We analyze convection in a rectangular box where two ‘‘substances,’’ such as temperature and a solute, are diffusing. The solutions of the Boussinesq theory depend on the thermal and solute Rayleigh numbers $R_T$ and $R_s$, respectively, in addition to other geometrical and fluid parameters. As $R_T$ is increased, the conduction state becomes linearly unstable with respect to steady (periodic) convection states if $R_s$${¯}_s$ (>R${¯}_s$). The critical value R${¯}_s$ is characterized by the frequency ω=0 appearing as a root of algebraic multiplicity two and geometrical multiplicity one of the linearized stability theory. Asymptotic approximations of the solutions of the nonlinear theory are obtained for $R_s$ near R${¯}_s$ by the Poincaré-Lindstedt method. It is found that a periodic (steady-state) solution bifurcates supercritically (subcritically) from the conduction state at $R_T$=$R_c^p$ ($R_T^s$), where $R_c^p$<$R_T^s$. The periodic branch joins the steady-state branch with an ‘‘infinite-period bifurcation’’ at $R_T$=$R_b$, where $R_c^p$<$R_b$<$R_T^s$. The shape of the resulting bifurcation diagram suggests the term, λ bifurcation. The infinite-period bifurcation corresponds to a heteroclinic orbit in the appropriate amplitude-phase plane. The stabilities of the bifurcation states are determined by solving the convection initial-value problem using the multiscale method.

A discrete-ordinate method [J. Comput. Phys. 55, 313 (1984)] based on nonclassical polynomials is applied to the solution of a large class of Fokker-Planck equations with nonlinear coefficients. These Fokker-Planck equations arise in the description of nonequilibrium processes in reactive systems, laser systems, and model systems with bistable potentials. This subject has received considerable attention in recent years in connection with stochastic processes in physics, cooperative phenomena, and synergetics. The present approach is based on an eigenfunction expansion of the time-dependent probability density function. A discrete-ordinate method is employed in a numerical calculation of the eigenvalues and corresponding eigenfunctions of the Fokker-Planck operator. A general procedure for determining the eigenvalue spectrum of such Fokker-Planck operators with the discrete-ordinate method based on nonclassical polynomials, constructed so as to give rapid convergence of the eigenvalues, is described. The method is applied to several systems which include a model problem for which an analytic solution is known, a model with a triple-well potential in the Schrödinger equation equivalent to the Fokker-Planck equation, and to a model for the the trans-gauche isomerization of n-butane in carbon tetrachloride. The present methods for studying eigenvalue and boundary-value problems should be applicable to a wide variety of problems in addition to those presented here.

To characterize fractals, a function is introduced which is a natural extension of the fractal dimension. This fractal dimension function is easily evaluated numerically and is amenable to a theoretical description. In particular, an intimate connection between fractal dimension and statistics is uncovered. As illustrative examples the concept is applied to sequences of random numbers and to the energy levels of coupled harmonic oscillators.

The extraction of the Kolmogorov (metric) entropy from an experimental time signal is discussed. Theoretically we stress the concept of generators and that the existence of an expansive constant guarantees that a finite-time series would be sufficient for the calculation of the metric entropy. On the basis of the theory we attempt to propose optimal algorithms which are tested on a number of examples. The approach is applicable to both dissipative and conservative dynamical systems.

The Bogoliubov prescription relevant to the equilibration of a gas is reformulated to describe dense fluids. The revised description assumes that in the ‘‘kinetic state’’ of a dense fluid, multiparticle distribution functions are functionals of the one- and two-particle distribution functions. This principle is applied to the Bogoliubov-Born-Kirkwood-Green-Yvon (BBKGY) sequence and a closed kinetic equation for the radial distribution function, g(x→,p→,t), is obtained relevant to a homogeneous, anisotropic fluid, where x→ and p→ are relative two-particle displacement and momentum, respectively. In the equilibrium limit the kinetic equation reduces to a linear integro-differential equation. A closed solution to this equation is obtained in operational form which, in the limit of weak interactions, reduces to the canonical exponential form, and, with interactions turned off, gives the correct unit value of g. These equilibrium equations are applied to the specific configuration of a fluid whose particles interact under point repulsion and Newtonian attraction. Asymptotic expressions for the radial distribution function for large and small values of interparticle displacement give oscillatory decay to unity and vanishing decay to zero, respectively. These findings are consistent with previously described behavior of the radial distribution.

An efficient discrete-ordinate method of solution of the time-dependent Boltzmann equation is employed in the calculation of the zero-field electron mobility and diffusion coefficients for hot-electron thermalization in rare-gas moderators. The discrete-ordinate method is modified to permit a rescaling of the quadrature points. This procedure is somewhat analogous to the two-temperature-moment methods employed in the theoretical analysis of electron swarms. The time-dependent transport coefficients are given as a sum of exponential decay terms characterized by the discrete eigenvalues of the Lorentz-Fokker-Planck operator for elastic electron-atom collisions. For argon, krypton, and xenon, the time dependence is strongly influenced by the Ramsauer-Townsend minimum and leads to maxima in the transient mobility and diffusion coefficient. Helium and neon with hard-sphere-like cross sections exhibit transient mobilities which initially are below the thermal zero-field mobility and then increase to the thermal mobilities as the electron distribution approaches equilibrium. The transient mobility for cross sections with Ramsauer minima are sufficiently sensitive to the details of the cross sections such that it may be feasible to distinguish between different cross sections experimentally. The calculations also indicate that the transient mobility is insensitive to the initial distribution function. A nonequilibrium phenomenon not previously recognized is the possibility of a negative transient mobility which occurs provided that the momentum-transfer cross section increases sufficiently rapidly with energy.

The long-time behavior of the solutions of classical scalar electrodynamics in a cavity are investigated numerically. It is found that time averages do not agree with microcanonical averages, the disagreement increasing as one approaches the continuum at given total energy and coupling constant.

Extensive and high-quality quasi-elastic incoherent neutron scattering data were obtained for water in the temperature range extending from room temperature down to -20 °C in the supercooled state. The analysis generally confirms findings of our previous experiment [S. H. Chen, J. Teixeira, and R. Nicklow, Phys. Rev. A 26, 3477 (1982)], but in particular three new results have been obtained: (a) two relaxation times are clearly identified, which are related to the short-time and intermediate-time diffusion of water molecules, respectively, and their temperature dependence has been determined; (b) one of these relaxation times is associated with jump diffusion of the protons, and the temperature dependence of the jump length has been qualitatively determined; (c) the Q dependence of the scattering intensity integrated over the quasi-elastic region gives a Debye-Waller factor which is temperature independent.

The total binding energy of an atom in the limit of large Z is given by (9/14) (3/2${)}^{1/3}$${\mathrm{Z}}^{7/3}$ according to independent approximate calculations by Sucher and by Shakeshaft and Spruch. This formula is shown to follow from the assumption of a Coulombic potential for each electron. Comparing it with the exact Thomas-Fermi result shows that inclusion of the non-Coulombic potential lowers the total energy by 4%.

The magnitudes of avoided crossings caused by an isolated pseudoresonant perturbation wandering through a Rydberg series are calculated as a function of the width of the pseudoresonance. Very simple formulas are derived for the limits of large and of small widths; for intermediate widths these formulas are still accurate to within 10–15 %.

For an N-fermion system, we prove the upper bound for the kinetic energy T(ψ)≤[1+C(N/q${)}^{2/3}$] F $d^3$r[∇→ √ρ(r→) ${]}^2$, where ρ(r→) is the single-particle density, q the number of spin state, and C=15(4π${)}^{2(}$${3/5\left)\right((2/5\mathrm{}))}^2$ ${}_{}{}^{/3\mathrm{}}{}_{}{}^{}$ in three dimensions.

The dielectronic-recombination cross sections involving 1s electrons are estimated for the ${}_{23}{}^{}{}_{}{}^{}\mathrm{V}$ target ions along the isonuclear sequence. Initial ionization charge states of the He-, Li-, and Be-like ions are examined. For a beam profile of width Δ$e_c$≊60 Ry, two overlapping peaks represent transitions of the active electrons from 1s+$k_c$$l_c$→2s2p or 2$p^2$ and 2snl or 2pnl (n≥3). The peak ratios are compared with results from a recent experiment on resonant-transfer excitation for the ${}_{23}{}^{}{}_{}{}^{}$V${+\mathrm{}}_2$He system.

Atomic fluorescence excited by photodissociation of a homonuclear diatomic molecule exhibits interference between emission events with a photon arising from atoms at different positions. When only one of the atoms is excited, this interference does not show up in the noise spectrum of the fluorescence, in contrast to what has previously been suggested. We indicate how the interference could be observed, and we describe its characteristics.

An explicitly soluble model of macroscopic quantum fluctuations of Stokes pulses is presented. The model deals with a small sample placed in a cavity and covers both linear and nonlinear regimes. The energy distribution of pulses narrows in the nonlinear regime, which is in agreement with the recent experiments.

The dynamical evolution of the electromagnetic field amplitudes is numerically calculated using an exciton-biexciton model of CuCl. For a bistable system we determine the switch-down and switch-up times and find them to be well explained by the polariton escape time using the group velocity for the polaritons. Our switching times, about 80 psec at photon energies of 3177 meV, are at least an order of magnitude larger than a previous theoretical estimate and we predict the switching times will become longer as the exciton resonance is approached.

A generalization of the Baranger-Mozer formalism is presented for a low-frequency thermal electric-microfield distribution. Correlation, proportion, temperature, and charge effects are included in this model for a three-component plasma (one electronic and two ionic). Results are presented for various ion mixtures in a range of temperature ratios encountered in dense and hot plasmas of inertial-confinement fusion produced by laser.

An upper bound on the magnitude of the inertial-transfer term in the von Kármán–Howarth equation, and hence an inequality relation involving the longitudinal correlation function f, is derived here by employing the experimentally established property of local Gaussianity in fourth order, 〈[$u_1$(x→,t${\left)\right]}^4$〉=3〈[$u_1$ (x→,t${\left)\right]}^2$${〉}^2$, for the fluid velocity u→(x→,t) in isotropic homogeneous turbulence. It is shown that the Ling-Huang empirical forms for weak turbulence, 〈[$u_1$(x→,t${)}^2$〉=$a^2$$t^{\mathrm{-}2}$, f=[1+(10νt${)}^{\mathrm{-}1}$$r^2$${]}^{\mathrm{-}1}$ with a,ν const, are admissible dynamically for (νt${)}^{1/2}$≤12.14a as a consequence of the inequality relation, and thus the Ling-Huang forms cannot remain valid throughout the entire final period of decay.

Equilibrium polymerization systems in which directed rings are formed correspond to O(2) vector models of magnetism and ones in which undirected rings form correspond to O(1) models. In this paper it is shown that a system in which polymers tend to form directed rings but there is a finite free energy for misdirected bonds corresponds to an O(2) model with a uniaxial anisotropy which causes it to crossover to an O(1) model.

A pair of equations are derived which govern propagation of the shear Alfvén wave in a low-β (β≪$m_e$/$m_i$) magnetized plasma. It is found that the (electron-) inertia-dominated shear Alfvén wave can exist as a two-dimensional solitary vortex.

The principle of detailed balance leads to a correlation inequality for surface-scattering operators. The equilibrium energy accomodation coefficient α is expressed in terms of the correlation coefficient of the scattering probability and has an upper bound considerably less than 2.