We consider the kinetics of a system where n distinct chemical species $A_i$ undergo the reactions $A_i$+$A_j$→inert (i≠j). In the rate-equation approximation, a conservation law for the reaction is derived, and an explicit solution for the case n=3 is given. At the level of the master equation, Van Kampen’s Ω expansion is employed to estimate the magnitude of local fluctuations in density which arise when the basic dynamical variable of the system is the (discrete) particle number, rather than a continuous particle density. We then develop a physical picture for the evolution of the n-species system, when the particles diffuse, which, together with the estimate of the magnitude of the local fluctuations, is used to deduce the form of the decay law when the initial densities of the n species are equal. For a d-dimensional n-species system below an upper critical dimension equal to 4(n-1)/(2n-3), the density is predicted to decay as $t^{\mathrm{-}\alpha \left(n\right)}$, with α(n)=1/2d{1-1/[2(n-1)]}, and this is verified by numerical simulations. In addition, the simulations reveal a striking symmetry breaking, where the equal-density initial state evolves to a long-time state where one species predominates.

It is shown that in standard perturbation theory the evaluation of the acceleration autocorrelation function can be used to get expressions for the memory function of the velocity autocorrelation function and for the diffusion constant. Applying this formalism, the inverse of the single-soliton diffusion constant D in any nonlinear Klein-Gordon theory is obtained as a power series in the temperature T and exponentials exp(-ħ${\omega }_j$/kT), where ${\omega }_0$ to ${\omega }_n$ are the characteristic phonon frequencies. This power series reduces in the classical limit to an expansion in simple powers of T. In particular D diverges at low temperatures at least as 1/T in the classical case and at least as $T^2$exp(ħ${\omega }_0$/kT) in the quantum case, where ${\omega }_0$ is the lower bound of the phonon frequency continuum. Application is made to the ${\phi }^4$ theory.

A procedure analogous to the Bogolyubov-Tyablikov approximation is proposed in order to bridge between the continuum theory and the Maier-Saupe model of nematic liquid crystals. The equation obtained for the order parameter, which results in a first-order transition, reduces to the Maier-Saupe equation in the limit of infinite-range interactions. A prescription for the evaluation of the free energy in the framework of the Bogolyubov-Tyablikov treatment is presented, and the resulting Maxwell construction is used to determine the first-order transition temperature. The transition temperature, the range of metastability, and the value of the order parameter at the transition are all considerably lower than in Maier-Saupe theory.

High-pressure volumetric measurements were used to investigate the importance of short-range smectic fluctuations in the nematic-isotropic transition. Three members of the homologous series of di-alkylazoxybenzenes were chosen as samples, since the nematic range varies considerably in these compounds. The investigation was aided by the discovery of a smectic-A–nematic–isotropic triple point at elevated pressure in the ninth member of the series. Theoretical work on pretransitional effects in the isotropic phase is not very useful in describing the effect of large smectic fluctuations on the nematic-isotropic transition itself. However, one recent theoretical proposal argues that the volume change at this transition should increase due to the presence of short-range smectic order, and our experimental results confirm this prediction.

The Runge-Gross version of the Hohenberg-Kohn-Sham density-functional theory is generalized to multicomponent systems.

The collisional relaxation of a bi-Maxwellian and a double-beam distribution is studied in order to assess the reliability of the Bhatnagar-Gross-Krook (BGK) operator in comparison with the full Fokker-Planck Coulomb collision operator. If a velocity-dependent relaxation rate is employed, qualitative agreement can be achieved between both models. Using the friction rate ${\nu }_F$ in the BGK model yields numerical results most similar to those of the exact Fokker-Planck model. However, in quantitative terms the collisional relaxation of higher moments is only poorly described. In particular, for the skewness or heat flux the numerical differences are intolerable. For this and other reasons detailed in the paper, the BGK model cannot serve as a reliable calculation of transport properties. Nevertheless, it may have merits in qualitatively assessing collisional effects in inhomogeneous systems which otherwise have to be treated as entirely collision-free.

The self-avoiding walk in the three-dimensional space with a topological obstacle—an infinite rod—is studied with the aid of a renormalization-group approach. Specifically, the mean winding number of the self-avoiding chain around the rod with both its ends fixed in space is calculated. The main interest of the problem is, however, a methodological one. Since the winding number is well defined only for no more than three dimensions, the ε-expansion method, so successful in the study of the self-avoiding chain, cannot be utilized. Instead, a variation of the method, the homotopy parameter expansion, is applied to the problem. This gives a nontrivial illustration of the method. The result suggests that the overall shape of the self-avoiding chain is less spherical than that for the simple random walk. This seems to be in conformity with the existing Monte Carlo result.

The theory of curvature elasticity in nematics described in earlier papers is reformulated and simplified. The extent to which the ‘‘splay,’’ twist, and ‘‘bend’’ constants ($K_1$,$K_2$,$K_3$) of nematic liquid crystals differ from one another and the way they depend upon the degree of alignment (as characterized by the nematic order parameters P${¯}_n$) are determined by the structure of the medium which in turn is characterized by the correlation functions. The direct correlation functions of the isotropic liquid in terms of which the elastic constants are expressed have broad functional similarities for systems with rather different intermolecular potentials. An approximate scheme is proposed to estimate the values of the direct pair correlation function. Numerical results for the elastic constants are presented for a range of hard-core sizes and shapes and dispersional strengths and anisotropies. It is found that the effect of the long-range dispersion interactions on the elastic constants is determined not only by the angle-dependent part of the forces, but, and more significantly, by the coupling between the isotropic part and anisotropic hard-core repulsions through the pair correlation function. The attractive interactions are found to make significant contributions to $K_i$ in ordinary nematics (nematics of rod-shaped molecules) when $x_0$ (length-to-width ratio) is small (≲3.0) and in discotic nematics when 1/$x_0$ is large (i.e., ≳3.0). Qualitative features exhibited by our calculation for both the ordinary and the discotic nematics are in agreement with the experiment.

The previously derived method of l-projected Hylleraas basis functions [for calculating the individual partial waves $R_l$($r_1$,$r_2$)$P_l$(costheta) of an atomic pair function of first perturbation order in 1/Z] is analyzed again and improved. It suffices to project, onto $P_l$(costheta), a subset of Hylleraas functions (the powers of u=$r_{12}$ and s=$r_1$+$r_2$ alone). The l-separated Hylleras functions give rapid convergence since they possess a similar discontinuity at $r_1$=$r_2$ as the exact partial wave [C. Schwartz, Phys. Rev. 126, 1015 (1962)]. The handling of the l projections is easy. The method is applied to the 1$s^2$ pair to obtain, at relatively low basis-set dimensions and without using any nonlinear parameter, the heretofore most accurate values of the partial-wave energies $E_l^2$ (l=0,1,2,. . .) and of the entire second-order energy $E_{\mathrm{He}{}^2}$.

Layering transitions of colloidal crystals confined between two smooth glass surfaces have been studied by transmission diffraction of light as well as direct-lattice imaging with an optical microscope. The polystyrene spheres we studied have diameter Φ=0.305 μm and a surface charge of ∼${10}^4$ electronic charges, and form three-dimensional (3D) colloidal crystals in completely deionized water at a volume concentration above ≊0.3%. A 3D suspension of these spheres in completely deionized water forms both body-centered-cubic (bcc) and face-centered-cubic (fcc) crystalline structures as a function of colloid density n≡(1/$a_s$${)}^3$, with lattice constants ranging between 0.7 and 1.5 μm. When the colloid is confined between glass plates separated by distances D∼0.5–1 μm, the spheres form a single-layer 2D fluid. For D near 1 μm, a transition occurs to a single-layer 2D hexagonal crystal. As D increases, the evolution from two- to three-dimensional crystals is observed as a series of structural transitions distinguished by the number of crystal planes between the plates and by the preferred crystal symmetry parallel to the glass boundaries. We present here a study of colloidal crystals confined in a wedge cell that allows diffraction and imaging from the same crystallite. We present diffraction and imaging measurements of structural phases of one- through seven-layer colloidal crystals confined between two smooth glass surfaces, for a range of densities such that 2<$a_s$/Φ<6. The sequence of structural phases we observe for this range of densities for clean samples is similar but not identical to that observed for $a_s$/Φ<2 by Pieranski, Strzlecki, and Pansu, which was modeled as a hard-sphere system. We also observe differences between the clean thin crystalline phases and ‘‘dirtier’’ thin phases to which a stray electrolyte has been added. The use of both diffraction and imaging was found necessary to fully characterize the system.

The symbolic dynamics of the circle map $f_{\lambda }$(x)=x+λ sin(2πx)+a (mod 1) is analyzed with use of the symmetries of the lines a=0 and a=(1/2), obtaining, by a projection method, factorization of the Artin-Mazur ζ function. The Milnor-Thurston kneading theory is shown to be of interest for both symmetric and nonsymmetric circle maps, where an obvious generalization is shown to be valid and certain calculations simplify the relationship between the Artin-Mazur ζ function and the itinerary.

Response of the basic period-doubling mode of the dissipative Fermi-Ulam model to a periodic perturbation is studied both experimentally and theoretically. An intuitively obvious explanation of the noisy-precursors phenomenon is presented. Possible practical implications of the resonant-response phenomenon are discussed. Connections with the virtual Hopf phenomenon predicted by Wiesenfeld are indicated.

When a quantum system has a chaotic classical analog, its matrix elements in the energy representation are closely related to various microcanonical averages of the classical system. The diagonal matrix elements cluster around the classical expectation values, with fluctuations similar to the values of the off-diagonal matrix elements. The latter in turn are related to the classical autocorrelations. These results imply that quantum perturbation theory must fail, for chaotic systems, in the semiclassical limit ħ→0: Two arbitrarily close Hamiltonians have, in general, completely different sets of eigenvectors.

In contrast to the H theorem in information theory, proved by Guiaşu and Watanabe, a new theorem is proposed in this paper and called the I theorem. The I theorem can be used to determine of which type is the ergodic Markov process characterized by the decrease of entropy; this is useful for the quantitative study of living systems.

We have measured, as a function of temperature and frequency, the velocity and attenuation of ultrasound near the glass transition of glycerol. The data are compared with that from frequency-dependent specific-heat measurements recently performed on the same sample. When the difference between adiabatic and isothermal processes is taken into account, we find that the relaxation time associated with the ultrasonic measurements is the same as that responsible for the dispersion seen in the specific-heat experiment. We also compare our results with recent hydrodynamic theories of the glass transition. The relaxation time that we measure can be fitted equally well either by a scaling form τ=${\tau }_0$[(T-$T_h$)/$T_h$${]}^{\mathrm{-}\alpha }$, as these theories predict, or by the Vogel-Tamman-Fulcher law τ=${\tau }_0$exp[E/$k_B$(T-$T_0$)], which has often been used to fit relaxation behavior in glasses. In the scaling-law fit, α=12.5, which is unexpectedly large. The recent theory of Marchetti, which includes the wave-vector dependence of the mode-coupling vertex, provides a good fit to the frequency dependence of the data at constant temperature.

The two-photon annihilation rate for positronium hydride is calculated with the use of extensive Hylleraas-type wave functions in which all six interparticle coordinates are used. The binding energy against dissociation into a positronium atom and a hydrogen atom is improved upon the previous best value by about 3.8%. The two-photon annihilation rate, calculated with a wave function that gives a positron-electron cusp value of -0.492$a_0^{\mathrm{-}1}$ (compared with the exact value of -0.5$a_0^{\mathrm{-}1}$), is determined as 2.459 ${\mathrm{nsec}}^{\mathrm{-}1}$.

The retarded nonrelativistic Schrödinger Green’s function for an electron in crossed electric and magnetic fields is derived here in closed form. The electric and magnetic fields are taken to be spatially uniform but the electric field may have arbitrary time dependence and arbitrary orientation with respect to the constant magnetic field.

By further developing an idea found in Goldstein’s book Classical Mechanics together with a trivial generalization of the virial theorem, an alternative and simple perturbative procedure for obtaining H=H(J) for certain one-dimensional potentials is presented. The anharmonic oscillator and the Stark effect for hydrogen, both in one dimension, are given as examples of the method.

The frequency-dependent dipole polarizabilities ${\alpha }_d$(ω) of the sodium isoelectronic sequence (Z=11,...,18) have been calculated within linear-response theory using the time-dependent coupled Hartree-Fock scheme. The excitation energies for the transitions 3 ${\mathrm{}}^2$S→n ${\mathrm{}}^2$P (n=3,...,7) have been estimated from the position of the poles of a frequency-dependent functional, and analytic representations of the excited Rydberg p-state wave functions have been obtained. The extrapolated polarizability values ${\alpha }_d$(ω${)}_{\omega \to 0}$ are compared with the available static data. The transition energies compared well with spectroscopic data. The oscillator strengths have been estimated in length form and the quantum defect values have been calculated using complete screening. They all compare favorably with data currently available.

Our previous calculations for the energy splittings of the states 2$s_{1/2}$-2$p_{1/2}$ and 2$s_{1/2}$-2$p_{3/2}$ for the muonic ions ${\mu }^{\mathrm{-}}$-Li, ${\mu }^{\mathrm{-}}$-Be, and ${\mu }^{\mathrm{-}}$-B are extended to include hyperfine-structure corrections. The results show that there is a rich spectrum of well-resolved hyperfine transitions lying throughout the visible and infrared parts of the spectrum. A measurement of the transition frequencies would provide a precise determination of the nuclear radius and the hyperfine-structure coupling constants.