We propose a dynamic mean-field approximation based on the master equations governing the time evolution of the singlet probability-distribution functions for kinetic lattice models. Both nonconserved-order-parameter (Glauber-dynamics) and conserved-order-parameter (Kawasaki-exchange-dynamics) cases are considered. These discrete spatiotemporal coupled equations can be regarded as a computationally efficient scheme for ‘‘simulating’’ the kinetics of phase ordering as well as for obtaining the equilibrium mean-field solutions to lattice models having complex spatial structures such as surfactant self-assembly. We illustrate the use of these cell-dynamics schemes by studying the three-state Potts model with both Glauber and Kawasaki dynamics.

This paper extends the methods of asymptotic analysis extensively developed for energy-level calculations of Rydberg states to the evaluation of matrix elements of δ(r). All terms up to ${\mathit{x}}^{\mathrm{-}8}$ in the asymptotic potential are systematically derived from a simple perturbation expansion. The formalism is developed in a way that closely parallels that for the corresponding energy expansion. The results are comparable in accuracy to the energy itself. Detailed numerical comparisons with high-precision variational calculations are presented for the states of helium up to nL=10K (i.e., L=7). For L≥5, the accuracy of the asymptotic expansion exceeds what has been achieved variationally. The asymptotic expansion for the specific mass correction to 〈δ(r)〉 is also obtained. Here the accuracy rivals the variational results even for L=3. Similar methods can be applied to the calculation of a wide variety of other atomic properties.

A general formula is presented that expresses the temporal evolution of compound systems. It connects the time dependence of the density matrix with the energy dependence of the scattering matrix. Using results of random matrix theory, we then study the decay behavior of stochastic compound systems with arbitrary coupling between bound states and decay channels. As an example we consider the case of one open channel and prove a nonexponential decay law for all times that asymptotically is of the form ${\mathit{t}}^{\mathrm{-}3/2}$.

We study free second-order processes driven by dichotomous noise. We obtain an exact differential equation for the marginal density p(x,t) of the position. It is also found that both the velocity Ẋ(t) and the position X(t) are Gaussian random variables for large t.

We report the results of Rayleigh-scattering experiments in liquid toulene that is subjected to a stationary temperature gradient, but which otherwise is kept in a quiescent state. The measurements were made with temperature gradients up to 224 K/cm and scattering angles down to 0.45°. They probe fluctuations with wavelengths up to 40 μm and demonstrate the presence of long-range fluctuations in a nonequilibrium liquid. The strength of the observed nonequilibrium heat-mode and viscous-mode fluctuations varies with the square of the temperature gradient and with the fourth power of the wavelength of the fluctuations, in agreement with the behavior predicted by fluctuating hydrodynamics.

A chaotic signal loses the memory of the initial conditions with time, and the future behavior becomes unpredictable. Here we propose a method to understand the loss of memory with time from a time series. This is done by introducing time-dependent generalized exponents. The asymptotic behavior of these exponents is interesting and can distinguish between chaotic systems that lose memory of the initial conditions completely, those that partially retain the memory, and those (borderline of chaos) that fully retain the memory. We discuss these features with some illustrative examples.

We treat the problem of D-dimensional tethered (or polymerized) membranes with long-range repulsive interactions (varying as 1/${\mathit{r}}^{\gamma }$) in a d-dimensional embedding space using both large-d and variational methods. We find three different regimes in the (γ,D) plane: for small D, the manifold is always crumpled, either Gaussian or swollen (with the exact exponent for the radius of gyration ν=2D/γ); for intermediate D, it undergoes a crumpling transition; and for large D, it is always flat. By extrapolating these results to the case of short-range interactions, we propose a phase diagram in the (d,D) plane for the still poorly understood problem of self-avoiding manifolds. The physical point D=2 and d=3 lies at the boundary of the always-flat region. This may provide an explanation for the recent numerical simulations that seemingly show that self-avoidance always makes two-dimensional membranes flat.

We present analytical and numerical studies of fluctuations about and transitions from the nonequi- librium dissipative steady state to the stable equilibrium state of a damped physical pendulum driven by constant torque and small white noise. We find the probability density function of the local flucutations about the nonequilibrium steady state and the transition rate from the nonequilibrium state to the equilibrium state by constructing an asymptotic solution to the Fokker-Planck equation by the WKB method. The solution to the eikonal (Hamilton-Jacobi) equation is constructed both analytically by an asymptotic series expansion, and numerically by integrating the set of characteristic equations. We compare the numerical results to the analytical calculations and determine the limits of validity of the asymptotic approximation. We apply the results to the case of the hysteretic Josephson junction and discuss a generalization of the numerical and analytical methods to other systems.

The existence of Feigenbaum-like sequences of inverted period doublings in four-dimensional reversible maps is pointed out. Starting with a pair of symmetrically coupled de Vogelaere maps and truncating at terms of degree 3, we solve numerically the renormalization equation in a 12-dimensional parameter space. In addition to the three known solutions obtained by previous workers with quadratic maps, which correspond to ‘‘normal’’ period-doubling sequences, interesting new solutions for the fixed map of the renormalization operator are found. A few possible physical implications are discussed.

We discuss the mean-field theory for a class of probabilistic cellular automata that can describe the dynamics of social impact. The models exhibit complex intermittent behavior.

This paper presents a renormalization approach to corrections to parameter scaling of the period-doubling transition to chaos. Starting from a suitable normal form for the doubling operator the multiple-scaling corrections numerically observed by Mao and Hu [J. Stat. Phys. 46, 111 (1987); Int. J. Mod. Phys. B 2, 65 (1988)] are derived from renormalization theory. In weakly dissipative period-doubling systems with a two-dimensional (2D) Poincaré map of uniform Jacobian, these corrections to scaling lead to a crossover between conservative and dissipative scaling. It is well known that this crossover can be described by a scaling function δ(${\mathit{B}}_{\mathrm{eff}}$). Starting once more from a normal form for the doubling operator, this scaling function is shown to be universal. Because this crossover is very sharp, the bifurcation parameters of these weakly dissipative systems universally scale like conservative 2D maps whenever ${\mathit{B}}^{2\mathit{n}\mathrm{-}2}$>0.8, where n is the order of the period-doubling bifurcation and B the Jacobian of the map. By relating this crossover to a multiple-scaling expansion for dissipative systems, a novel feature of the scaling function can be explained: δ(${\mathit{B}}_{\mathrm{eff}}$) shows in its dissipative limit a logarithmic singularity.

Galilean space-time symmetries are discussed in the context of the Liouville space bra-ket formalism and the Wigner-Weyl representations. Also considered is Liouville conjugation (changing operators into their Hermitian conjugates), owing to which any symmetry transformation can be performed by both a unitary and an antiunitary superoperator. Special reference is made to the symmetries of relaxation superoperators for open systems, emphasis being on the practical significance of Wigner-Weyl as symmetry-adapted epresentations. As a side result, symplectic Fourier transforms are shown to be equivalent to inversions.

Second derivatives of the total energy with respect to nuclear coordinates and self-consistent-field parameters within the linear-combination-of-atomic-orbitals approach to local-density-functional theory are discussed. We explicitly include the effects of the fitting procedures for both the electrostatic and the exchange-correlation energies of the electrons in the evaluation of derivative expressions by solving for the fitting coefficients in the appropriate Lagrangian before any differentiation. The advantages and disadvantages of fitting vis–à–vis accurate three-dimensional numerical integration are discussed.

The statistics of eigenvector elements is studied for random band matrices as a function of band and matrix sizes. It is shown that the statistics obey a scaling law similar to that found for the mean localization length of eigenvectors. To describe the statistics we propose an expression for the probability distribution of elements based on the assumption of an exponential form of eigenvectors. We demonstrate a fundamental role of fluctuations of the localization length.

A large number of statistical models of current interest in physics can be characterized as differentially quenched systems. They are prepared by successively introducing into a volume one fraction after another of the particles in a many-body system, each successive fraction being equilibrated while all the earlier fractions are kept frozen in place. Examples of current interest include self-avoiding walks, chemical association models, growth models, and models of random sequential adsorption (RSA). In this paper, we develop generalizations of the replica method adequate for calculating the properties of such systems. These are applied to RSA, resulting in both virial series and integral equations for the physical quantities describing this system. We extend the virial series for the adsorption rate to give excellent agreement with simulation results at all densities.

We consider the theoretical problem of transmission of a pulse of particles through a transversely infinite slab of thickness L, in which they travel at speed v and are isotropically and elastically scattered with mean-free path length λ. In the case λ/L=∞ there is no scattering, the pulse is transmitted intact, and the transmission-time moments are simply 〈${\mathit{t}}_{\mathrm{trans}}^{\mathit{n}}$〉=(L/v${)}^{\mathit{n}}$. However, we find that in the weak-scattering limit λ/L→∞, the transmission-time probability density develops a long tail resulting in anomalous scaling of the moments. The relevance of these results to recent experiments is discussed.

The recently observed kinetic effect [R. W. M. Hoogeveen and L. J. F. Hermans, Phys. Rev. Lett. 65, 1563 (1990)] due to a light-induced velocity-selective heating or cooling is incorporated into the standard kinetic theory based on a generalized Boltzmann equation. The measured pressure difference results from a spatially inhomogeneous heat flux analogous to the thermal pressure which Maxwell had anticipated for rather different experimental conditions.

This paper investigates the motion of a Brownian particle experiencing both a friction (biased) force and a randomly fluctuating force with a long-time-correlation function ${\mathit{C}}_{\mathit{f}}$(t)∼${\mathit{t}}^{\mathrm{-}\mathrm{\beta }}$, 0<β<1, 1<β<2, and β=1, instead of a Dirac δ function. The generalized Langevin equation and Fokker-Planck equation and corresponding solution are presented. It is shown that when 0<β<1 or 1<β<2, the diffusion motion of the Brownian particle is the anomalous diffusion that is related to fractal Brownian motion (FBM). But when β=1 the diffusion motion is anomalous diffusion with no connection to FBM. The effects of friction retardation result in a probability density function for finding the particle at displacement X at time t that depends on the initial value of velocity of the particle. The approach in this paper may provide a systematic method for the study of particles diffusing in fractal media.

The (2+1)-dimensional Burgers equation is obtained as the equation of motion governing the surface perturbations of a shallow viscous fluid heated from below, provided the Rayleigh number of the system satifies the condition R≠30. A solution to this equation is explicitly exhibited and it is argued that it describes the nonlinear evolution of a nearly one-dimensional kink.

We report scaling relations of thermal turbulence in cells of aspect ratio 0.5, 1.0, and 6.7.