Strict Equilibrium Research Articles

Interstellar cloud shapes - A minimum-hypothesis account involving a purely gravitational effect

The effect produced by the anisotropic gravitational field on the shape of a nonspherical, self-gravitating interstellar cloud is examined. The projected axial ratios observed for molecular clouds can be accounted for if clouds are not generally in a state of strict dynamical equilibrium. For clouds having spectral line widths dominated by supersonic turbulent motions, this purely gravitational effect predicts a mean true axial ratio q about 0.3 for both oblate and prolate cloud geometries. Clouds having p less than about 0.3 are likely to be prolate and quite elongated. The observational data on molecular cloud cores, which have essentially thermal line widths, can be fitted equally well to either oblate or prolate spheroids having q between 0.1 and 0.4 with q about 0.2 providing the best match: highly elongated cores are again more likely to be prolate. While other processes certainly operate throughout the Galaxy to influence cloud shape, it appears plausible that at least some clouds may be shaped principally by their anisotropic gravitational field.

Credible assignments in coordination games

This paper uses the experimental method to examine an arbiter's ability to determine the outcome of two-person coordination games. All of the arbiter's assignments in the experiments were strict equilibrium points, but some assignments violated payoff-dominance or symmetry. An assignment that corresponds to the game's outcome is a credible assignment. The experiments test the hypothesis that an assignment to a strict equilibrium is a credible assignment. Our subjects did not find the individual rationality and mutual consistency of an equilibrium assignment to be sufficient reason for implementing the assignment when doing so conflicts with payoff-dominance or symmetry.

Dynamics of Competition among Microalgae in Variable Environments: Experimental Tests of Alternative Models

Two species of green microalgae, Scenedesmus quadricauda var. longispina and chlorella sp., were grown together in cntinuous cultures under conditions of phosphorus-limitation. This resource was supplied as a series of discrete periodic pulse, preventing achievment of a strict equilibrium. Two mathematical models, based on the equations of Monod and Droop were parameterized from earlier, single-species experiments. These models were then tested for their ability to predict competitivedynamics in these and similar competition cultures. Chlorella outcompeted Scenedesmus in all cultures, regardless of the period of phosphorus pulses

Nonaxisymmetric instabilities in a slender torus - Two- and three-dimensional simulations

Nonaxisymmetric instabilities in accretion disks are investigated in terms of the slender, non-self-gravitating, ideal fluid torus model. Nonlinear simulations are carried out in two and tree dimensions, using Cartesian-grid finite differencing. The fastest growing instability is the principal mode. This mode saturates with the formation of ellipsoidal density distributions, designated as 'planets'. The simulations provide evidence for nonlinear mode-mode coupling. When several wavenumbers m are unstable, multiple 'planets' form that subsequently merge. Sources of numerical error are examined, and the effects are contrasted with those of the physical instability. The same qualitative evolution is seen in both the two- and the three-dimensional simulations, even though strict vertical hydrostatic equilibrium is no longer rigorously maintained after mode saturation.

Photometric geodesy of main-belt asteroids: II. Analysis of lightcurves for poles, periods, and shapes

Twenty-five of the twenty-six asteroids included in the “photometric geodesy” program (S.J. Weidenschilling, C.R. Chapman, D.R. Davis, R. Greenberg, D.H. Levy, and S. Vail, Icarus 70, 191–245, 1987) have been studied on the assumption that asteroids can be modeled as smooth, featureless, triaxial ellipsoids rotating about their shortest axes. Using all lightcurves available, rotational poles have been obtained by three independent methods. With these poles each asteroid's sidereal period and triaxial ellipsoid axial ratios have been determined, along with the associated photometric parameters. The studied asteroids seem to have rotational poles that do not lie near their orbital planes. Two distinct types of asteroids may be indicated by a stronger solar phase angle-amplitude dependence (perhaps due to a rougher surface?) and a weaker phase-amplitude (smoother?) relation. Although no strict hydrostatic equilibrium shape is found, several asteroids are close enough to equilibrium figures to allow an estimate of their densities to be made under certain assumptions.

Diffusion profiles associated with the onsager matrix in non-equilibrium A-A ∗-vacancy and A-B-vacancy solutions

A new derivation of the Onsager matrix is presented based on diffusion correlation effects in a non local equilibrium ternary solution of conserved vacancies with a binary isotope mixture. In this calculation, we proceed deductively, thus avoiding the strong construction of Howard and Lidiard which conjectures a non-equilibrium diffusion thought experiment at a constant isotopic ratio and subjects the result to local equilibrium constraints, thereby generating a phenomenological proof of Onsager Reciprocity for A-A ∗-vacancy solutions. By contrast, we introduce correlation cross-effects, together with detailed balance, within a strict non local equilibrium structure, and from this derive a symmetric Onsager matrix for the A-A ∗-V case identical to that of Howard and Lidiard. Solutions of the corresponding Pick equations are for the first time evaluated and profiles presented for infinite diffusion couples. These solutions demonstrate that the eigen-values of the diffusion matrix are D V and D ∗ A⪡D V , so that profiles in infinite diffusion couples are generally very steep near the origin. Furthermore, each isotope flows down the other's gradient (positive off-diagonal D-coefficients). For the more general A-B-V case, Manning's approximate generalization of the Howard-Lidiard L-matrix in a conserved vacancy interpretation is expressed as a Fickian D-matrix which has the eigenvalues λ 1 = D V = D ̄ X Vf and λ 2 = D∗ AD∗ B D ̄ where D ̄ is the average tracer coefficient. The corresponding diffusion profiles illustrate quantitatively the nature of the vacancy pump in a countercurrent A-B experiment.

The Stability of Rotating Stars

It was first shown by von Zeipel (1924) that strict radiative equilibrium is impossible in the radiative zones of a uniformly rotating star. This result lead Eddington (1925) and Vogt (1925) to independently suggest the existence of large-scale meridional currents in the radiative zones of rotating stars. Eddington argued that since these currents will transport angular momentum, the condition of strict radiative equilibrium would result in meridional currents combined with differential rotation.

The oblique rotator

Two models are presented for rotating upper main sequence stars in strict radiative equilibrium and possessing a magnetic field which is not axisymmetric with respect to the rotation axis of the star. One of the models involves the parameter χ which is the angle between the rotation axis and the symmetry axis of an inclined poloidal magnetic field. For |$\chi\lt55^\circ$| the results are analogous to the results found for the aligned models |$(\chi=0^\circ)$| which have been reported previously in the literature. For |$\chi\gt55^\circ$| the models exhibit the behaviour of the perpendicular rotator |$(\chi=90^\circ)$| with the surface field increasing with increasing rotation rate; though as for the case of the perpendicular models the more general oblique models with |$\chi\gt55^\circ$| fail to converge for rotation speeds and fields relevant to the magnetic stars. The second model involves a certain level of truncation in the infinite series representation of a general magnetic field. We discuss this model in terms of the simple oblique dipolar model. Further, the system of equations to be solved for this model is overdetermined and we compare solutions obtained from particular sets of equations.

A remark on the Harsanyi-Selten theory of equilibrium selection

We consider a class of 3-person games in normal form with two pure strategies for each player and two strict equilibrium points. To select one of these two strict equilibrium points as the solution, the equilibrium selection theory of Harsanyi and Selten is applied. The games are constructed in such a way that the a priori probabilities reflect somewhat poorly the risk situation of the players. It is argued and illustrated by examples that this might yield unreasonable results. The a priori probabilities would describe the risk situation of the players more completely if their definition were not based on the expectation of correlated decision behavior.

Efficiency properties of strategies market games: An axiomatic approach

The paper investigates the conditions under which an abstractly given market game will have the property that if there is a continuum of traders then every noncooperative equilibrium is Walrasian. In orther words, we look for a general axiomatization of Cournot's well-known result. Besides some convexity, continuity, and nondegeneracy hypotheses, the crucial axioms are: anonymity (i.e., the names of traders are irrelevant to the market) and aggregation (i.e., the net trade received by a trader depends only on his own action and the mean action of all traders). It is also shown that the same axioms do not guarantee efficiency if there is only a finite number of traders. Some examples are discussed and a notion of strict noncooperative equilibrium for anonymous games is introduced.

On the charge polarization of cosmic systems

The positive electric charge density associated with the internal electric fields of self-gravitating systems in hydrostatic equilibrium can be canceled by nearby external flowing plasmas, such as winds. For example, it can be shown that the positive electric charge of a star is likely to be completely screened by its stellar wind. Because winds and other nonstatic phenomena are widespread, the electrical polarization due to the positive charge on static systems such as stars should occur on relatively local scales, in contrast to the universal scale recently suggested by Bally and Harrison. The latter viewpoint would be correct only if the entire universe were in strict hydrostatic equilibrium.

Models of magnetic stars - IV. The perpendicular rotator with meridional circulation

First-order perturbation theory models for uniformly rotating stars with a predominantly dipolar magnetic field with axis inclined at an angle π/2 to the rotation axis have been computed, both in strict radiative equilibrium and with the inclusion of finite resistivity and meridional circulation. In some ways the results are similar to the previous calculations with aligned rotation and magnetic axes. However, a positive correlation between rotation speed and surface flux for a given internal magnetic flux is found. This effect, although small for much of the parameter space investigated, is in contrast to the increasing ratio of interior to surface flux found as the rotation speed increases in the aligned models.

Zero-point motion and superfluid helium

We propose that He II exhibits macroscopic (sigma/sub P//Napprox.O(1)) quantum zero-point motion in momentum space, i.e., that a nonzero root-mean-square superfluid velocity exists even in an equilibrium superfluid system at rest. At absolute zero, using coherent states, we relate the uncertainty sigma/sub P//N in the total momentum P (per particle) to the long-range-order (LRO) part of the phase gradient correlation function, which is proposed as an order parameter. The local equilibrium equation for the superfluid velocity potential derived by Biswas and Rama Rao yields, in the strict equilibrium limit, the equation determining this order parameter in terms of fluctuation correlations that remain to be determined. The order parameter is interaction dependent, nonzero at T=0 if mu-tilde (0)-rho/sub 0/V/sub 0/>0, and can vanish at some transition temperature T/sub lambda/when fluctuation terms become comparable to the T=0 value. (Here V/sub 0/ rho/sub 0/, and mu-tilde (0) are the uniform parts of the potential, density, and chemical potential with shifted zero of energy, respectively.) A characteristic length ..lambda.. (T), diverging at T=T/sub lambda/appears naturally, with its defining relation reducing to a macroscopic uncertainty relation sigma/sub P//N) ..lambda.. (0) =h/2 at T=0. With certain assumptions it is shown that at T=0, LRO in themore » phase gradient correlation function is incompatible with off-diagonal long-range order (ODLRO) in the correlation function, and with nonzero condensate function.« less

The Ginzburg-Pitaevskii equation and microscopic quantum hydrodynamics

We show that for local equilibrium the microscopically derived phase and continuity equations, neglecting certain terms, yield an equation similar to the Ginzburg-Pitaevskii equation at absolute zero. The complex wave function is ψ=√ρ(r) × exp{i(〈φ(itr)〉+μt/ h ̷ )} where μ is the uniform chemical potential in strict equilibrium. A length scale ξ(0)= h ̷ /√ nρ 0V 0 appears naturally, where V 0 is the spatially integrated potential. With experimental values for the uniform total number density ρ 0 and a hard sphere radius of 1 Å, ξ(0)∼1.35 Å, comparable to the phenomenological GP length ∼4 Å. Deviations of macroscopic quantities from their equilibrium values must be small, e.g.| ρ( r)− ρ|/ ρ 0 ⪡1 and we must have (2mξ 2(0)/ h ̷ ρ 0)|▽·{ j(r)-ρ(r)v s(r)}|⪡1 , where v s(r)=( h ̷ /m▽〈φ( r)〉 . In the above, j( r), ρ( r) and 〈φ( r)〉 are the total number current and number density, and the mean phase, respectively, in local equilibrium.

Convection in a closed cavity heated from below when equilibrium conditions are disrupted

The equilibrium of a fluid is possible in a closed cavity in the presence of a strictly vertical temperature gradient (heating from below) [1]. There is a distinct sequence of critical Rayleigh numbers Ri at which this equilibrium loses its stability relative to low characteristic perturbations. The presence of different finite perturbations, unavoidable in an experiment, leads to the absence of a strict equilibrium when R < R1. The problem of the influence of the perturbation on the convection conditions near the critical points arises in this context [2, 3]. The case in which the cavity is heated not strictly from below is investigated in [2] and the case in which the perturbation of the equilibrium is due to the slow movement of the upper boundary of the region is investigated in [3]. In [2, 3] the perturbation has the structure of a first critical motion and thus the results of these papers coincide qualitatively. The perturbation of the temperature in the horizontal sections of the boundary, which creates a perturbation with a two-vortex structure corresponding to the second critical point R2, is examined in this paper. A similar type of perturbation is characteristic for experiments in which the thermal conductivity properties of the fluid and the cavity walls are different. The nonlinear convection conditions are investigated numerically by the net-point method.

Experimental technique for the precise determination of H and D concentration in rutile (TiO2)

Results of a precise determination of the ir absorption strength of H and D in rutile are presented, which permit determination of H or D concentration to an accuracy of 3%. The integrated absorption per ion was (2.51±0.09)×10−17 and (1.33±0.05)×10−17 cm for H and D, respectively. The procedure for calculating the H or D concentration in a particular sample from the ir absorption peaks is illustrated. Used in conjunction with the results of the theoretical analysis of the preceding paper, this technique makes possible the accurate determination of the energy and density of all electron trapping levels in TiO2 crystals. We also describe results showing that heat treatment in a dry H2 atmosphere at ∼800 °C results in the introduction of Ti intersititials but negligible H in the crystal, as predicted by the preceding paper. A technique for measuring diffusion parameters of H and D under conditions of strict chemical equilibrium, thus avoiding the large departures from Fick's law which usually plague such measurements in materials such as TiO2, is also described.

A Generalized Nash Solution for Two-Person Bargaining Games with Incomplete Information

The paper extends Nash's theory of two-person bargaining games with fixed threats to bargaining situations with incomplete information. After defining such bargaining situations, a formal bargaining model (bargaining game) will be proposed for them. This bargaining game, regarded as noncooperative game, will be analyzed in terms of a certain class of equilibrium points with special stability properties, to be called “strict” equilibrium points. Finally an axiomatic theory will be developed in order to select a unique solution from the set X of payoff vectors corresponding to such strict equilibrium points (as well as to probability mixtures of the latter). It will be shown that the solution satisfying the axioms proposed in this paper is the point where a certain generalized Nash product is maximized over this set X.

The membrane potential of Acetabularia mediterranea.

The cytoplasm of an Acetabularia cell is normally at a potential of about -170 mv relative to the external solution; the vacuole is also at this potential. Although there is strict flux equilibrium for all ions, the potential is more negative than the Nernst potentials of any of the permeating ions. Darkness, CCCP, low temperature, and reducing [Cl(-)](o) by a factor of 25 all rapidly depolarize the membrane and inhibit Cl(-) influx. Some of these treatments do not inhibit the effluxes of K(+) and Na(+). Increasing [K(+)](o) also depolarizes the membrane both under normal conditions and at low temperature; in the latter case the membrane is partially depolarized in normal seawater (low [K(+)](o)) and in high [K(+)](o) positive potentials of up to +15 mv are attained. It is concluded that the membrane potential is controlled by the electrogenic influx of Cl(-), and also, at least in some circumstances, by the diffusion of K(+). In addition, it is suggested that electrogenic efflux of H(+) may be important in transient nonequilibrium situations. An Appendix deals with the interpretation of simple nonsteady-state tracer kinetic data.

Depth-dependent line blanketing by neutral and ionized metals in a homogeneous model solar photosphere

Line blanketing due to both the neutral and ionized metals is treated here by the modified picket-fence method and applied to the case of a homogeneous model solar photosphere in strict radiative equilibrium. Improvement in fitting the observed photospheric radiation field is noted over a model in which the line absorption coefficient is constant with depth.

A model for the penumbra of sunspots

A penumbra model in hydrostatic equilibrium is presented. The model accounts for the continuum observations as well as the observations of Fraunhofer lines in the penumbra. The uncertainty in the model in deeper layers is discussed. It is shown that the penumbra is probably not in strict radiative equilibrium.