An alternative approach is proposed to treat the critical point for general Ising spin systems. From the analytic properties of the free energy, we are in principle able to calculate analytically the critical temperature ${\mathit{T}}_{\mathit{c}}$(l) for (d+1)-dimensional Ising films to any order cumulant as a function of the number l of hyperlayers in the hyperfilm. Explicit expressions up to the fourth order are given. It is shown from the general expression for ${\mathit{T}}_{\mathit{c}}^{\left(\mathit{m}\right)}$(l) that [${\mathit{T}}_{\mathit{c}}$(l)-${\mathit{T}}_{\mathit{c}}$]/${\mathit{T}}_{\mathit{c}}$=b/(l+a) in the limit l→∞. Comparisons with existing results for d=2 are discussed.

A thermodynamic description of the photorefractive effect is developed and used to evaluate thermal fluctuations in the space-charge field of a photorefractive medium responsible for light-scattering noise. Equilibrium grating properties are obtained through minimization of the Helmholtz free energy and are found to be in agreement with results obtained by other methods. Thermal fluctuations in the space-charge field are obtained from the curvature of the free-energy surface. These fluctuations give rise to light-scattering fluctuations in the medium refractive index through the electro-optic effect and are the fundamental source of light-scattering noise. Light-scattering fluctuations associated with the optical Kerr effect are also examined. The signal-to-noise ratio and dynamic range of a photorefractive medium are determined. Stochastic noise model calculations are presented for ${\mathrm{BaTiO}}_3$ under conditions for which the dominant noise contribution is associated with the photorefractive effect. Our results suggest a very large dynamic range for photorefractive materials (120–140 dB) that should prove useful for optical signal processing applications.

The dynamic usage of the technique of effective potentials is motivated and established. The one-loop effective potential and the Gaussian effective potential are derived from Ehrenfrest’s theorem by using adiabatic elimination. An application is made to the Hénon-Heiles problem, and comparison is made with previous results; it is shown that quantum effects destroy chaos in two ways: (a) quantum fluctuations make the curvature more positive and (b) tunneling dominates the dynamics.

We present a two-dimensional continuous cellular automaton that is equivalent to a driven spring-block model. Both the conservation and the anisotropy in the model are controllable quantities. Above a critical level of conservation, the model exhibits self-organized criticality. The self-organization of this system and hence the critical exponents depend on the conservation and the boundary conditions. In the critical isotropic nonconservative phase, the exponents change continuously as a function of conservation. Furthermore, the exponents vary continuously when changing the boundary conditions smoothly. Consequently, there is no universality of the critical exponents. We discuss the relevance of this for earthquakes. Introducing anisotropy changes the scaling of the distribution function, but not the power-law exponent. We explore the phase diagram of this model. We find that at low conservation levels a localization transition occurs. We see two additional phase transitions. The first is seen when moving from the conservative into the nonconservative model. The second appears when passing from the anisotropic two-dimensional system to the purely one-dimensional system.

A general and exact expression for x-ray and neutron reflectance and transmittance is given in terms of an integral of the real-space scattering-length-density profile fluctuation of the film, with respect to an arbitrary constant reference density level, over the wave function inside the film. Various special cases and approximations are then derived from this exact form by suitable approximations of the wave function. In particular, two practical approximate formulas are derived which are improvement over the corresponding distorted-wave Born approximations. One is for an arbitrary film deposited on a known substrate and the other for a free liquid surface. Numerical results are used to illustrate the accuracy of these formulas.

Based on an ideal system under external random forces, the dynamical process of random activation is studied. The time evolution of the system is described by the Langevin equation, and a scaling rule is introduced to generalize the system. The generalized system predicts the fractional power spectrum 1/${\mathit{f}}^{\mathrm{\alpha }}$ from a white spectrum to a Lorentzian. The exponent α is a function of the fractal dimension of the scaling rule. It is found that the fractal dimensions of 2, 1, and about 0.47 specify the particular mode of the generalized system, where the total power of the fractional power spectrum is minimum. The values indicate a Lorentz spectrum, a 1/f spectrum, and a power spectrum of 1/${\mathit{f}}^{1.53}$ type, respectively. The system of the minimum total power in this study is equivalent to one in the minimum potential energy, where the system is in the steady state. Therefore the random-activated system in the steady state gives a 1/f spectrum, and a 1/${\mathit{f}}^{\mathrm{\alpha }}$ spectrum is considered to represent the fluctuation of the complex system from the steady state.

A fractal array of coupled maps, where space is nonuniform, is considered as a dynamical system. The stability and bifurcations of spatially synchronized, periodic states on the Sierpinski gasket are studied. The matrix that expresses the coupling among neighboring elements exhibits a spectrum of eigenvalues with multifractal properties, and their global scaling behavior is characterized by the function f(α). The multifractal character of the eigenvalues affects the stability boundaries of the synchronized, periodic states in the parameter plane of the system. The boundary structure allows access to regions of stability and gives rise to bifurcations that are not present in regular lattices.

The Apollonian tiling of the plane into circles is analyzed with respect to its group properties. The relevant group, which is noncompact and discrete, is found to be identical to the symmetry group of a particular geometric tree graph in hyperbolic three-space. A linear recursive method to compute the radii is obtained. Certain modifications of the problem are investigated, and relations to other problems, such as the universal scaling of circle maps, are pointed out.

We study the time evolution of the initial instability toward the final steady state for two pattern-forming systems: explosive crystallization and the viscous-fingering problem. We show analytically that a scaling solution exists for the power spectrum of the interface shape in the case of explosive crystallization. The scaling exponents are computed exactly. For the viscous-fingering problem, a similar scaling solution is obtained, and the scaling exponents are determined numerically. We also study the robustness of the scaling solutions against external perturbations. We find that local perturbation on the surface tension does not alter the scaling exponents, although it has dramatic influence on the late-stage steady-state patterns.

We categorize the possible large-scale behaviors of one-dimensional nearest-neighbor hopping ‘‘sandpile’’ slope models on a circle of N sites with nonconserving noise, via decimation. Defining the height at site i to be ${\mathit{h}}_{\mathit{i}}$, we find a continuous infinity of fixed points of the decimation, parametrized by an index n (1≤n≤2) with 〈(${\mathit{h}}_0$-${\mathit{h}}_{\mathit{N}/2\mathrm{}}$${)}^2$〉∼${\mathit{N}}^{2/\mathit{n}}$. The fixed point n=2 corresponds to our ordinary notion of a rough interface, while the others are new and far rougher.

Starting from a distribution of time scales a quantum time-autocorrelation function (TAF) of a fluctuating force acting in a fractal environment is proposed as well as its associated mode density. The assumption of a stochastic Ornstein-Uhlenbeck process for every time scale gives rise to power laws for both the mode density and the TAF at frequencies and times of interest in the problem. Asymptotic analysis of the mode density and the TAF are made for different frequency and time ranges. The temperature dependence of the quantum TAF is also studied. Characteristic long-time tails appear in the low-temperature regime.

Based on the field-theoretical renormalization-group theory for O(N) symmetric systems in 4-ɛ dimensions, the free energy of critical films confined between two parallel plates at distance L is analyzed. For five different boundary conditions and for temperatures above and at the bulk critical temperature ${\mathit{T}}_{\mathit{c},}$b, the analytic expressions of the universal scaling functions are derived and discussed. At bulk criticality, the scaling functions reduce to universal amplitudes commonly known as Casimir amplitudes. In addition it is shown that the scaling functions can be expanded in a series of powers of L/${\xi }_{\mathrm{bulk}}$, in which all expansion coefficients are universal numbers with the Casimir amplitudes characterizing the leading terms. Special attention is paid to the specific heat of the films at bulk criticality as a function of L. One of those universal expansion coefficients of the scaling functions enters the experimentally accessible amplitude of the specific heat. In order to identify the finite-size contributions to the free energy of the films, the surface free energies are discussed too and expressed in terms of universal surface amplitudes. In view of experimental applications such as, e.g., for ${}_{}{}^3{}_{}{}^{}\mathrm{-}_{}^4{}_{}{}^{}$He mixtures, tricritical films are also analyzed. The implications of these results for surface forces and wetting films are pointed out.

Close to continuous phase transitions with a diverging correlation length ξ, the thermal singularities of films with thickness L are governed by a universal finite-size scaling function Θ(L/ξ) of their free energy. We propose several experiments for magnetic and fluid films as well as numerical simulations which allow one to determine Θ(L/ξ). Based on field-theoretical results for Θ(L/ξ), we make quantitative predictions for the specific heat of magnetic films, of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ films, and of films composed of ${}_{}{}^3{}_{}{}^{}\mathrm{-}_{}^4{}_{}{}^{}$He mixtures close to the onset of superfluidity. The critical fluctuations within fluid films lead to a long-range Casimir force between the confining walls which can be studied by an atomic-force microscope. We discuss the strength and the temperature dependence of this force for various fluid systems. The study of wetting films close to critical end points gives direct access to the full scaling function Θ(L/ξ). In this respect we discuss in detail the wetting of a substrate by ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^3{}_{}{}^{}\mathrm{-}_{}^4{}_{}{}^{}$He mixtures close to their λ transitions as well as by binary liquid mixtures close to their critical end points.

In this paper we present the results of a study of the mean-square displacement of a Brownian particle in a harmonic potential and of a Brownian particle in shear flow. We have focused on the long-time behavior of the mean-square displacement. In contrast with earlier results, presented by others who studied the Stokes limit of these problems, we have studied this problem using the time-dependent linearized incompressible Navier-Stokes equations to describe the fluid motion. Then we see that the mean-square displacement is strongly influenced by backflow effects in the fluid, resulting, among other things, in long-time tails of correlation functions. We have compared our results with those calculated in the Stokes limit; important differences exist between them. The main differences are the long-time tails in correlation functions and, related with them, the larger time scales that should be considered to obtain diffusive behavior in the case of a Brownian particle in a harmonic potential or to obtain the cubic regime in the mean-square displacement of a Brownian particle in shear flow. Furthermore, we have studied the velocity autocorrelation function of a Brownian particle in a harmonic potential. In the overdamped case we have shown a ${\mathrm{\tau }}^{\mathrm{-}7/2}$ long-time tail instead of the exponential tail that can be obtained in the Stokes limit. Also the sign of both tails differ.

The purpose of this paper is to gain insight into the dynamical origins of intermittent fluctuations in turbulent flow. To this end stochastic models of turbulence are developed, which borrow their interaction structure from fractal geometry. The dynamics conserve energy and ensure equipartition of energy in equilibrium. The resulting stochastic process models the chaotic motion of Navier-Stokes-generated cascades [J. Eggers and S. Grossmann, Phys. Lett. 156, 444 (1991)]. In all cases we find small-scale intermittency, but the strength of the fluctuations depends crucially on the geometry of the cascade. We calculate the variance of the energy fluctuations in a low-noise expansion and evaluate its anomalous scaling exponent to lowest order in the noise strength. The result agrees well with numerical simulations.

We have used the Evans-Cummings (EC) nonequilibrium-molecular-dynamics (NEMD) heat-flow algorithm for liquid mixtures to calculate the thermal conductivity and the Dufour coefficient of a binary Lennard-Jones mixture. A consistency check has been carried out by applying the NEMD color-conductivity algorithm to evaluate the mutual diffusion and the Soret coefficients, which, according to the Onsager reciprocal relations, are equal. Further checking was carried out by comparing these NEMD results with equilibrium Green-Kubo calculations. This work extends that recently carried out by us [S. Sarman and D. J. Evans, Phys. Rev. A 45, 2370 (1992), hereafter referred to as I] to a much lower density and higher temperature than that studied in I. At this state point the Soret and Dufour coefficients are much larger than their corresponding triple-point values in I. This enables considerably more accurate checks of algorithm validity. The NEMD and the Green-Kubo estimates for the thermal conductivity and the mutual diffusion coefficients agree within statistical uncertainties of about 2%. The difference between the cross-coupling coefficients obtained by the various methods is more than twice as accurate as in I, namely, 6% compared to 15%. This provides a convincing check of the correctness of the EC algorithm. We have also developed a way of applying the EC algorithm that increases the accuracy of estimates of the Dufour coefficient. When this method was applied to the triple-point fluid, the results agreed with those in I; however, the errors were reduced by a factor of 3.

We formulate a lattice Boltzmann model which simulates compressible fluids. By choosing the parameters of the equilibrium distribution appropriately, we are able to select the sound speed (which may be set arbitrarily low), bulk viscosity, and kinematic viscosity. This model simulates compressible flows and can include shocks. With a proper rescaling and zero-sound speed, this model simulates Burgers’s equation. The viscosity determined by a Chapman-Enskog expansion compares well with that measured from simulations. We also compare the exact solutions of Burgers’s equation on the unit circle to solutions of our lattice Boltzmann model, again finding reasonable agreement.

The Rayleigh-Bénard problem has been studied using discrete-particle simulation of a two-dimensional fluid in a square box. The presence of temporal periodicity in the convective roll structure was observed, but, more significantly, different simulation runs under identical conditions but with initial states that differed in ways that are seemingly irrelevant at the macroscopic level exhibited very different forms of pattern evolution. The final state always consisted of a horizontally adjacent pair of rolls, but not all initial states evolved to produce well-established periodic behavior, despite the fact that very long runs were undertaken. Results for both hard- and soft-disk fluids are described; the simulations included systems with over ${10}^5$ particles.

In this paper, a two-dimensional model for water-oil-surfactant mixtures, previously studied by the Monte Carlo method, is now investigated by examining (4×∞) strips of the lattice using the transfer-matrix method. The correlation and structure functions in the disordered phase are calculated accurately, as demonstrated by comparison with Monte Carlo results. We find the behavior of these functions to be similar to their counterparts on the more realistic three-dimensional lattice, justifying the use of two-dimensional lattices for studying the disordered phase. From these functions we calculated the disorder and Lifshitz lines which are used to divide the disordered phase into a region of ordinary fluid and a region of microemulsion. We also illustrate that two-dimensional models may not be able to produce the three-phase coexistence between microemulsion, the water-rich phase, and the oil-rich phase obtained with three-dimensional models.

The mode-coupling theory (MCT) of the glass transition can exhibit stretching (nonexponential) behavior observed in the primary relaxation of supercooled liquids near the glass transition. Götze first showed that MCT can exhibit this stretching behavior in the α-relaxation regime by phenomenologically introducing a linear term into the mode-coupling integral for the dynamic viscosity. We show in this paper how this linear term can be generated using fundamental considerations. In particular, we extend fluctuating nonlinear hydrodynamics of compressible liquids, developed by Das and Mazenko [Phys. Rev. A 34, 2265 (1986)], to include another slow variable in addition to the conserved slow variables such as the mass and momentum densities and show that the extended theory naturally generates this linear term. The additional slow variable is defined as the distortion field of the local structure inherent in dense liquids. The suppression of the cutoff mechanism due to Das and Mazenko is also discussed.