-type operators on a compact manifold X with boundary are partly Dirichlet, partly (oblique) Neumann conditions, where the partitioning is provided by a pseudodifferential projection; they have an interest in string and brane theory. Relying on pseudodifferential methods, we give sufficient conditions for the existence of the associated resolvent and heat operator, and show asymptotic expansions of their traces in powers and power-log terms, allowing a smearing function . The leading log-coefficient is identified as a non-commutative residue, which vanishes when =1. The study has new consequences for well-posed (spectral) boundary problems for first-order, Dirac-like elliptic operators (generalizing the Atiyah-Patodi-Singer problem). It is found e.g. that the zeta function is always regular at zero. In the selfadjoint case, there is a stability of the zeta function value and the eta function regularity at zero, under perturbations of the boundary projection of order -dim X.     

Nonstatic gas spheres in general relativity

A model for a time-dependent relativistic gas sphere is given. Both pressurep and density are positive inside the outer boundary of the sphere, and they both drop to zero at the boundary itself. Their respective gradients are negative. For expanding models the pressure and the density are decreasing, and for contracting models the pressure and the density are increasing. The adiabatic sound speed is thus real, and for the layers close to the surface of the sphere it is found that the adiabatic sound speed is less than the speed of light. For these layers the condition>3p is also fulfilled. However, the area 4R ² for a sphere of constantr is not an increasing function of comoving radial coordinater.     

Finite-Size Effects for the Potts Model with Weak Boundary Conditions

Using the Pirogov–Sinai theory, we study finite-size effects for the ferromagnetic q-state Potts model in a cube with boundary conditions that interpolate between free and constant boundary conditions. If the surface coupling is about half of the bulk coupling and q is sufficiently large, we show that only small perturbations of the ordered and disordered ground states are dominant contributions to the partition function in a finite but large volume. This allows a rigorous control of the finite-size effects for these weak boundary conditions. In particular, we give explicit formulæ for the rounding of the infinite-volume jumps of the internal energy and magnetization, as well as the position of the maximum of the finite-volume specific heat. While the width of the rounding window is of order L ^–d , the same as for periodic boundary conditions, the shift is much larger, of order L ^–1. For strong boundary conditions—the surface coupling is either close to zero or close to the bulk coupling—the finite size effects at the transition point are shown to be dominated by either the disordered or the ordered phase, respectively. In particular, it means that sufficiently small boundary fields lead to the disordered, and not to the ordered Gibbs state. This gives an explicit proof of A. van Enter's result that the phase transition in the Potts model is not robust.     

Fluctuating boundary layers in combined convection flows along a vertical surface

The effect of buoyancy force on laminar boundary layer in two dimensional flow and heat transfer along a semi-infinite vertical surface, when the velocity of the on-coming stream oscillates in magnitude about a steady mean, is analysed. Two separate solutions valid for low and high frequency ranges are developed. It is found that for low frequency oscillations the phase angles of oscillatory components of skin friction and the rate of heat transfer increase as the Grashof number increases. For very high frequencies, the velocity field is of shear-wave type unaffected by the mean flow; the phase lead in the skin friction is, then, /4 and the rate of heat transfer fluctuation indicates a phase lag of /2 over that of the free stream oscillations.The authors are thankful to referee for many useful suggestions.     

Neutrino radiation in spherically-symmetric gravitational fields I. The energy-momentum tensor for the one-neutrino and many-particle neutrino field

The physical situation of a star emitting neutrinos is considered. Some difficulties in the classical theory are mentioned, and a more detailed approach to the properties of neutrino radiation in general relativity is given. The classical theory is here regarded as a one-particle theory, and by summing over many particles propagating in randon radial directions, the energy-momentum tensor of the total radiation field is shown to approximate to the geometrical optics type satisfying other conditions defining its radial and time dependence.     

The Spectral Gap of the 2-D Stochastic Ising Model with Mixed Boundary Conditions

We establish upper bounds for the spectral gap of the stochastic Ising model at low temperatures in an l×l box with boundary conditions which are not purely plus or minus; specifically, we assume the magnitude of the sum of the boundary spins over each interval of length l in the boundary is bounded by l, where <1. We show that for any such boundary condition, when the temperature is sufficiently low (depending on ), the spectral gap decreases exponentially in l.     

Influence of magnetic fields on taylor vortex formation in magnetic fluids

The flow of a magnetic fluid placed inside a small gap between concentric rotating cylinders is investigated for axial, radial and azimuthal magnetic fields. An equation of motion is derived phenomenologically to describe the hydrodynamics of magnetic fluids. Studied are the changes in the critical Taylor numberT _c and wave numberT _c which characterize the instability of primary circular Couette flow towards Taylor vortices. It is found that all above magnetic fields have a stabilizing effect on circular Couette flow and thatT _c increases or decreases, depending on the direction of the magnetic field. Besides this, the influence of the magnetic fields on the correlation length ₀, the wave number of maximal growthk _m and the linear growth rate amplitude ₀ is determined.     

Specifications and Martin boundaries forP(Φ)2-random fields

It is shown thatP()₂-Gibbs states in the sense of Guerra, Rosen and Simon are given by a specification. The construction of the specification is based on finding a proper version of the interaction density given by the polynomialP. The existence of this version follows from the fact that all powers of the solution of a Dirichlet problem for an open bounded setU with boundary data given by a distribution are integrable onU. As a consequence the Martin boundary theory for specifications can be applied toP()₂-random fields. It follows that anyP()₂-Gibbs state can be represented in terms of extreme Gibbs states. In certain cases the extreme Gibbs states are characterized in terms of harmonic functions. It follows, in particular, that for any given boundary condition introduced so far the associated cutoffP()₂-measure has a representation as an integral over harmonic functions.     

Analysis and Experiments for a Computational Model of a Heat Bath

A question of some interest in computational statistical mechanics is whether macroscopic quantities can be accurately computed without detailed resolution of the fastest scales in the problem. To address this question a simple model for a distinguished particle immersed in a heat bath is studied (due to Ford and Kac). The model yields a Hamiltonian system of dimension 2N+2 for the distinguished particle and the degrees of freedom describing the bath. It is proven that, in the limit of an infinite number of particles in the heat bath (N), the motion of the distinguished particle is governed by a stochastic differential equation (SDE) of dimension 2. Numerical experiments are then conducted on the Hamiltonian system of dimension 2N+2 (N1) to investigate whether the motion of the distinguished particle is accurately computed (i.e., whether it is close to the solution of the SDE) when the time step is small relative to the natural time scale of the distinguished particle, but the product of the fastest frequency in the heat bath and the time step is not small—the underresolved regime in which many computations are performed. It is shown that certain methods accurately compute the limiting behavior of the distinguished particle, while others do not. Those that do not are shown to compute a different, incorrect, macroscopic limit.     

Classical fluids of negative heat capacity

It is shown that new parametersX can be defined such that the heat capacity C_xT(S/T)_x is negative, even when the canonical ensemble [i.e., at fixed T=(U/S)_Y and YX] is stable. This implies an extension of the classical theory of polytropes from ideal gases to general fluids. As examples of negative heat capacity systems we treat blackbody radiation and general gas systems with nonsingular _T. For the case of a simple ideal gas we even exhibit an apparatus which enforces a constraint X(p, V)=const that makes C_x<0. We then show that it is possible to infer the statistical mechanics of canonicallyunstable systems-for which even the traditional heat capacities are negative-by imposing constraints that stabilize the associated noncanonical ensembles. Two explicit models are discussed.     

On the temperature of nonequilibrium optical fields

The dependence of the temperature of single-mode optical radiation on the relative dispersion of the number of photons is investigated. It is shown that the amplitude-stabilized field oscillator has a temperature twice as high as the oscillator excited by a heat source for the same mean energy. Further increase in the temperature of the stationary radiation field is possible by transition to nonclassical states not described by a positive definite quasi-probability function P().Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 7–13, June, 1972.     

Analytical solutions of the lattice Boltzmann BGK model

Analytical solutions of the two-dimensional triangular and square lattice Boltzmann BGK models have been obtained for the plane Poiseuille flow and the plane Couette flow. The analytical solutions are written in terms of the characteristic velocity of the flow, the single relaxation time , and the lattice spacing. The analytic solutions are the exact representation of these two flows without any approximation. Using the analytical solution, it is shown that in Poiseuille flow the bounce-back boundary condition introduces an error of first order in the lattice spacing. The boundary condition used by Kadanoffet al. in lattice gas automata to simulate Poiseuille flow is also considered for the triangular lattice Boltzmann BGK model. An analytical solution is obtained and used to show that the boundary condition introduces an error of second order in the lattice spacing.     

The Hermite-like description of two-magnon states of 1D quantumS = 1/2 chains with elliptic exchange is complete

In this Letter, we give an example of Navier-Stokes flow with a viscosityv, which converges to Euler flow inL ² norm asv 0. Our Navier-Stokes flow has a discontinuity on the boundary at the initial time, the so-called initial layer, and its initial data has singularity asymptotically asv 0 near the boundary. Even with these singularities, it can be said that our Navier-Stoke flow can converge to some Euler flow inL ² norm.     

Another Return of “Return to Equilibrium”

The property of return to equilibrium is established for a class of quantum-mechanical models describing interactions of a (toy) atom with black-body radiation, or of a spin with a heat bath of scalar bosons, under the assumption that the interaction strength is sufficiently weak. For models describing the first class of systems, our upper bound on the interaction strength is independent of the temperature T, (with 0<TT₀<), while, for the spin-boson model, it tends to zero logarithmically, as T0. Our result holds for interaction form factors with physically realistic infrared behaviour. Three key ingredients of our analysis are: a suitable concrete form of the Araki-Woods representation of the radiation field, Mourres positive commutator method combined with a recent virial theorem, and a norm bound on the difference between the equilibrium states of the interacting and the non-interacting system (which, for the system of an atom coupled to black-body radiation, is valid for all temperatures T0, assuming only that the interaction strength is sufficiently weak).Acknowledgements We thank V. Bach and I.M. Sigal for countless discussions on related problems and spectral methods without which this work would never have been done. M.M. is grateful to V. Jaksi for illuminating discussions. We have enjoyed the hospitality of IHES during the initial and final stages of this work.     

Intense spontaneous amplified emission of Li2 diffuse violet bands in the 4400 Å region

We report observation of intense spontaneous amplified radiation of Li₂ diffuse violet band in the 4100–4900 Å region. The radiation is strongly enhanced when the lithium vapor in a heat pipe is optically pumped with a pulsed dye laser with the output wavelength tuned to near the Li 2s–4s two-photon resonance transition. The diffuse violet band can probably be assigned to a recently reported triplet bound-free (2³ _g a ³ _u ⁺ ) transition. It is found that the productions of the molecular diffuse band are contributed from Li atoms as well as Li₂ molecules. The excitation functions and their dependence on laser power density are presented and the mechanisms for producing the diffuse bands are discussed.Department of Physics, National Central University, Chung-Li, Taiwan, Republic of Chinabl]References