Pattern selection in binary-fluid mixtures heated from below is studied for separation ratios S in the regime 0<$S_{\infty }$-S≪1, where $S_{\infty }$ is the critical value for which the wavelength 2π/$k_c$ of the instability first becomes infinite. The basic equations are reduced to a scalar equation for the horizontal planform whose coefficients can be determined analytically for boundary conditions of experimental interest. Equivariant bifurcation theory is used to study pattern selection on both square and hexagonal lattices. The results depend strongly on the parameter β describing asymmetry in the boundary conditions at top and bottom. When β=0, squares are stable on the square lattice but are replaced by rolls with increasing β. The transition between stable squares and stable rolls takes place via a stable branch of new solutions called crossrolls. On the hexagonal lattice, hexagons are stable if β=0, but rolls are stable when β≠0.

In the case of a nonlinear stationary response resulting from the application of an alternating field superimposed on a continuous field in a dielectric liquid, the birefringence of that medium is well expressed as real and imaginary parts of two normalized complex birefringence functions Δ$N_j^{\mathrm{*}}$=$X_j$-${\mathrm{iY}}_j$, where j=1,2. By eliminating the angular frequency ω in the parametric equations $X_j$(ω) and $Y_j$(ω), Cartesian equations are obtained, which provide new quantitative interpretations of $Y_j$($X_j$) experimental plots. Moreover, for each harmonic term, a polar equation is established indicating that, in the complex plane, the end of the vector Δ${\mathrm{N}}_j$=($X_j$,$Y_j$) describes a set of quasiconchoids of circles whose radii vary with P, the ratio of the induced dipole moment to the permanent dipole moment. For any value of P, changes in phase angle ${\theta }_j$ with frequency are widely considered because of their very interesting use in Kramers-Kronig relations.

We consider the linear-response properties of composites made of two isotropic materials. We concentrate on linear-response phenomena, with nonvanishing cross coefficients, such as the magnetoelectric effect, the thermoelectric effect, and coupled, multispecies diffusion. We find that the moduli of the composite must obey a number of compatibility relations, involving the moduli of the components, but totally independent of the mixture ratio of the components and the microstructure of the composite. The compatibility conditions take the form C(L,$L_a$,$L_b$)==${\mathrm{LL}}_a^{-1}$$L_b$-$L_b$$L_a^{-1}$L=0, where $L_a$,$L_b$ are the response matrices of the components, and L is that of the composite. These come about because there exists a choice of driving forces and fluxes—an eigenbasis—that are decoupled in both components.

When the composite itself is isotropic, these constitute n(n-1)/2 relations that its moduli must obey (for real response matrices), leaving only n independent coefficients to be specified—n being the number of driving fields. When the composite is anisotropic, the elements of L are spatial tensors and the same conditions apply as relations between these tensors. In the latter case one can eliminate the moduli of the components, and obtain relations that the moduli of the composite must satisfy among themselves. The mere fact that the composite is made of two isotropic components, without any further information, imposes strong relations among its moduli. For example, we show that all the tensor moduli of such a composite are symmetric. The same compatibility conditions must also be obeyed by the response matrices of any three composites of the same two isotropic components; so, knowledge of the moduli of two composites supplies constraints on any third, or, for that matter, on the components themselves. We discuss the general properties of the compatibility conditions and their possible relation to parallel and series construction of composites, and demonstrate our findings for the most common case of two driving fields in three dimensions. The compatibility conditions apply not only for the effective response matrices of well-homogenized composites, but also for the local response—at any point within some region that is filled with an arbitrary two-phase mixture—to the potentials on the boundary of the region. All the above apply, with little change, to the case where the response matrices are complex-symmetric or Hermitian. We have nothing to add to the lore on composites, in the case where all the cross coefficients vanish; the compatibility conditions are identities in this case. All our results remain intact for multiphase composites, when all the components but two are either perfect conductors or perfect insulators.

It is the purpose of this paper to point out that the creation of fractal basin boundaries is a characteristic feature accompanying the intermittency transition to chaos. (Here ‘‘intermittency’’ transition is used in the sense of Pomeau and Manneville [Commun. Math. Phys. 74, 189 (1980)]; viz., a chaotic attractor is created as a periodic orbit becomes unstable.) In particular, we are here concerned with type-I and type-III intermittencies. We examine the scaling of the dimension of basin boundaries near these intermittency transitions. We find, from numerical experiments, that near the transition the dimension scales with a system parameter p according to the power law d≃$d_0$-k‖p-$p_{\mathrm{I}{\Vert }^{\beta }}$ with β=(1/2, where $d_0$ is the dimension at the intermittency transition parameter value p=$p_{\mathrm{I}}$ and k is a scaling constant. Furthermore, for type-I intermittency $d_0$$d_0$=D, where D is the dimension of the space. Heuristic analytic arguments supporting the above are presented.

The stochastic relaxation in a class of bistable soft potentials f(x)=-2 ln[cosh(x)]+2 ln[${\cosh }^2$(x)+${\sinh }^2$(R)] is investigated. The structure of these bistable potentials varies from a smooth parabola to a wedge-shaped behavior as the bistability parameter R increases. It is pointed out that the original (white-noise) Kramers formula for the decay rate applies to such bistable potentials in the weak-noise limit, i.e., for small diffusion constants only. An improved analytical prediction for the decay rate valid for small Arrhenius factors is presented. The novel calculation predicts a prefactor that depends on the noise intensity D. This prefactor changes continuously from the Kramers value for small D to the expression for a W-shaped wedge potential for large D. Thus the traditional Kramers theory is extended by this approach. The comparison of our analytical prediction with the numerical results for the smallest nonvanishing eigenvalue is quite convincing for small Arrhenius factors. Moreover, a detailed discussion of various mean-first-passage-time (MFPT) and correlation-time expressions in different types of potentials reveals that in the limit of small Arrhenius factors (i) all MFPT’s, if applied properly, converge to the smallest nonzero eigenvalue of the corresponding Fokker-Planck equation, and (ii) the structure of the bistable potential at infinity has no dominant influence on the decay rate.

A formalism is developed to study refraction by a spherical nematic bubble. It is applicable to bubbles that are larger than light wavelengths, but smaller than the dimensions for excitation of director-fluctuation-induced scattering. The technique yields a nonlinear differential equation and an associated integral which govern the trajectory of a ray inside a nematic region for an arbitrary director configuration. Explicit solutions are provided for five simple interior arrangements—isotropic, onion skin, radial star, horizontal (bottle brush), and vertical. It is then demonstrated that for extraordinary-ordinary refractive-index difference small compared to either, interfacial refraction at the bubble surface is the dominant contribution; deviations from a rectilinear path are small. When ranked in terms of decreasing scattering effectiveness, the sequence is horizontal, onion, isotropic, radial, and vertical if the light is linearly polarized and coupling optimally to the extraordinary index component; for unpolarized incoherent light the order becomes isotropic, horizontal, onion, radial, and vertical.

A global interfacial wave theory is developed to demonstrate the essence and origin of the complicated dendritic structure of a growing needle crystal. In this paper, by using the multiple-variable-expansion method, the local dispersion relation for normal modes is derived in a paraboloidal coordinate system. The local instability mechanism of the interface is explored. Further, in a companion paper [J. J. Xu, following article, Phys. Rev. A 40, 1609 (1989)], a global instability mechanism is established. The existence and significance of a simple turning point in the solidification system is first identified. We obtain uniformly valid solutions in the vicinity of the turning point and investigate the interaction of the interfacial waves in that region. A remarkable mechanism of wave emission and reflection at the turning point is discovered.

In this paper, we investigate the global instability of a needle crystal growth. It is shown that in the solidification system there exists a simple turning point, which plays a central role for the formation of self-sustaining dendritic structure of a growing needle crystal. A most remarkable signal reflection and wave-emission mechanism at the turning point is explored.

We consider kinetic models for irreversible processes that exhibit nontrivial steady states and phase transitions between these states. We study the steady states of these nonequilibrium systems using the methods of equilibrium statistical mechanics. To accomplish this, we use two methods for associating an effective Hamiltonian $H_{\mathrm{eff}}$ with a given steady state. Varying the kinetic rate parameters changes $H_{\mathrm{eff}}$, which can lead to phase transitions. Since $H_{\mathrm{eff}}$ is defined indirectly, the transitions may occur via new mechanisms not possible in equilibrium systems. We apply these methods to several one-dimensional lattice models relevant to certain aspects of catalysis. Two of these models exhibit second-order phase transitions, one to a catalytically inactive state, the other to a catalytically active state. The transitions are in different universality classes. We employ a model used in polymer unbinding and wetting transitions to investigate these transitions.

A high-resolution synchrotron x-ray study of oriented, supported, hydrated dilauryl phosphatidylcholine in the lyotropic $L_{\alpha }$ (smectic-A or ‘‘fluid’’) phase shows that thermal fluctuations give rise to algebraic decay of positional order in stacking of the bimolecular lamellae, the so-called ‘‘Landau-Peierls’’ state. For planar monocrystalline samples of thickness 10–20 μm, the exponents derived from intensity profiles are in accord with harmonic theory. When combined with previous measurements of the splay elastic constant K in this lyotropic system, the exponents indicate a bulk compressional elastic constant B=(1.9±1.0)×${10}^8$ erg/${\mathrm{cm}}^3$, consistent in magnitude with values measured in thermotropic liquid crystals. Samples of thickness ∼1–2 μm show anomalous behavior in the wings of the longitudinal intensity profiles, but are not in accord with harmonic theory, suggesting that surface effects partially quench the thermal fluctuations. Grandjean terraces on sample surfaces produce striking refractive effects at low Bragg angles.

The Jahn-Teller theorem has been modified on the basis of statistical physics: The ensemble average of the molecular energy is supposed to be minimal with respect to distortions of the nuclear framework. The consequences of this formulation are discussed for the $D_{5d}$-coordinated cobaltocene having the degenerate electronic ground state ${}_{}{}^2{}_{}{}^{}\mathrm{E}_{1\mathrm{g}}^{}{}_{}{}^{}$. At low temperatures T<155 K, the symmetrical structure is unstable, and the strength of the rhombic distortion depends on temperature. At moderate temperatures T≥155 K, however, the symmetrical configuration $D_{5d}$ is stable. The three adjustable parameters of the theory were successfully fitted to susceptibility data reported in the literature.

The generalized time-dependent one-dimensional Ginzburg-Landau equation (GLE) is considered with a frequency parameter that is a slow function of the spatial coordinate. In the case of nonzero coefficients of the spatial and nonlinear dispersions, within the geometrical optics approximation the equation is shown to have a family of solutions describing stable stationary nonlinear waves modulated spatially. In the dispersionless case (GLE proper) all the stationary-wave solutions are unstable provided the frequency takes different values at x=+∞. Numerical calculations reveal in this case complex (apparently chaotic) behavior. Comparison is made with a similar problem arising in the theory of superconducting links.

A method for finding the state distribution in a physical system is presented, on the basis of a detailed analysis of the experimental response of the system to an external probe. As the study of the signals with which the experimenter probes the system of interest necessarily consists of a finite number of measurements, the maximum entropy principle is employed in order to reconstruct the state distribution on the basis of partial information. The formalism is illustrated by recourse to an elementary problem. A rather complicated x-ray diffraction problem is also discussed as it neatly depicts the power of this new approach.

The multiconfiguration time-dependent Hartree-Fock method is applied to compute the hyperpolarizabiltiy of the neon atom and its frequency dependence. For the static limit γ=γ(0;0,0,0) we obtain γ=86.5 a.u. This value agrees very well with a recent experimental value of 75 a.u. <γ<93 a.u. obtained by Shelton and Lu [Phys. Rev. A 37, 3813 (1988)]. The results for the Kerr hyperpolarizability γ(ω;0,0,ω)=${\gamma }^{\mathrm{Kerr}}$ provide an estimate of the dispersion effects. Approximating our results with the formula ${\gamma }^{\mathrm{Kerr}=\gamma (1+\mathrm{A}{\omega }_L^2}$), where for the Kerr effect ${\omega }_L^2$=2${\omega }^2$, we find A≊0.50×${10}^{\mathrm{-}10}$ ${\mathrm{cm}}^2$.

The low-density, collision-induced, depolarized, translational Raman spectrum of gaseous parahydrogen has been measured at 36 K and compared to quantum line-shape calculations. The calculations were made in the isotropic potential approximation using a modified dipole–induced-dipole model of interaction.

We have obtained electron affinities (shown below in parentheses) of the 11 lowest bound excited configurations for ${\mathrm{Cr}}^{\mathrm{-}}$, ${\mathrm{Fe}}^{\mathrm{-}}$, ${\mathrm{Co}}^{\mathrm{-}}$, and ${\mathrm{Ni}}^{\mathrm{-}}$; viz., ${\mathrm{Cr}}^{\mathrm{-}}$ 3$d^4$4s4${\mathrm{p}}^2$ ${}_{}{}^8{}_{}{}^{}\mathrm{P}_{5/2}^{}{}_{}{}^{}$ (0.718 eV), ${\mathrm{Cr}}^{\mathrm{-}}$ 3$d^4$4s4$p^2$ ${}_{}{}^8{}_{}{}^{}F_{1/2}^{}{}_{}{}^{}$ (0.629 eV), ${\mathrm{Fe}}^{\mathrm{-}}$ 3$d^6$4s4$p^2$ ${}_{}{}^8{}_{}{}^{}\mathrm{P}_{9/2}^{}{}_{}{}^{}$ (0.313 eV), ${\mathrm{Fe}}^{\mathrm{-}}$ 3$d^6$4s4$p^2$ ${}_{}{}^8{}_{}{}^{}F_{13/2}^{}{}_{}{}^{}$ (0.106 eV), ${\mathrm{Fe}}^{\mathrm{-}}$ 3$d^5$4s4$p^3$ ${}_{}{}^{10}{}_{}{}^{}\mathrm{S}_{9/2}^{}{}_{}{}^{}$ (1.271 eV), ${\mathrm{Co}}^{\mathrm{-}}$ 3$d^6$4s4$p^3$ ${}_{}{}^9{}_{}{}^{}D_6^{}{}_{}{}^{}$ (0.888 eV), ${\mathrm{Co}}^{\mathrm{-}}$ 3$d^7$${(}^3$F)4s4$p^2$ ${}_{}{}^7{}_{}{}^{}\mathrm{D}_5^{}{}_{}{}^{}$ (0.197 eV), ${\mathrm{Co}}^{\mathrm{-}}$ 3$d^7$4s4$p^2$ ${}_{}{}^7{}_{}{}^{}G_7^{}{}_{}{}^{}$ (0.126 eV), ${\mathrm{Ni}}^{\mathrm{-}}$ 3$d^8$4s4$p^2$ ${}_{}{}^6{}_{}{}^{}\mathrm{F}_{11/2}^{}{}_{}{}^{}$ (0.206 eV), ${\mathrm{Ni}}^{\mathrm{-}}$ 3$d^8$4s4$p^2$ ${}_{}{}^6{}_{}{}^{}G_{13/2}^{}{}_{}{}^{}$ (0.272 eV), and ${\mathrm{Ni}}^{\mathrm{-}}$ 3$d^7$4s4$p^3$ ${}_{}{}^8{}_{}{}^{}F_{13/2}^{}{}_{}{}^{}$ (1.044 eV).

Elastic scattering of electrons from hydrogen in the energy range 30–400 eV is studied. The calculations are performed using an optical potential to improve the close-coupling approximation. The calculation at 30 eV employs an 18-state basis set emphasizing long-range pseudostates, while at 100 and 400 eV an 11-state basis containing more compact pseudostates is found to be more suitable. Differential elastic cross sections are presented and compared with experiments. Integrated elastic and total inelastic cross sections are also reported and compared with values estimated semiempirically.

Absolute cross sections for the emission of Lyman-α radiation have been measured for H+${\mathrm{H}}_2$ and ${\mathrm{H}}^{+}$+${\mathrm{H}}_2$ collisions. The projectile energies ranged from 2.5 to 0.05 keV for H projectiles, and from 2.0 to 0.08 keV for ${\mathrm{H}}^{+}$ projectiles. The polarizations of the emitted Lyman-α radiation were also measured. Arguments are made to suggest that most of the observed Lyman-α emission from these reactions results from production of fast excited hydrogen atoms during the collisions, as opposed to dissociative excitation of the target ${\mathrm{H}}_2$ molecules.

The extreme ultraviolet spectrum of foil-excited Li has been recorded at ion energies 100–300 keV with higher spectral resolution than before. Previously assigned decays of core-excited three-electron doublet states (some of which autoionize rapidly) are confirmed with higher wavelength precision.

We present photon-counting distributions of multimode squeezed light of bandwidth B. Such light, for example, is generated by the intracavity parametric down-converter of Wu et al. [Phys. Rev. Lett. 57, 2520 (1986)], which has now become a standard source for generating highly squeezed light. By expanding the electric-field operator into relevant modes over the counting interval T and applying the Bogoliubov transformation that generates squeezing, we calculate the photon-counting distributions under both BT≫1 and BT≪1 conditions. We find that the probability for counting an odd number of photons is identically zero under the former condition, whereas the latter is equivalent to having a photodetector with quantum efficiency that is much less than unity.