Correlation functions of second-rank tensors for a system of freely rotating asymmetric-top molecules are given in the form of functional series. A rotation matrix is determined first with the help of a set of pseudoelliptic functions. These functions are expanded next into Fourier series. A reduced thermal average then gives the correlation functions in the form of series of integrals involving spherical-top correlation functions. The series are rapidly converging, and their Fourier transforms are presented. This development is exact and is well adapted to numerical treatment. It can be easily employed to study the extended rotational diffusion models for molecular liquids.

Thermodynamic properties, such as pressure entropy and configurational energy, of fluid ${\mathrm{H}}_2$ are evaluated where the first-order quantum correction is taken into account. Agreement for the pressure and entropy is found to be good at high temperature (T≥100 K).

We show that the mode locking in coupled circle maps with random phases is very different from that in a single circle map. A finite nonlinearity $K_c$ is needed for a step to appear. The width of the step behaves as (K-$K_c$${)}^2$. The complete mode locking (at K=1 for uncoupled maps) behaves singularly as the coupling is turned on. We argue that our model describes the mode locking in charge-density-wave materials. Our results are in qualitative agreement with the experimental observation by Sherwin and Zettl that only few true steps exist in the I-V characteristics and that in addition to these there are some ‘‘incomplete’’ steps.

We study growth oscillations in a class of deterministic models of growth which may be regarded as mean-field-like approximations to certain more realistic stochastic models. We demonstrate that the general morphology of the stochastic and deterministic models are similar. We then show that growth oscillations produce a rich multiperiodic structure in the deterministic models, as they do in the related stochastic systems. In both cases the multiperiodic oscillations are due to an induced incommensuration between a fundamental length scale (in this case the lattice spacing) and a dynamical length scale (in this case, the average distance grown per time step). We show how to estimate the effects of the growth oscillations and how to solve the deterministic equations analytically. This solution leads to a series of approximations, successive terms of which include the effects of longer-wavelength contributions to the multiperiodic growth oscillations. Finally, we present a sequence of approximations, the first one of which is the deterministic model studied here, which approach the related stochastic models.

The chaotic solution of dissipative scalar-delay-differential equations with a nonlinear feedback periodic with respect to its argument is shown to behave as a Gaussian-Markovian process in a large time scale. The short time scale is shown to be defined by the correlation time of the delayed feedback. The dimension of the chaotic attractor is shown to be approximately equal to the number of short times that are contained inside the delay.

We consider the problem of selection of symmetry-breaking states as the system evolves through the critical point at a rate γ in the presence of fluctuations and a small biasing interaction g. We show that due to the dynamics in the vicinity of the critical point the system becomes extremely sensitive to g as γ becomes smaller. Such sensitivity is a general phenomenon valid for a large class of symmetry-breaking transitions. Under an approximation that is valid over a wide range of conditions, we obtain analytic expressions for the branch-selection probabilities and show their validity by comparison with numerical simulations.

Using the newly developed incomplete enumeration method, the number $A_N$(C) and the mean-square radii of gyration 〈$R_N^2$(C)〉 of N-site lattice animals with cyclomatic number C have been studied up to N=70,C=7 on the square and N=30,C=5 on the simple cubic lattices. These are compared with extended exact series data up to N=17,C=4 on the square and N=12,C=4 on the simple cubic lattices. The result is consistent with ${\lambda }_C$=${\lambda }_0$, ${\mathrm{theta}}_C$=${\mathrm{theta}}_0$-C, and ${\nu }_C$=${\nu }_0$, for all C, where ${\lambda }_C$ is the growth parameter, and the exponents ${\mathrm{theta}}_C$ and ${\nu }_C$ characterize the number and the mean-square radii of gyration, respectively, of lattice animals with C cycles.

Hornreich et al. predicted that blue phases I, II, or III should undergo structural transformations into anisotropic uniaxial blue phases in electric or magnetic fields. It is shown here that, besides a hexagonal blue phase BP $H^{3\mathrm{D}}$ discovered previously, a novel blue phase, called BP X, can be obtained either from BP I for E∥[110] or from BP II for E∥[100]. Using the method of Kossel diagrams, clear-cut evidence is presented that blue phase X has tetragonal symmetry. The transformation of BP I into BP X involves a homogeneous strain in the plane perpendicular to E and a dilation in the field direction.

The positronium negative ion is studied using an adiabatic treatment in hyperspherical coordinates. The potential curves of five different symmetries, ${}_{}{}^1{}_{}{}^{}\mathrm{S}_{}^{\mathrm{e}}{}_{}{}^{}$, ${}_{}{}^1{}_{}{}^{}\mathrm{P}_{}^{\mathrm{o}}{}_{}{}^{}$, ${}_{}{}^3{}_{}{}^{}\mathrm{P}_{}^{\mathrm{o}}{}_{}{}^{}$, ${}_{}{}^3{}_{}{}^{}\mathrm{P}_{}^{\mathrm{e}}{}_{}{}^{}$, and ${}_{}{}^1{}_{}{}^{}\mathrm{D}_{}^{\mathrm{e}}{}_{}{}^{}$, are discussed along with the corresponding resonant states. A new method for calculating the body-frame probability density is used to study the different symmetries and to distinguish between them.

The dynamics of a model, which has recently been proposed to simulate parallel pumping experiments in ferromagnets, has been investigated. With increasing pumping amplitude the system displays periodic, quasiperiodic, and chaotic motion. In addition, the influence of noise at a particular transition point is discussed in connection with the experimental results.

The dissipation inherent in the time evolution of a composite system consisting of a number of subsystems is characterized. We present two thermodynamic quantities which serve as general lower bounds to the minimum entropy production in a composite system. These lower bounds are easier to compute than the minimum entropy production and the difference between the bounds and the minimum entropy production quantifies a mismatch in the coevolution of subsystems. This mismatch has important implications for process control and design.

The relation between the log-normal distribution and the 1/f noise is experimentally studied by use of cascade electrical networks. Results show that the 1/f spectrum can be obtained only in networks consisting of resistances and inductances, whereas the log-normal distribution for bifurcated currents is always accompanied for cascade networks in general. The log-normal distribution is observed even in the case where parts of branches of cascade networks are coupled to each other.

A neural network which is capable of recalling without errors any set of linearly independent patterns is studied. The network is based on a Hamiltonian version of the model of Personnaz et al. The energy of a state of N (±1) neurons is the square of the Euclidean distance—in phase space—between the state and the linear subspace spanned by the patterns. This energy corresponds to nonlocal updatings of the synapses in the learning mode. Results of the mean-field theory (MFT) of the system as well as computer simulations are presented. The stable and metastable states of the network are studied as a function of ‘‘temperature’’ T and α=p/N, where p is the number of embedded patterns. The maximum capacity of the network is α=1. For all α (0≤α<1) the embedded patterns are not only locally stable but are global minima of the energy. The patterns appear, as metastable states, below a temperature T=$T_M$(α). The temperature $T_M$(α) decreases to zero as α→1. The spurious states of the network are studied in detail in the case of random uncorrelated patterns. At finite p, they are identical to the mixture states of Hopfield’s model. At finite α, a spin-glass phase exists as a metastable state. According to the replica symmetric MFT the spin-glass state becomes degenerate with the patterns at α=${\alpha }_g$=1-2/π and disappears above it. Possible interpretations of this unusual result are discussed. The average radius of attraction R of the patterns has been determined by computer simulations, for sizes up to N=400. The value of R for 0<α<1 depends on the details of the dynamics. Results for both parallel and serial dynamics are presented. In both cases R is unity (the largest distance in phase space by definition) at α→0 and decreases monotonically to zero as α→1. Contrary to the MFT, simulations have not revealed, so far, any singularity in the properties of the spurious states at an intermediate value of α.

We develop a procedure based on the averaging method of Bogoliubov, Krylov, and Mitropolsky to obtain analytical approximate solutions of the equation for the longitudinal motion of a particle in a radio-frequency-quadrupole (RFQ) accelerator under the Kapchinskii-Teplyakov assumption. Our analytical results, which fairly agree (within a few per thousand) with those coming from the numerical integration of the equation under consideration, are exploited in the case of an RFQ device with N=150 cells in which the accelerated particles are protons and for values of the efficiency A from 0.05 to 0.5. Starting from an injection energy of 10 keV, a final energy of about 1.5 MeV is achieved. Our procedure may be applied to other fundamental problems arising in the project of an RFQ accelerator, such as the study of the equation for transversal oscillations.

The role of phase diffusion for quantum chaos in the quantum-mechanical model of the laser in the Haken limit is discussed. Fractal properties of the support of the asymptotic attracting probability distribution for the system are studied.

The equilibrium configuration of several models with long-range interaction is discussed by maximizing the entropy S with the fields constrained by the field equations. The calculations include a scalar field model (with Newtonian gravity as a special case) and general relativity, as well as a model in which a field is coupled to a nonconserved charge. The results illustrate the connection between temperature (particle number density) -dependent terms in the field equations and the appearance of nonuniform temperature (chemical potential) in equilibrium. The model with the nonconserved charge illustrates the likelihood of ‘‘miraculous’’ cancellations in phenomenological coarse-grain theories.

The equilibrium state of a system is determined by requiring the entropy to be maximum under an infinitesimal variation. Properly speaking, the variation of the thermodynamic variables of a system under a long-range force should be accompanied by a variation of the long-range force fields so as to maintain the field equations (self-consistent-field variation). These problems are, however, often treated by restricting the variations to a small region and ignoring the corresponding field variations (fixed-external-field variation). The equivalence between the two formulations is found to be nontrivial and sometimes only of a limited nature.

Relativistic contribution to the hyperfine structure of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ are reexamined in order to resolve inconsistencies in published values. The orbit-orbit and diamagnetic screening contributions are recomputed and are found to contribute less than one part per million (ppm), contrary to previous results. A new value (318 ppm compared to the perturbation result of 327 ppm) is obtained for the relativistic velocity correction using recently available relativistic Hartree-Fock wave functions. New values of the hyperfine-structure splitting of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ in the 1S2S state and the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ ion in the 1S and 2S states are presented. Comparison with experiment suggests that the relativistic velocity correction should be 323 ppm and the nuclear structure correction should be 184.2 ppm.

Anisotropic objects are known to acquire an induced moment in a direction different than that of the field, which leads to periodic motion under certain resonance conditions. The numerical solution of the nonlinear equation of motion which has the form of parametric resonance is performed. When the external field is increased the solution shows Feigenbaum frequency doubling and strange-attractor behavior.

An equation of motion for the reduced density matrix for the field in a single-mode laser is derived when the single- and two-atom interactions are included. An exact steady-state solution for the photon distribution function is presented and some aspects of the photon statistics are discussed.