The observable consequences of percolation models of immiscible displacement in porous media are discussed, with emphasis on the critical behavior. At the microscopic level, these include the fractal nature of the nonwetting fluid configuration at breakthrough in drainage, and the size distribution of the residual nonwetting clusters in imbibition. At the macroscopic level, it is suggested that percolation ideas are consistent with the usual multiphase Darcy equations, and critical behaviors of the relative permeability and capillary pressure curves are obtained. By using these results, predictions are made for the shape of the fluid saturation profiles near the percolation thresholds in the presence of buoyancy or viscous pressure gradients. Finally, it is pointed out that very close to the percolation thresholds, the diverging correlation length requires these macroscopic ideas to be modified. A simple way of doing this is suggested.

We derive analytically the spectrum for the Schrödinger equation for quasiperiodic systems with two length scales: one large ‘‘macroscopic’’ scale [e.g., a cos(2πx/λ)] and one small ‘‘microscopic’’ scale [e.g., v cos(2πx)]. The phase diagram includes regimes with exponentially narrow gaps due to the slowly varying potential, regimes where the rapidly varying potential amplifies these narrow gaps, and regimes with exponentially narrow ‘‘Landau bands.’’ The full ‘‘devil’s-staircase’’ spectrum with gaps at wave vectors q=mπ+nπ/λ develops in a hierarchical manner as a increases. The results apply to systems with superlattices, to celestial orbits with two periodic perturbations, to systems with slowly varying lattice distortions, and, in particular, to quasi-one-dimensional magnets such as bis(tetramethyltetraselenafulvalene) perchlorate [(TMTSF${)}_2$${\mathrm{ClO}}_4$] in magnetic fields, where our findings may provide insight into the experimentally observed cascade of phase transitions.

A new method is proposed to evaluate the probability density function ρ over the Lorenz attractor. The conservation statement within the continuity equation for ρ is equivalently replaced by the requirement that all elements of an infinite set of integrals involving ρ are equal. Two approximations for ρ are constructed, after considering a finite number s of such integrals and a function Π depending upon a set of parameters. The values these parameters assume in the approximation for ρ are given by the solution of a system of equations which arises from the requirement that the elements of the subset of s integrals are equal. The results are compared with the probability density function coming from the numerical integration of the continuity equation for ρ.

We reexamine the problem of the δ-kicked rotator in the light of some numerical results which support the thesis of the existence of a singular continuous component in the quasienergy spectrum, in accordance with previously obtained analytical results. In particular, we discuss to what extent a memory of classical chaos may survive in the quantum motion.

We report high-resolution x-ray scattering measurements of the critical fluctuations in mixtures of hexylphenyl cyanobenzoyloxy benzoate (${\mathrm{DB}}_6$) and terephtal-bis-butylaniline (TBBA). The phase sequence exhibited on cooling pure ${\mathrm{DB}}_6$ (or mixtures with a low concentration of TBBA), is nematic (N) to smectic-$A_d$ ($A_d$) to smectic-$A_2$ ($A_2$). Mixtures with ≳12 molar percent (mol %) of TBBA have a smectic-$A_1$ ($A_1$) phase between the nematic and smectic-$A_2$ phases. For each of the second-order transitions the critical-temperature dependence of the susceptibility and correlation lengths are fit to power laws of the form $t^x$ where t=(T-$T_c$)/$T_c$.

For the N-$A_d$ transition in pure ${\mathrm{DB}}_6$ the susceptibility exponent γ=1.29±0.05 and the parallel and perpendicular correlation-length exponents are ${\nu }_{?}$=0.67±0.03 and ${\nu }_{\perp }$=0.52±0.03, respectively. Close to the multicritical point (12 mol?TBBA) where the second-order N-$A_1$ line meets the first-order portion of the $A_1$-$A_2$ line, the N-$A_1$ exponents are γ=1.09±0.05, ${\nu }_{?}$=0.57±0.03, and ${\nu }_{\perp }$=0.43±0.03. The correlation length anisotropy (${\nu }_{?}$≠${\nu }_{\perp }$) persists along the entire N-$A_1$ line, with the observed exponents decreasing as the concentration approaches the multicritical point. The $A_1$-$A_2$ line has both first-order and second-order regions.

All the measured exponents (γ,${\nu }_{?}$,${\nu }_{\perp }$) were invariant along the second-order portion of the $A_1$-$A_2$ line and the correlation-length exponents were isotropic (${\nu }_{?}$=${\nu }_{\perp }$). The measured exponents were γ=1.46±0.05, and ${\nu }_{?}$=${\nu }_{\perp }$=0.74±0.03. These numbers also held close to the tricritical point where the $A_1$-$A_2$ transition became first order.

Consider irreversible cooperative filling of sites on an infinite lattice where the filling rates $k_i$ depend on the number, i, of occupied sites adjacent to the site(s) being filled. If clustering is significantly enhanced relative to nucleation ($k_1$/$k_0$≡ρ≫1), then the process is thought of as a competition between nucleation, growth, and (possible) coalescence of clusters. These could be Eden clusters with or without permanent voids, Eden trees, or have modified but compact structure (depending on the $k_i$, i≥1).

Detailed analysis of the master equations in hierarchial form (exploiting an empty-site shielding property) produces results which are exact (approximate) in one (two or more) dimensions. For linear, square, and (hyper)cubic lattices, we consider the behavior of the average length of linear strings of filled sites, $l_{\mathrm{av}}$=${\mathcal{J}}_{s=1\mathrm{}}^{\infty }$ ${\mathrm{sl}}_s$/${\mathcal{J}}_{s=1\mathrm{}}^{\infty }$ $l_s$, where $l_s$ is the probability of a string of length s [$l_{\mathrm{a}\mathrm{v}=(1\mathrm{}\mathrm{-}\mathrm{CTHETA}{)}^{\mathrm{-}1}}$ for random filling, at coverage CTHETA].

In one dimension, $l_s$=$n_s$ gives the cluster size distribution, and we write $l_{\mathrm{av}}$=${\mathrm{n}}_{\mathrm{av}}$. We consider the scaling $l_{\mathrm{av}\mathrm{\sim }\mathrm{A}\left(\mathrm{CTHETA}\right){\rho }^{\omega }}$ as ρ→∞ (with CTHETA fixed), which is elucidated by the introduction of simpler models neglecting fluctuations in cluster growth or cluster interference. For an initially seeded lattice, there exists an upper bounding curve $l_{\mathrm{av}{}^{+}}$ for $l_{\mathrm{av}}$ (as a function of CTHETA), which is naturally obtained by switching off nucleation (setting $k_0$=0). We consider scaling of $l_{\mathrm{av}{}^{+}}$ as the initial seed coverage ε vanishes. The divergence, $l_{\mathrm{av}\mathrm{\sim }\mathrm{C}(1\mathrm{}\mathrm{-}\mathrm{CTHETA}{)}^{\mathrm{-}1}}$ as CTHETA→1, is also considered, focusing on the cooperativity dependence of C. Other results concerning single-cluster densities and $l_s$ behavior are discussed.

We use the Evans method [Phys. Lett. 91A, 457 (1982) to simulate heat flow in the Lennard-Jones fluid. With use of the standard Lennard-Jones representation of argon, the results are in broad agreement with experimental argon data except in a small region close to the critical point. The results indicate that away from phase boundaries there is little N dependence of the computed results. Close to the critical point the computed effective thermal conductivity is a nonmonotonic function of the external field. The long-time tail of the heat flux autocorrelation function is somewhat larger than mode-coupling predictions.

A dilute gas under a constant heat flux is studied with use of a Monte Carlo simulation based on the Boltzmann equation. Results for several spatial correlation functions of equal-time fluctuations are reported and compared qualitatively with previous fluctuating hydrodynamics calculations for liquids.

The equilibrium configuration of a dielectric fluid when subjected to a static electric field is discussed under the assumption of maximum entropy S. However, the maximization must be constrained by the field equations, a complication which renders the usual textbook treatments inapplicable. The Helmholtz ansatz, recently justified on statistical mechanical grounds, is recovered. Dielectric fluids, though relatively simple, illustrate the general problem of thermodynamic equilibrium under long-range forces, which has not hitherto been properly treated in the literature.

This paper is concerned primarily with the development of an analytic approach to the theory of steady-state velocity selection in the boundary-layer model of dendritic solidification. We consider the two-dimensional version of this model with a fourfold crystalline anisotropy α in the surface tension. By extending a WKB method introduced in an earlier paper, we are able to determine the α dependence of the selected growth rate in the limit of small α; and we are also able to study this rate for larger α’s in the limit in which the dimensionless undercooling approaches unity. Portions of the paper are devoted to a reinterpretation of the mathematical structure of the solvability condition in problems of this kind.

The colloid configurational contribution to the excess stress tensor of a dilute suspension interacting via a screened Coulomb potential in a fluid undergoing Couette flow is analyzed. The structure factor for the nonequilibrium system obtained by solving a fluctuating diffusion equation with a wave-vector-dependent diffusion constant is used to obtain shear-rate-dependent viscosity and normal stress functions. The viscometric functions are nonanalytic in the limit of vanishing shear rate, although the nonanalytic parts are very small. At high shear rates, the viscosity due to the colloid is inversely proportional to the shear rate. In general, the principle of material objectivity, often evoked in rheology, is not satisfied.

We develop a method to obtain first-passage-time statistics for non-Markovian processes driven by dichotomous fluctuations. The fluctuations themselves need not be Markovian. We calculate analytic first-passage-time distributions and mean first-passage times for exponential, rectangular, and long-tail temporal distributions of the fluctuations.

The dynamical equations for the motion of classical mechanical particles at constant temperature T and pressure p, derived by Nosé and by Hoover, are solved for a two-dimensional Lennard-Jones system. The molecular-dynamics isothermal-isobaric (MDTP) simulations of the solid-fluid transition show that the T-p thermostat works well. The MDTP results agree with the corresponding MD data and the pressure thermostat does not destroy the long-range bond correlation in the fluid near the freezing point.

The problem of nonlinear response is considered by employing a general time-evolution equation, and a Green’s function which is the transition or conditional probability density for an unperturbed system. Expansion of the Green’s function in terms of orthonormal functions enables us to express the distribution function describing the nonlinear behavior by means of matrix products whose elements are composed of correlation functions in the absence of the perturbation. In other words, it is shown how the distribution function induced by a strong perturbation may be calculated by knowing the Green’s function without the perturbation. As the special case of the linear response, we have obtained Kubo’s relation. The Laplace-transform technique with respect to time is found quite useful in developing the present theory in which the transient effect is also taken into account. As an application of the theory, a new relation valid in the region of the second-order perturbation connecting the transient rise and decay with the stationary alternating perturbations has been obtained.

A local HNC/HNC and a HNC/HNC (hypernetted-chain) treatment of the double layer for 2:1 electrolytes is presented. The local HNC/HNC treatment produces improved structure information about the double layer. For divalent counter-ions, nonmonotonic co-ion profiles and improved charge inversions near the electrode are obtained as well as improved diffuse layer potentials; for monovalent counter-ions, packing layers build up at high surface charges. These are effects due to the treatment of the ionic correlations in non-neutral homogeneous systems with the local composition of the electrolyte in front of the electrode. At zero surface charge, a nonzero surface potential is obtained.

We report on experiments with a forced nonlinear oscillator which reveal, for the first time, the existence of systems with a large variety of one-dimensional maps which do not exhibit universality. These new classes of maps are characterized by the local and global asymmetry of the return map which exhibits different functional forms at either side of its maximum. The local asymmetry leads to a new type of nonuniversal classes without convergence to a single metric constant. The general features observed are reproduced in simulations with maps analogous to those observed experimentally.

Molecular-dynamics computer simulations of dense krypton gas have been carried out for ρ=13.8 atoms/${\mathrm{nm}}^3$ and T=297 K, with use of the best available pair potentials. The results are compared to published experimental data at this state to discover the role of the many-body potentials of the atoms on their dynamics. It is found that these effects produce a marked slowing down of many-body collision processes.

Simulations of Eden clusters with more than ${10}^9$ sites by the use of a Cray Research Cray-2 supercomputer show that the thickness of the surface layer of round two-dimensional clusters, averaged over all directions, increases as the radius for sizes above 50?0?0 sites. In simulations of flat surfaces, on the other hand, the thickness increases as the square root of the height for strip widths above 500. The initial growth of such strips is not described by one single power law and asymptotically has a ‘‘dynamic’’ exponent close to (3/2), i.e., it increases as predicted theoretically with (height${)}^{1/3}$. In the range covered, our three-dimensional flat-surface thickness is not consistent with a logarithmic law, and also in four dimensions the thickness does not approach a constant. Round clusters with more than ${10}^9$ sites on the square lattice confirm the expected behavior due to anisotropy. Higher dimensions up to d=12 were also studied.

Magnetic spin models and resistor networks are studied on two types of fractals: X fractal and checkerboard fractal. At finite temperature there is no long-range order. The transition at zero temperature shows a discontinuity in the magnetization, and the associated magnetic exponent is equal to the dimension d¯ of the fractals. The q-state Potts model and percolation on these fractals are also studied.

Two-dimensional oscillatory convection in a binary fluid mixture in an infinite plane porous layer heated from below is studied. Small-amplitude nonlinear solutions in the form of standing and traveling waves are found and their relative stability is established. Stable traveling waves are preferred near onset. The interaction of the two types of wave with steady overturning convection is also studied. As the Rayleigh number is increased the period of each type of wave approaches infinity, standing waves as -ln($R_c^{\mathrm{SW}\mathrm{-}\mathrm{R}}$) and traveling waves as 1/($R_c^{\mathrm{TW}\mathrm{-}\mathrm{R}}$), where $R_c$ is the critical Rayleigh number at which the transition to finite amplitude overturning convection occurs. This transition is hysteretic. The presence of modulated traveling waves (i.e., waves with two distinct frequencies) is also predicted. These predictions are made on the basis of analyses of multiple bifurcations in the presence of O(2) symmetry. This symmetry is present in two-dimensional problems with periodic boundary conditions and a reflection symmetry in a vertical plane. The relevance of the results to recent experiments on binary fluids, both in bulk mixtures and in a porous medium, is discussed.