With the help of the renormalization method we have calculated the low-temperature expansion series of free energy for the Ising models that are based on a fractal structure and with zero critical temperature. The results show that there exists a critical value of vd¯ such that, when vd¯ is larger than the critical value, scaling theory fails and the hyperscaling law is broken. We also found that the Ising models with zero critical temperature belong to two series of universality classes, respectively, one of which mainly corresponds to the Ising models based on the lattices generated in a site self-similar manner, which are named site-hierarchical lattices. Another series involves all Ising models that are based on other fractal structures. These two universality classes have different critical conditions of scaling and approach different thermal exponents α when vd¯ becomes larger than the critical value.

Starting from quantum Langevin equations for operators we study thermal properties of a one-dimensional harmonic chain to whose ends independent heat baths are attached. In this paper, we mainly discuss the thermal equilibrium state that the chain eventually approaches if the heat baths are at equal temperatures. In the classical limit, this state is determined by the Gibbs ensemble of the free chain, whereas in the quantal case, this is only true if the strength of coupling between chain and heat baths is made infinitely small. We find that corrections for finite coupling strength are appreciable only in boundary layers near both ends of the chain. The thickness of the boundary layers depends only on the temperature and not on the damping constant. Outside these boundary layers we find an analogy between thermal properties of the chain and a discrete random walk.

We study nonequilibrium properties of a one-dimensional harmonic chain to whose ends independent heat baths are attached which are kept at different temperatures. Using the quantum Langevin equation approach, we determine the stationary nonequilibrium state for arbitrary temperatures and coupling strength to the heat baths. This allows us to discuss several typical nonequilibrium properties. We find that the heat flux through the chain is finite as the length of the chain goes to infinity, i.e., we recover the well-known fact that the lattice thermal conductivity of the perfect harmonic chain is infinite. In the quantal case, the heat flux ${\mathit{j}}_{\mathrm{qm}}$ is reduced compared with its classical value ${\mathit{j}}_{\mathrm{cl}}$, ${\mathit{j}}_{\mathrm{qm}}$∝(T¯/${\mathit{FTHETA}}_{\mathit{D}}$${)}^3$${\mathit{j}}_{\mathrm{cl}}$, where T¯ is the average temperature of the heat baths, and where ${\mathit{FTHETA}}_{\mathit{D}}$ is the Debye temperature of the chain. Furthermore, we investigate the variance of the displacement operators and the temperature profile along the chain. In accordance with the infinite thermal conductivity we find a vanishing temperature gradient in the chain except in boundary layers at its ends.

The understanding of some important space-physics problems (e.g., the Jeans escape of light atoms from a planetary atmosphere and the ion escape in the terrestrial polar wind) is related to the problem of the escape of a minor species through a nonuniform background. This latter problem was studied for different interparticle collision models (Maxwell molecule and hard sphere), for different minor-to major-species mass ratios, and for different values of the escape velocity (${\mathit{v}}_{\mathit{c}}$). The gravitational force was simulated by a critical escape velocity (${\mathit{v}}_{\mathit{c}}$), and the Lorentz forces were ignored. First, a simple variable change was used to transform the problem into a simpler one where the background medium is uniform. This simplified problem, which is similar to the ‘‘Milne’’ problem, was solved by a Monte Carlo simulation. In the collisionless region and for the case of zero escape velocity (${\mathit{v}}_{\mathit{c}}$=0), the velocity distribution function of the minor species (${\mathit{f}}_{\mathit{t}}$) showed large deviations from a Maxwellian, with large temperature anisotropy (parallel temperature less than the perpendicular temperature) and asymmetry in the parallel direction (upward tail), for both of the collision models. In the collision-dominated region the normalized density gradient was found to be independent of the mass ratio for the Maxwell molecule collision. For the hard-sphere cross section, it was reduced by a mass-ratio-dependent factor. This is due to the heat-flow contribution to the momentum balance, which vanishes only for the former collision model. The Monte Carlo results were compared with the moment-equations approach and the direct solution of the Boltzmann equation. This comparison confirms that the Monte Carlo method is a viable alternative and complementary to other techniques, especially in recent years, due to the large increase in the available computation power.

A general variational approach to the quantum statistics of many-particle systems is developed, based on the Gibbs-Bogoliubov variational principle and the ‘‘thermodynamically equivalent Hamiltonian’’ method first applied to the BCS model by Bogoliubov, Zubarev, and Tserkovnikov [Sov. Phys.—Dokl. 2, 535 (1957)] and subsequently generalized by Wentzel [Phys. Rev. 120, 1572 (1960)]. The method retains a judiciously chosen discrete set of momentum transfers in the second-quantized interaction Hamiltonian. It is expected to be applicable to a variety of systems including both crystalline solids and quantum fluids such as liquids and dense plasma phases.

A kinetic surface-reaction model, based upon the reaction of carbon monoxide and oxygen on a catalyst surface, is presented. The equations governing the reaction are obtained. A bistability zone is found: One is a reactive stable state producing carbon dioxide, and in the other the catalyst is poisoned. The relation between this reactive stable state and kinetic parameters of the reaction is studied.

An alternative approach for treating strongly interacting statistical systems with strongly singular potentials is developed. This approach combines both the advantages of decoupling procedures for Green functions, which eliminate divergences caused by singular potentials, and the advantages of perturbation theory, which is a regular procedure that makes it possible to construct subsequent approximations above the chosen decoupling. The resulting iterative sequence of Green functions generates corresponding sequences of observable quantities. To improve the convergence of the latter sequences a method is suggested based on the renormalization-group ideas. A simple illustration is given.

A Lyapunov function is defined for two-dimensional differential systems. The relationship between the Lyapunov function and the stationary probability distribution of the corresponding Fokker-Planck equations in the weak-noise limit is analyzed. Based on the structure of the Lyapunov function we classify two types of singularities. The classification of different kinds of singularities is shown to be significant for the intrinsical dynamics of the deterministic flow. The bifurcation behavior of periodically forced planar systems that contain distinctive kinds of singularities in the autonomous cases is investigated. It is found that the responses of planar systems to the external force, i.e., the resulting bifurcation sequences, are substantially different if the trajectories circle around different types of singularities.

Certain probabilistic cellular automata (PCA) have been shown to exhibit nonergodic behavior (i.e., possess two or more distinct stable states) over finite regions of their parameter spaces, i.e., generically. To demonstrate that this behavior is a consequence of irreversible (or ‘‘nonequilibrium’’) dynamics and not special to the discrete space, time, and variables of PCA, we define and study a class of noisy partial differential equations (Langevin models), which are constructed to have spatially anisotropic domain-wall kinetics. The models are simple nonequilibrium generalizations of ordinary equilibrium time-dependent Ginzburg-Landau theories. We present analytic and numerical arguments to show that these models exhibit the same generic nonergodic behavior as do their discrete PCA counterparts.

The influence of noise on the delay of a bifurcation point in the presence of a swept control parameter has been investigated theoretically, by digital simulation and by analog electronic experiment. The results obtained in an earlier paper [N. G. Stocks, R. Mannella, and P. V. E. McClintock, Phys. Rev. A 40, 5361 (1989)] have thereby been extended and complemented. In particular, exact analytic expressions have been derived for the time-dependent probability densities P(x,t), and these have been used to obtain the mean first-passage time ${\mathit{t}}_{\mathrm{MFPT}}^{\mathrm{*}}$ for ${\mathit{x}}^2$(t) to reach a threshold under the influence of Gaussian fluctuations, in several contexts: additive external white noise, additive external exponentially correlated noise, additive internal white noise, additive internal exponentially correlated noise, multiplicative white and colored noise. Based on Zeghlache, Mandel, and Van den Broeck’s [Phys. Rev. A 40, 286 (1989)] alternative definition of the bifurcation time ${\mathit{t}}_{\mathrm{moment}}^{\mathrm{*}}$ in terms of the evolution of the second moment 〈${\mathit{x}}^2$(t)〉, an expression is derived for ${\mathit{t}}_{\mathrm{moment}}^{\mathrm{*}}$ for the general case of combined additive and multiplicative noises. The calculations are tested by comparison with the results of analog experiments and digital simulations, with which they are shown to be in excellent agreement.

A computer-simulation model is introduced to study the transport properties of a binary fluid mixture in which the constituents of one fluid (the tracer particles) carry charges in a linear charge-density gradient in the background charge of the second fluid. In the steady-state equilibrium, an effective conductivity ${\mathrm{\sigma }}_{\mathit{c}}$(p,r) is estimated as a function of the carrier concentration p and the range of interactions r. The conductivity is observed to vary nonmonotonically with the concentration p, and it exhibits a maximum at a characteristic value ${\mathit{p}}_{\mathit{c}}$. The conductivity decreases when the range of interaction is increased until a characteristic value ${\mathit{r}}_{\mathit{c}}$, beyond which (r>${\mathit{r}}_{\mathit{c}}$) it begins to saturate as the onset of screening spans with r-${\mathit{r}}_{\mathit{c}}$.

For thermodynamically ideal two-component systems, as for example isotopic systems, the mass effect on the mutual diffusion coefficient is negligibly small for mass ratios of order 2. For higher mass ratios a significant influence on the diffusion coefficient may be expected. We present here accurate molecular-dynamics calculation results for the hard soft sphere and the Lennard-Jones liquids, which indicate quantitatively the mass-ratio effect on the mutual diffusion coefficient. The predictions of the Enskog kinetic theory are found to give the correct trend and order of magnitude of these effects but overestimate them at all the mass ratios studied. Both systems, the hard soft sphere liquid (the softness of the interaction is so small that it can be regarded as a hard one) and the Lennard-Jones liquid, show appreciable mass-ratio dependence of the diffusion coefficient only for ratios larger than 10, in agreement with our previous numerical work [Schoen and Hoheisel, Mol. Phys. 52, 1029 (1984); 53, 1367 (1984); 57, 65 (1986)].

A growing array of numerical results obtained in our laboratory indicates that, in certain situations, the Maxwellian velocity distribution for a subensemble of low-mass test particles in equilibrium with a heat bath is not valid. This paper provides a theoretical framework in which the observed non-Maxwellian distributions can be understood. The basic arguments are as follows. When the mass of a test particle is small compared with the mass of the heat bath particles, and when this particle is subjected to a strong systematic force, the resulting dynamical motion of the test particle is subjected to a friction force that is nonlinear in the velocity of the test particle. The dynamics of the test-particle motion is then governed by a nonlinear Langevin equation, and the probability density of the stochastic variables must accordingly be obtained from a related nonlinear Fokker-Planck equation. The steady-state solutions of this differential equation are seen to correspond generally to non-Maxwellian velocity distributions.

We propose a simplified version of the free-energy density functional for the inhomogeneous hard-sphere fluid mixture recently derived by Rosenfeld [Phys. Rev. Lett. 63, 980 (1989)]. This new functional, which requires four distinct weight functions, generates a triplet-direct-correlation function for the one-component fluid, which is in good agreement with Monte Carlo simulation results. It also performs well in describing the density profile of the hard-sphere fluid in contact with hard and soft walls. Yet, it is not suitable for studying the freezing transition.

Rayleigh-Bénard convection is studied in quasi-one-dimensional geometries. Fixed and periodic boundary conditions are imposed using a rectangular and an annular cell, respectively. The destabilization process of the homogeneous convective pattern is studied for increasing Rayleigh number scrR. The first time-dependent behaviors are given by the appearance of coupled oscillators. At larger scrR values, the spatial breakdown appears through the propagation of spatial defects, which appear to be solitary waves. This spatiotemporal destabilization is followed at higher scrR by a spatiotemporal intermittent regime, which corresponds to a dramatic decrease of the spatial coherence and to a mixing of turbulent patches within laminar domains. This last regime is studied within the frame of phase transitions. The statistical analysis evidences a second-order phase transition at least in the rectangular geometry (fixed boundary conditions), while this transition looks imperfect in the annular geometry (periodic boundary conditions). Nevertheless, the essential qualitative features shown by theoretical and numerical models are observed in both geometries. Comparison with a simple model of directed percolation shows that the imperfect nature of the transition in the annulus could be the consequence of some mechanism of self-generation of the turbulent domains. This mechanism is, however, unknown but is probably related to the influence of the boundaries.

We study the linear stability of an arbitrary number of spherical concentric shells undergoing a radial implosion or explosion. The system consists of N incompressible fluids with small amplitude perturbations at each of the N-1 interfaces. We derive the evolution equation for the perturbation ${\mathrm{\eta }}_{\mathit{i}}$ at interface i; it is coupled to the two adjacent interfaces via ${\mathrm{\eta }}_{\mathit{i}\pm 1}$. We show that the N-1 evolution equations are symmetric under n⇆-n-1, where n is the mode number of the spherical perturbation, provided that the first and last fluids have zero density (${\mathrm{\rho }}_1$=${\mathrm{\rho }}_{\mathit{N}}$=0). In plane geometry this translates to symmetry under k⇆-k. We obtain several analytic solutions for the N=2 and 3 cases that we consider in some detail. As an application we derive the shock timing that is required to freeze out an amplitude. We also identify ‘‘critical modes’’ that are stable for any implosion or explosion history. Several numerical examples are presented illustrating perturbation feedthrough from one interface to another. Finally, we develop a model for the evolution of turbulent mix in spherical geometry, and introduce a geometrical factor G relating the mixing width h in spherical and planar geometries via ${\mathit{h}}_{\mathrm{spherical}}$=${\mathit{h}}_{\mathrm{planar}}$G. We find that G is a decreasing function of R/${\mathit{R}}_0$, implying that in our model ${\mathit{h}}_{\mathrm{spherical}}$ evolves faster (slower) than ${\mathit{h}}_{\mathrm{planar}}$ during an implosion (explosion).

We have performed extensive measurements of shear viscosity, ultrasonic absorption, and sound velocity in a ternary system consisting of water–decane–sodium di(2-ethylhexyl)sulfo- succinate(AOT), in the one-phase region where it forms a water-in-oil microemulsion. We observe a rapid increase of the static shear viscosity in the dense microemulsion region. Correspondingly the sound absorption shows unambiguous evidence of a viscoelastic behavior. The absorption data for various volume fractions and temperatures can be reduced to a universal curve by scaling both the absorption and the frequency by the measured static shear viscosity. The sound absorption can be interpreted as coming from the high-frequency tail of the viscoelastic relaxation, describable by a Cole-Cole relaxation formula with unusually small elastic moduli.

A coupling of the transverse fluctuations of an elastic sheet in a shearing solvent is described. When applied to a dilute colloidal crystal under shear it is found to lead to a mechanical instability which results in the continuous decrease in the coherence length of crystalline order in the sliding layers as the shear rate increases. Such behavior has been observed in light-scattering studies of colloidal suspensions.

We have studied the structure that arises when a low-viscosity fluid displaces a high-viscosity fluid in a transparent three-dimensional porous medium. The viscosity ratio m is not far from 1 (m≃14) in this system. The fluids we use have equal densities so that there are no gravity effects. The injected fluid forms a ramified structure when an intermediate displacement rate is used (${\mathit{N}}_{\mathrm{Ca}}$≃${10}^{\mathrm{-}4}$). We have measured the projected area A of these growing structures. The cluster mass M is found to scale with a characteristic length ${\mathit{R}}_{\mathit{A}}$ obtained from this area, as M∼${\mathit{R}}_{\mathit{A}\mathit{A}}^{\mathit{D}}$. We find the exponent to be ${\mathit{D}}_{\mathit{A}}$=2.5±0.1. The physical model was simulated on the computer using the three-dimensional off-lattice diffusion-limited aggregation (DLA) algorithm. Projections of these clusters show features in common with the experimentally generated patterns. Quantitatively we find reasonably good agreement in terms of the exponent ${\mathit{D}}_{\mathit{A}}$, although the experiments are outside the DLA regime of viscous fingering.

We analyze a system of hard parallel molecules that are composed of hard cylindrical cores and infinitely long and infinitely thin tails. We show that this system exhibits a continuous nematic–smectic-${\mathit{A}}_{\mathit{d}}$ phase transition at zero packing fraction. The transition is studied in the framework of the virial expansion of the free energy. Keeping only the second virial coefficient in the expansion we find the transition at ${\mathit{B}}_2$ρ=1.80 and ${\mathit{d}}_1$/L=1.56, where the density ρ is measured in units of the second virial coefficient ${\mathit{B}}_2$, and the basic smectic period ${\mathit{d}}_1$ in units of the total length L of a molecule.