A theory is presented for the transverse intensity distribution bistability of a Gaussian optical beam after its passage through a nonlinear thin film. The equations governing the intensity distribution are cast in the form analogous to optical bistability in a longitudinal cavity (a Fabry-Perot interferometer), i.e., into two coupled transcendental equations from which multiple solutions are obtained. This formalism allows one to examine various physical approximations in obtaining the equations, and to improve on these approximations. It also lucidly illustrates the mechanisms of transverse intensity distribution bistability. The theoretical predictions are verified with quantitative experimental results on thin films of nematic liquid crystals.

Based on a microscopic theory developed recently, a dynamical model of density fluctuations in simple fluids and glasses is proposed and analyzed analytically and numerically. The model exhibits a liquid-glass transition, where the glassy phase is characterized by a zero-frequency pole of the longitudinal and transverse viscosities indicating the systems' stability against stress. This also implies an elastic peak in the density-fluctuation spectrum. Approaching the glass transition the slowing down of density fluctuations is controlled by the increasing longitudinal viscosity, which in turn is coupled via a nonlinear feedback mechanism to the slowly decaying density fluctuations. This causes a divergence of the structural relaxation time at a certain critical coupling constant ${\lambda }_c$. At the glass transition density fluctuations decay with a long-time power law $\Phi \mathrm{}\left(t\right)\mathrm{}\sim t^{-\alpha }$ with $\alpha =0.395\mathrm{}$ and approaching the transition the viscosity diverges proportional to ${\epsilon }^{-\mu }$ and ${\epsilon }^{-\mu }$, where $\epsilon =|1-\frac{\lambda }{{\lambda }_c}|$ and $\mu =\frac{(1+\alpha )}{2\alpha }$, ${\mu }^{\prime }=\mu -1\mathrm{}$ below and above the transition, respectively. The long-time tail "paradox" in dense fluids is briefly discussed.

It is observed that the Navier-Stokes space-time dilatation invariance ($\overrightarrow{\mathrm{x}}\to \lambda \overrightarrow{\mathrm{x}},t\to {\lambda }^{2}t$) implies that the random motility and chemotactic flux function take the forms observed experimentally for motile Escherichia coli attracted by low concentrations of oxygen; moreover, the rate function for $E$. coli consumption of dissolved oxygen is required to have the form (const) × ${\mathrm{}\left(\mathrm{local}\mathrm{oxygen}\mathrm{concentration}\right)\mathrm{}}^{2/3}$. It is also noteworthy that the Schrödinger-Bloch function for redistribution of chemotactic bacteria cells is invariant under the space-time dilatation transformations if and only if the chemotactic flux coefficient-random motility ratio equals 2, a value in the range 1.1 to 2.5 observed experimentally by Holz and Chen in the oxygen chemotaxis of motile $E$. coli. Suitably specialized governing equations for the phenomenological transport theory also admit a Galilean transformation invariance symmetry if and only if the chemotactic flux coefficient-critical substrate diffusivity ratio equals -2 and the consumption rate function is simply linear in the local oxygen concentration. Applicable to the regime of viscous incompressible flows with Reynolds numbers much less than unity, the Navier-Stokes superposition invariance may also give rise to a corresponding invariance symmetry in equivalent but linear phenomenological transport equations.

A general, formal structure of the Kirkwood coupling-parameter expansion method for the distribution function of a fluid is presented. In this method, no closure approximation, such as the superposition approximation, is required. Various properties of this method are discussed.

We generalize the Arnold functional to discuss Lyapunov stability of rotational water waves.

We present a new approach for estimating line broadening. It relies upon the Thomas-Fermi model at finite temperature and on the assumption of local thermodynamic equilibrium. This method can predict general features without requiring detailed computations of specific effects, usually encountered in accurate calculations of line broadening. Obviously, no very precise results can be expected from our nonspecific approach, but rather plausible estimates of the general features can be obtained with only modest effort. Results for a Hf plasma are presented and compared with experimental observations.

This paper deals with two types of rotational average which arise in a number of physical processes in fluids. The first are termed "phased averages," and occur in the theory of bimolecular interactions with radiation; the second are "Boltzmann-weighted" averages, which occur in the theory of anisotropic media. A general method for obtaining these results is presented, and analytic results are given for tensorial interactions up to and including the fourth rank. Several applications are discussed including laser-induced forces, cooperative two-photon absorption, and electric-field-induced harmonic generation. The proof of a relation involving conventional rotational averages is also given.

The critical surface can be unstable to coherent rippling perturbations due to the action of negative pressure induced by the random magnetic field associated with ion-acoustic turbulence. The negative magnetic pressure occurs if there exists a preferential orientation of the random magnetic field (anisotropy of the ion-acoustic turbulence) when the nonpotential component of the magnetic pressure more than compensates the potential part.

Adiabatic elimination of stochastic variables is formulated in a systematic and largely rigorous manner, and applied to a variety of examples. Ambiguities in naive methods are resolved, and it is shown that the resultant stochastic differential equation is not always to be interpreted as an Ito or as a Stratonovich stochastic differential equation. A class of nonlinear examples is also treated in which naive methods fail completely.

We develop a stochastic theory of rapidly diffusing spatially distributed systems. Discussion is within the framework of the cell model, in which the system is described in terms of a lattice of $n$ cells. Utilizing projector-operator techniques, we formalize the method of homogenization of such systems. That is, by projecting out high-$q$ Fourier modes in the adiabatic limit of large diffusion, we map the system to one defined on a coarser-grained lattice. We thus demonstrate a "blocking" procedure in the cell model. Finally we consider a simple hydrodynamic model and show that near the point of convective instability projection-operation methods predict the same amplitude equations for the slow hydrodynamic modes as does the method of multiple scales.

We present a stochastic theory of the renormalization-group transformations of the time-dependent Ginzburg-Landau model, describing a system which exhibits the characteristics of a second-order phase transition. We eliminate a certain range of the Fourier components of the magnetic spin variables via a projection-operator method. This effectively changes the scale of the system and transforms the coupling constants. The procedure is shown to be equivalent to the integration over short-wavelength modes as in the renormalization-group transformation performed by Wilson and Kogut. This equivalence shows that the projection-operator method is a valid procedure for scaling critical systems, and in particular it indicates that the treatment of fluctuations is systematic. We suggest that such a method should also provide a straightforward approach to the dynamic renormalization group.

The emergence of spatial correlations around a nonequilibrium steady state is studied by means of a stochastic description based on a multivariate master equation. The dependence of the strength and range of the correlations on the distance from equilibrium is determined. The formalism is applied to chemically reacting systems and to simple fluids submitted to a temperature gradient.

The direct correlation function of the one-dimensional lattice-gas (Ising) model with nearest-neighbor and next-nearest-neighbor interactions is calculated exactly. It is shown that, depending on the strengths and signs of the coupling constants, the direct correlation function can either have a finite range equal to that of the interactions, or decay exponentially in a monotonic or oscillatory fashion. At the critical point, which occurs at zero temperature, the direct correlation function is found in all cases to have exactly the range of the interactions, while its values become unbounded; this is in contrast to earlier nonrigorous results. The second moment of the direct correlation function is found to diverge at the critical point, in the neighborhood of which it becomes proportional to the correlation length of the density (spin) fluctuations, confirming a prediction of scaling theory in one dimension.

Under parameter fluctuations an erratic motion can settle into a nearby approximate cycle, which eventually decays if the orbit is perturbed. Such temporary patterns are studied in the framework of iterative maps of probability distributions. The usefulness of such a structure in modeling the mortality of living systems is pointed out.

The specific heat of a single one-dimensional quantum oscillator associated with a potential proportional to ${\mathrm{}\left|x\right|\mathrm{}}^{\nu }(\nu >0)$ is discussed for the limiting cases $T\to 0$ and $T\to \infty$, and numerically calculated for intermediate temperatures. To perform these calculations for arbitrary values of $\nu$, we use, for the energetic spectrum, an approximate formula quite superior to the standard WKB one; namely, that formula recently established by Turschner.

Featuring the modified hypernetted-chain (MHNC) scheme as a variational fitting procedure, we demonstrate that the accuracy of the variational perturbation theory (VPT) and of the method based on additivity of equations of state is determined by the excess entropy dependence of the bridge-function parameters [i.e., $\eta \mathrm{}\left(s\right)\mathrm{}$ when the Percus-Yevick hard-sphere bridge functions are employed]. It is found that $\eta \mathrm{}\left(s\right)\mathrm{}$ is nearly universal for all soft (i.e., "physical") potentials while it is distinctly different for the hard spheres, providing a graphical display of the "jump" in pair-potential space (with respect to accuracy of VPT) from "hard" to "soft" behavior. The universality of $\eta \mathrm{}\left(s\right)\mathrm{}$ provides a local criterion for the MHNC scheme that should be useful for inverting structure-factor data in order to obtain the potential. An alternative local MHNC criterion due to Lado is rederived and extended, and it is also analyzed in light of the plot of $\eta \mathrm{}\left(s\right)\mathrm{}$.

A careful search has been made for ${\mathrm{H}}_2^{-}$, ${\mathrm{H}}_3^{-}$, ${\mathrm{N}}^{-}$, and ${\mathrm{N}}_2^{-}$ ions formed by two-step electron capture, with negative results. These ions, all of which have been reportedly observed, should be most readily produced by this method, yet no convincing trace of any of them was found. The evidence strongly supports the nonexistence of metastable ${\mathrm{H}}_2^{-}$ and ${\mathrm{H}}_3^{-}$. In addition, the theoretically predicted ${\mathrm{Mg}}^{-}$ ion could not be observed, although the Mg${\mathrm{H}}^{-}$ molecular ion was readily produced and found to be stable.

The connection between the three-dimensional hydrogen atom and a four-dimensional harmonic oscillator obtained in previous works, from a hybridization of the infinitesimal Pauli approach to the hydrogen system with the Schwinger approach to spherical and hyperbolical angular momenta, is worked out in the case of the zero-energy point of the hydrogen atom. This leads to the equivalence of the three-dimensional hydrogen problem with a four-dimensional free-particle problem involving a constraint condition. For completeness, the latter result is also derived by using the Kustaanheimo-Stiefel transformation introduced in celestial mechanics. Finally, it is shown how the Lie algebra of SO(4,2) quite naturally arises for the whole spectrum (discrete plus continuum plus zero-energy point) of the three-dimensional hydrogen atom from the introduction of the constraint condition into the Lie algebra of Sp(8,R) associated with the four-dimensional harmonic oscillator.

Accurate eigenvalues (eight to six significant figures) and critical screening parameters are calculated for a two-particle system interacting through (a) a static screened Coulomb potential (SSCP), and (b) a cut-off Coulomb potential (COCP). A comparison of the results shows that as far as bound states are concerned it is not possible to simulate a SSCP by a COCP by a suitable scaling of the screening length.

The method of composite maps, which consists of a sequence of maps, is applied to several members of the Klein-Gordon family of nonlinear differential equations to produce new kink or soliton solutions. This yields, in particular, new kink solutions of the double-sine-Gordon model as well as solitons of the ${\phi }^6$ model, which are of importance in field theory, condensed-matter physics, and optics.