Measurements of radial fingering patterns have been performed over a range of dimensionless force parameter 1.1≤C≤35. All observed flows show a power law dependence of radius of gyration on area with exponent (1/1.79). Local curvatures provide the best characterization of pattern ramification. The power spectra of these curvatures are used to define an average wave number K¯. A scaling is observed to collapse dimensionless average wave number (K¯’) versus dimensionless area of the mixing zone onto one function which rises early in the flows and then coarsens as the flows become more developed.

We investigate the role of plasma screening in fine-structure excitation of excited (n=2) hydrogenic ions, ${\mathrm{He}}^{+}$, ${\mathrm{Ne}}^{9+}$, ${\mathrm{Ar}}^{17+}$, and ${\mathrm{Fe}}^{25+}$ in collisions with protons. Within the context of the semiclassical, impact-parameter formalism, both ion-sphere and Debye-Hückel static screening models have been used. For certain plasma conditions, the former model is easily justified by arguments based on the ordering of various time scales; however, the latter model is more appropriate for many plasmas presently subject to spectroscopic analysis. Substantial (e.g., factor-of-10) reductions from unscreened cross-section values are readily achievable. We discuss numerous qualitative trends exhibited by the screened cross sections, and obtain empirical relations that roughly scale the cross sections for different charges of the target ion.

The results of hydrodynamics and atomic kinetic simulations of laser-irradiated carbon fibers are presented. The hydrodynamics are used to compare the results of the critical density surface as a function of time for different formulations of the equation of state (EOS). In particular two local-thermodynamic-equilibrium EOS cases and one non-local-thermodynamic-equilibrium EOS are compared. Results are then compared to data from a carbon-fiber experiment performed by Burgess et al. at the Australian National University. Next the atomic kinetics is coupled to the hydrodynamics to produce a detailed accounting of the level populations of the carbon ions as the fiber is heated, expanded, and cooled. The study of the formation of the inversion in the hydrogenic ion stage is presented. These calculations are used to draw comparisons with laboratory experiments.

A translation-invariant state (a quantum Markov chain) is associated with a nearest-neighbor interaction on a one-dimensional lattice by a new technique which provides closed forms for all the correlation functions. When applied to an Ising-type perturbation of a chain of harmonic oscillators, the dynamics can be computed explicitly. The resulting translation-invariant distribution is substantially different from the Planck distribution when the temperature and the coupling constant are large. For the evolution of the field operators on a given mode, we obtain a natural nonlinear generalization of the theorem which states that the free evolution of the field operators is obtained by second quantization of the classical free evolution.

The translational diffusion coefficient D of a single polymer molecule embedded in a d-dimensional space is evaluated in the frame of the Kirkwood and Riseman method. The influence of the dimensionality d on the original ‘‘diagonal approximation’’ is estimated and a full second-order renormalization-group calculation for a Gaussian chain is performed. These results are compared with the exact diffusion coefficient in the non-free-draining limit.

A general self-similarity relation is shown to exist, expressing the Renyi-dimension function in terms of local expansion rates both for flows and maps. For the particular case of the information dimension, such an implicit equation yields the well-known Kaplan-Yorke relation. Moreover, it can be explicitly solved in some interesting cases, among which are two-dimensional maps with constant Jacobian. Detailed measurements are performed for the Hénon attractor, with a very accurate estimate of its capacity. Finally, an expansion around the information dimension allows recovery of the Grassberger-Procaccia estimates in an easy way.

The threshold field for flexoelectric instability in nematic liquid crystals, and the upper limit of the maximum tilt angle connected with this instability are rigorously evaluated. The nematic liquid crystal, with positive dielectric anisotropy, is assumed to be in the homeotropic configuration, without external field. The surface anchoring energy is considered infinitely strong on one wall and finite on the other. The stability of the distorted configuration is discussed from the energy point of view. The influence of the surface polarization on the threshold field is also reported. It is shown that the surface polarization changes not only the effective flexoelectric coefficient, in agreement with phenomenological theories, but also the effective anchoring energy, from the coupling with the flexoelectric effect.

The problem of stirred percolation, which has been proposed to model various systems with annealed disorder (including globular-phase microemulsions), is formulated as a continuous-time random walk. Transport quantities, such as the dc diffusion constant, depend both on the statistics of the instantaneous clusters near the percolation threshold and on their rearrangement times. The scaling of these rearrangement times is discussed, and it is shown that both small clusters and large ones affect the scaling behavior of various physical quantities. Qualitatively, the diffusion process resembles a Levy flight, with the duration of large steps being shorter than that of small steps. For some range of parameters, the diffusion constant scales differently than it does in quenched percolation. The natural scaling variables, and therefore the predicted scaling behaviors, are found to differ from previous predictions.

We present a detailed study of the application of Grassberger-Procaccia-type algorithms for estimating entropy in experimental systems. It is shown that this method can clearly distinguish a chaotic regime from a quasiperiodic one, and quantify the degree of chaos. We demonstrate this on the data from two different Rayleigh-Bénard experiments.

A method for calculating properties of biased random walks on nontrivial networks is described. It is a generalization of a method previously developed by the authors for unbiased walks. The essence of the method is the identification of certain basic types of walks. Generating functions corresponding to complicated walks are expressible in terms of the generating functions corresponding to simple walks. Moreover, we shall demonstrate that generating functions for walks on complex structures can be expressed in terms of generating functions corresponding to substructures. For example, the properties of walks on a network composed of coupled one-dimensional objects (like a resistor network) can be calculated, once the generating functions for certain walks on a one-dimensional ‘‘segment’’ have been computed. These combination rules are not as simple as the Kirchoff rules for resistor networks. Still, they are rather straightforward. The role of bias is examined. We consider the cases of a finite segment, a bent segment (in an external field), a segment with a dangling bond, and two segments in parallel. The crossover from the diffusive (weak-bias) regime to the drift (strong-bias) regime is obtained.

Nonequilibrium potentials are constructed which serve as macroscopic generalized thermodynamic potentials in dissipative systems far from thermodynamic equilibrium undergoing a local bifurcation of codimension 2 of a fixed point. The cases of two vanishing linear stability coefficients, of one vanishing and one purely imaginary pair, and of two purely imaginary pairs of linear stability coefficients are treated. As a result we establish the existence and form of a nonequilibrium potential for systems sufficiently close to codimension-1 or codimension-2 bifurcations for all cases where locally stable attractors exist in the phase diagram in parameter space. The attractors of the system, their basins of attraction in configuration space, and their bifurcations are determined by extremum properties of the nonequilibrium potentials.

The ensemble theory of Gibbs has been universally accepted as a theoretical foundation for dealing with thermodynamic systems in the equilibrium state. No comparable theory has gained wide acceptance in nonequilibrium statistical mechanics. We would like to propose a formalism for nonequilibrium systems which follows the spirit of Gibbs’s ensemble theory. We will show that such a nonequilibrium ensemble dynamics may be employed to derive, in a straightforward manner, (1) formal equations for the time evolution of the macroscopic features of the system, (2) the existence of a nonequilibrium entropy function which is a nondecreasing function of time, (3) the Onsager reciprocity theorem, and (4) the fluctuation-dissipation theorem. Just as Gibbs’s ensemble dynamics may be justified by demonstrating that the mean-square fluctuation of the macroscopic properties of certain thermodynamic systems is small, so we will show that the mean-square fluctuation in the time evolution of the macroscopic properties is small for certain classes of thermodynamic systems.

Nonstationary phase-space distributions ρ(t) are shown to sweep out phase space at a rate scrR which can be determined from the autocorrelation function, P(t)=$h^N$Tr[ρρ(t)]. It is demonstrated that, after the initial decay of P(t), subsequent recurrences (increases) of P(t) cause permanent slowdowns in the rate scrR. Such slowdowns may cause localization of quantum eigenstates, whereas no such implications apply to the classical stationary distributions. The localization is discussed in terms of the fraction of the available phase space which the dynamics accesses, and the phase-space distribution of individual eigenfunctions.

Specific-heat–versus–temperature curves are calculated across the reentrant (N-$A_d$-N-$A_1$) phase diagram of liquid crystals, using the microscopic ‘‘spin-gas’’ model. The transition enthalpies are found to be much weaker on each side of the partial bilayer smectic phase ($A_d$) than at the onset of the monolayer smectic phase ($A_1$). This qualitative difference, which agrees with experimental results obtained on octyloxybenzoyloxycyanostilbene (T8), is readily explained by the dipolar frustration reentrance mechanism of our model.

A simple conjecture on the behavior of the canonical ensemble partition function for a system of rigid spheres and disks, as random close packing is approached, is supported in a suggestive manner by definite trends that emerge from ‘‘derivative-logarithm’’ analyses of the low-density virial series with the presently known coefficients. The conjecture predicts that the inverse pressure at random close packing should become zero with infinite slope, whereas it very probably does so with finite slope as the ordered closest-packing limit is approached.

We present mechanisms for sidebranch generation in dendritic growth, stressing the importance of nonlinear effects. One such mechanism, which we call solvability-induced sidebranching, relies on the fact that the steady states form a discrete family. In addition, time-dependent simulations of crystal growth in the boundary-layer model are performed. We find that there is a critical value of the anisotropy strength below which the needle crystals are linearly unstable to tip splitting; above this value, the needle crystals have a finite-amplitude instability.

The surface phases of a nematic liquid in the presence of a smooth substrate are studied using Landau–de Gennes theory. The alignment of the nematic director is restricted to be either parallel or perpendicular to the substrate, although methods of treating more general structures are indicated. We describe extensions and a synthesis of earlier studies which obtained analytic solutions of the theory in certain limits, including surface-induced biaxiality, to determine the domains of wetting by planar and homeotropic nematic films as well as by the isotropic phase, as a function of surface-interaction parameters. The relation of wetting to orientational transitions in the surface-induced bulk alignment is discussed. The ‘‘parabolic approximation’’ recently introduced in related contexts is used to obtain solutions of the theory in more general circumstances where exact analytic solutions are precluded. It is found that, for realistic values of the ratio of model elastic constants, complete wetting by planar nematic films exhibits ‘‘reentrant’’ behavior and does not occur for arbitrarily strong substrate potentials. We argue that this outcome is not an artifact of the parabolic approximation, despite several deficiencies of the method. The relevance to experimental studies of nematic-liquid–substrate interfaces is described.

Phase transitions on more complex Sierpiński carpets are studied. A detailed discussion of previous expressions for lacunarity is given, and a new expression is proposed. Variations of the critical exponents with various geometric factors and the scaling of resistor networks are examined. It is shown that the critical properties depend so much on the detailed construction of the carpet that the validity of universality is called into question. New ways of characterizing universality, if it exists, may have to be introduced.

A ‘‘fictitious’’ time parameter for the two-body Coulomb problem has recently been used to perform the sum over classical paths in the Feynman form of the path integral, as well as to formulate related coherent-state wave functions. It is pointed out that this time variable is actually the eccentric anomaly of the corresponding classical orbit, which has a known significance in terms of the orbit parameters.

The Hamilton-Jacobi and Schrödinger equations for a particle moving in N (>1) dimensions in a concentric superposition of a Coulomb potential and an anisotropic 2:1 axially symmetric harmonic-oscillator potential together with an axial linear potential separate in parabolic coordinates. The separation constant is an eigenvalue of an operator which is a generalization of the axial component of the Runge-Lenz vector of the Coulomb problem in the Schrödinger case. Also for the Schrödinger case, methods of solving the joint eigenvalue problem and the separated differential equations are studied and specific results for the two-dimensional case presented.