We investigate the statistical mechanics of a set of randomly cross-linked macromolecules. Using the replica method, we are led to consider a field theory for an order parameter which describes the breaking of the symmetry of independent translations and rotations between replicas. We construct the Landau free energy in this order parameter by systematically integrating out the polymer degrees of freedom. We find that the system can exhibit two transitions. The first corresponds to the collapse of a swollen rubber, while the second is a continuous transition to an equilibrium amorphous solid once the number of crosslinks exceeds a critical value. Very close to the latter transition, the Landau free energy is of the same form as that for the long-range Ising spin glass, enabling us to use Parisi’s replica–symmetry-breaking scheme to describe the ordered state.

A model dynamical system is described which generates spatial patterns based only on connectivity and screening. The growth probability at sites on the pattern is determined by the extent to which the sites are ‘‘screened’’ by points on the exterior of the pattern. The scaling properties were found to depend on the strength of the imposed screening constraint. Information theory was used to measure the rate of entropy production of the different patterns. A critical point in the screening constraint was observed where the pattern geometry changed from two dimensional (compact) to one dimensional. At this critical point, random fractals are produced and the rate of entropy production is a maximum. Analogies are drawn between the patterns produced by our model system, and similar patterns generated by diffusive processes.

A triple expansion of the scattering kernel in terms of Legendre polynomials is used to transform the nonlinear Boltzmann equation into a system of moment equations. The expansion is based on a vector representation of the scattering kernel in the laboratory system, which can be applied to arbitrary collision laws. Low-order approximations result in a remarkable simplification of the moment equations. The rate of convergence of the scattering-kernel expansion is investigated for isotropic scattering in the center-of-mass system. A comparison between the differential scattering and deflecting rate and its approximations define useful truncation indices for practical application.

Electric-field-induced molecular reorientation is studied in the magnetically biased ferronematic liquid-crystal (FNLC) film. A theory derived from the free energy of the system is given. To investigate the field-induced distortion, we have measured the optical birefringence which depends on the integration of the director orientation. In our experiment, the electric-field-induced molecular reorientation of FNLC biased at different magnetic fields has been obtained. The electric-field-induced molecular reorientation of FNLC films biased at fixing magnetic field with different thicknesses has also been measured. Furthermore, hysteresis phenomena have been observed. At higher magnetic field intensities, it seems that the magnetic particles flocculate into large clumps.

A general theory of the one-variable Fokker-Planck equation is presented. The emphasis is placed on the calculation of the Green function, from which the nonstationary solution with arbitrary initial condition can be obtained. The frequency component of the Green function is expressed in a simple form. The behavior of the time-dependent Green function is studied on the basis of this expression. The short-time behavior is expressed as a power series in terms of time. The long-time behavior is also explored. The eigenvalue problem and other properties of the Fokker-Planck equation are discussed as well.

For a two-level system coupled stochastically to a bath with dichotomic colored noise the relaxation behavior is investigated. An exact numerical calculation shows that it can be described by two (longitudinal and transverse) relaxation rates only when correlations of the stochastic process decay rapidly enough as compared to the amplitude of the fluctuations. The numerical evaluation of the model also shows that the relation $T_2^{\mathrm{-}1}$≥0.5$T_1^{\mathrm{-}1}$, well known from textbooks, is generally not satisfied, in agreement with a recent fourth-order perturbative calculation. Finally, it is shown that this relaxation model for a δ-correlated bath is mathematically equivalent to the Haken-Strobl-Reineker model for the coupled coherent and incoherent motion of a quantum particle in the special case of transport in a dimer.

Long-range interatomic interactions between distant parts of a DNA helix are known to be dominant factors in determining longitudinal DNA motions. A local field approach has been developed to account for these long-range forces that both simplifies the numerical complications involved in the previous treatment of these interactions and rationalizes the physics. The method leads to excellent agreement with the experimental value for the speed of longitudinal sound (≊2.0 km/s), where agreement was the chief virtue of the earlier method. A new result of this theory is the prediction of one-dimensional plasmon excitations, analogous to plasmons in higher-dimensional charged systems. This one-dimensional plasmon has an electromagnetic character in contrast to the mostly mechanical nature of the slower compressional sound-wave phonon modes. The dispersion for this plasmon mode is soundlike, having a speed of propagation of some 36.0 km/s. Analysis of the effect of viscous frictional forces shows that the one-dimensional plasmon waves are well-defined resonant states for parameter values where the mechanical phonon modes are strongly overdamped.

The critical heat capacity associated with the nematic (N) to smectic-A (Sm-A) phase transition has been studied using high-resolution ac calorimetry in two mixtures of polar octyloxycyanobiphenyl (8OCB) and nonpolar terephtal-bis-butylaniline (TBBA). The slope ${\mathrm{dT}}_c$/dX of the N–Sm-A phase boundary, where X is the mole fraction of TBBA, is large enough to cause Fisher renormalization of the critical exponent α. A sample with X=0.15 exhibited a first-order transition, and a sample with X=0.088 was very close to tricritical. In the latter case, the renormalized heat-capacity exponent ${\alpha }_R$ was -0.90. Since ${\alpha }_R$=-α/(1-α), this corresponded to α=0.47, which agrees well with the tricritical value α=0.5.

The problem of relaxation is studied via the microscopic Hamiltonian model of an impurity (or particle of interest) embedded in a linear chain of harmonic oscillators. When the mass of the particle of interest is sufficiently larger than that of the ‘‘bath’’ particle and the system is classical, the velocity autocorrelation function of the particle of interest is known to consist of the sum of an exponentially decaying term and a nonexponential contribution with a slow tail of oscillatory nature. The damping of the exponential decay is determined by using a renormalization procedure within the context of the generalized Langevin equation. By expressing the ‘‘bath’’ coordinates in terms of normal modes and using a scalar product of the Kubo type, it is shown that the classical Liouvillian becomes formally equivalent to the quantum-mechanical Hamiltonian introduced by Friedrichs to study unstable quantum-mechanical states. In the case of a finite number N of particles (or 2N normal modes) the excited state of the Friedrichs model largely overlaps an ‘‘eigenstate’’ ‖S〉, the ‘‘eigenvalue’’ of which is straightforwardly expressed in terms of the model parameters. It is shown that in the continuum limit (N=∞) this ‘‘eigenvalue’’ becomes complex and its imaginary part coincides with the renormalized damping above. It is also shown that the projection approach to the Fokker-Planck equation leads precisely to the same renormalized damping coefficient.

The major conclusion is that the Fokker-Planck description refers to the stochastic dynamics of a sort of dressed variable rather than that of the merely bare velocity. Furthermore, we conclude that whereas the decay of the autocorrelation function of the bare velocity cannot be exponential in accordance with the general remarks by Lee [Phys. Rev. Lett. 51, 1227 (1983)], the decay of the dressed variable is an exact exponential. It is argued that to establish a contact with real experiments, the Kubo approach to the linear response to an external excitation should be reformulated and that this might naturally lead to the basic tenets of the subdynamics of Prigogine and co-workers. The relations between the present and a former approach to the Fokker-Planck equation to be associated to non-Markovian processes are studied.

There are many definitions of the fractal dimension of an object, including box dimension, Bouligand-Minkowski dimension, and intersection dimension. Although they are all equivalent in the continuous domain, they differ substantially when discretized and applied to digitized data. We show that the standard implementations of these definitions on self-affine curves with known fractal dimension (Weierstrass-Mandelbrot, Kiesswetter, fractional Brownian motion) yield results with significant errors. An analysis of the source of these errors leads to a new algorithm in one dimension, called the variation method, which yields accurate results. The variation method uses the notion of ε oscillation to measure the amplitude of the one-dimensional function in an ε neighborhood. The order of growth of the integral of the ε oscillation (called the ε variation), as ε tends toward zero, is directly related to the fractal dimension. In this paper, we present the variation method for one-dimensional (1D) profiles and show that, in the limit, it is equivalent to the classical box-counting method. The result is an algorithm for reliably estimating the fractal dimension of 1D profiles; i.e., graphs of functions of a single variable. The algorithm is tested on profiles with known fractal dimension.

By considering the Frank free energy, we show that the electric-field-induced twist and bend Freedericksz transitions can be first order. This agrees with the prediction of Arakelyan, Karayan, and Chilingaryan [Sov. Phys.–Dokl. 29, 202 (1984)] from Landau theory for the bend geometry. We have carried out dielectric measurements on the liquid-crystal 5CB (4-cyano-4’-n-pentylbiphenyl) that provide the first experimental evidence that these transitions are first order. The effects of a competing magnetic field on the threshold voltage and on the width of the transition region are compared with theory. In the presence of a competing magnetic field, the bend transition in 5CB is to a modulated rather than a uniform phase.

Recently DePristo and Kress [Phys. Rev. A 35, 438 (1987)] have presented a Padé approximation formula for the kinetic-energy functional of a many-electron system, such as an atom. This functional has been adopted in the present work in variational density-functional calculations of the total atomic binding energy, using the Ne atom as an example. For this atom similar calculations have been done earlier by the author using the sum of the first two terms of the gradient expansion of the kinetic-energy functional. The total-energy functional used in the present work contains the kinetic-energy term, the term describing the interaction of the electrons with the atomic nucleus, the term describing the interaction among the electrons, and the exchange-energy term. The electron (number) density of the Ne atom is modeled by using hydrogenlike one-electron wave functions (with the 2s function orthogonalized to the 1s function) containing three variational parameters. The result of the present calculations is compared with the previous calculations of the author and it is concluded that the DePristo-Kress Padé approximation formula for the kinetic-energy functional gives a better agreement with the Hartree-Fock kinetic energy for the Ne atom than the sum of the first two terms of the gradient expansion of the kinetic-energy functional.

We predict the ${}_{}{}^1{}_{}{}^{}\mathrm{S}$ two-electron ionization ladder (TEIL) of ${\mathrm{Li}}^{\mathrm{-}}$, for n=3–9. This special class of resonances can be observed via electron-lithium scattering experiments. In conjunction with our previous work on such doubly excited states, we conclude that the TEIL spectrum as n→∞ is determined essentially by the rigid-rotor correlated motion of the two valence electrons. It appears that no core-induced, quantum-defect concept is necessary to explain the spectrum.

The determination of cross sections for the relaxation of atomic multipole moments from the intensities of the components of a resolved Zeeman fluorescence spectrum may be carried out using the density-matrix formalism. However, the spherical-tensor basis is not well adapted to state-selected measurements and can allow a significant propagation of error. We can reduce the error by determining the Q(j,m→j’,m’) cross sections directly from the data taken at the lowest pressures. A comparison is given between recent experimental and theoretical relaxation cross sections for collisions of 5 ${}_{}{}^2{}_{}{}^{}P$ potassium atoms with ground-state He, Ne, and Ar atoms.

The first-order capture theory satisfying Coulomb boundary conditions has been extended to describe electron screening for a multielectron target in a consistent and realistic manner. The results of explicit calculations for p+He, C, and for ${\mathrm{He}}^{2+}$+Li are presented and are compared with those of standard Coulombic screening procedures.

We have measured the electron energy spectra near 0° produced in collisions of ${\mathrm{D}}^{\mathrm{-}}$ with Ar. Using a 400-keV ${\mathrm{D}}^{\mathrm{-}}$ beam and with good experimental energy and angular resolution we have found structure in the ejected electron energy spectra which is due to the decay of the ${}_{}{}^1{}_{}{}^{}\mathrm{P}$ ° shape resonance. The doubly differential cross sections (DDCS’s) have been measured as a function of angle and it was found that this structure disappeared for laboratory angles greater than 1° as expected. A resonance contribution to the DDCS’s was extracted at ${\theta }_L$=0°, transformed to the projectile frame, and fit with a Breit-Wigner shape. Our resonant energy is in reasonable agreement with other experiments. We also find a small asymmetry in the two resonant structures in the laboratory measurements at ${\theta }_L$=0°.

Elastic differential, integral, and momentum-transfer cross sections are calculated for the e-${\mathrm{NH}}_3$ system in the energy range of 0.1–1.0 keV. A parameter-free spherical optical potential is constructed from near-Hartree-Fock one-center expansion ammonia wave functions. The optical potential is then treated in a partial-wave analysis to extract various cross sections. Several versions of parameter-free polarization and exchange potentials are employed. The calculated differential cross sections are compared with the available experimental data.

The intensity ratio of L${\beta }_2$+L${\beta }_5$+L${\beta }_6$ to L${\alpha }_1$+L${\alpha }_2$ was measured in seven elements (Z≥70) through a selective excitation of the $L_{\mathrm{III}}$ subshell using an x-ray tube. A high-resolution Si(Li) detector system coupled to a computer controlled multichannel analyzer was used in these measurements. The results show that the relative transition probabilities Lβ and Lα agree with the relativistic Hartree-Fock-Slater calculations in the case of Yb and U, while a discrepancy of 3.1–8.7 % between experiment and theory is found for the other elements.

We describe a very efficient basis-set method for calculating cross sections for high-order above-threshold ionization of a hydrogenic atom in perturbation theory. We present results for up to 20-photon ionization of H(1s) and D(1s) by 1064 nm circularly, linearly, or elliptically polarized radiation. We show the rapid decrease in the integrated cross sections as the polarization passes from linear to circular, and we illustrate the asymmetry in the angular distributions for elliptic polarization.

We present perturbational formulas necessary to calculate the amplitude of the rotating nuclear magnetization and its angular frequency in a ruby NMR laser with an injected signal exhibiting subcritical Hopf bifurcation [R. Holzner, B. Derighetti, M. Ravani, and E. Brun, Phys. Rev. A 36, 1280 (1987)]. These formulas apply, in general, to any system reducible to a two-dimensional system with quadratic nonlinearity through application of the adiabatic elimination technique. For this laser model, in particular, we find that near the bifurcation point, the magnetization exhibits a hard transition without bistability or hysteresis.