We study unbiased random walks in discrete time n on a square lattice, in the form of a strip of finite width N in the y direction, with a family of boundary conditions parametrized by a stay probability Γ per time step at the edge sites. The diffusion coefficient K=${\lim }_{\mathit{n}\to \mathrm{\infty }}$〈${\mathit{X}}_{\mathit{n}}^2$〉/n is computed analytically to exhibit its dependence on N and Γ. The result is generalized to the case of a strip with side branches attached to the boundary sites to simulate the effect of rough edges. A further generalization is made to obtain K for a random walk in d dimensions on a lattice bounded in one of the directions. Thus, K serves as a probe of both the transverse size of the region in which diffusion takes place and the nature of the bounding surfaces.

A dynamically corrected variational transition-state theory is formulated for the thermally activated escape of a particle trapped in a potential well separated from a different well or continuum by a barrier and coupled to a heat bath. The theory is based on the Hamiltonian-equivalent formulation of the generalized Langevin equation. The dynamical corrections are obtained by utilizing the reactive-flux method in which the choice of dividing surface is guided by minimization of the transition-state flux. Analytic correction formulas, valid for memory friction, are obtained for the Kramers-Grote-Hynes estimate of the rate in the range from moderate friction to the large-friction limit. The analytic expansion is in terms of the inverse barrier height (1/β${\mathit{V}}^{\mathrm{‡}}$). For the special case of an extended Smoluchowski equation containing finite damping corrections, the exact expansion is also obtained using the mean-first-passage-time formulation. The dynamically corrected variational transition-state-theory expansion is shown to be identical to the mean-first-passage-time result.

It is shown how to include free energies not symmetric in the ordering field in a recent Riemannian geometric theory of critical phenomena [G. Ruppeiner, Phys. Rev. A 44, 3583 (1991)]. A mixing of coordinates scheme, as is conventionally used to deal with asymmetry, is proved to be consistent with the geometric theory. Furthermore, with scaled forms of the free energy and the assumption of universality, this appears to be the only scheme for including asymmetry in this theory which does not introduce a singularity near the critical isochore.

We study the two-dimensional dilute Bose gas using the ladder or T-matrix approximation, which is valid if ln(1/${\mathit{na}}^2$)≫1, and show that this approximation can describe also the critical region of the Kosterlitz-Thouless transition accurately. In particular, we consider the superfluid properties of the gas and find that the superfluid density shows the square-root cusp together with the universal jump at the critical temperature. The theory is applied to spin-polarized atomic hydrogen adsorbed on a superfluid helium film, since this appears to be one of the most promising experimental setups to achieve the Kosterlitz-Thouless transition in a weakly interacting Bose gas.

The Ziff-Gulari-Barshad model for the reaction: CO+O→${\mathrm{CO}}_2$ on a catalytic surface exhibits a first-order transition to a CO-poisoned state, as the CO adsorption rate is varied. If the model is modified to include desorption of CO at rate k, the first-order transition line terminates at a critical point ${\mathit{k}}_{\mathit{c}}$. Here we present detailed simulations of the critical behavior. The results support theoretical arguments placing the transition in the Ising-model universality class.

We have studied quadric type energy (frequency) recursion relations resulting from fractal lattices. We found that the allowed energy intervals in successive levels form two similar two-scale Cantor sets. If we imagine the iterating procedure as a dynamical process, the iterating results in different levels generate a ‘‘time’’ sequence, and we can introduce an equal probability measure ${\mathit{p}}_{\mathit{i}}$ on a Cantor set and construct, following Helsey et al. [Phys. Rev. A 33, 1141 (1986)], a partition function Γ(q,τ)= ${\mathit{tsum}}_{\mathit{i}}$${\mathit{p}}_{\mathit{i}}^{\mathit{q}}$/${\mathit{l}}_{\mathit{i}}^{\mathrm{\tau }}$; finally, we obtain ${\mathit{D}}_{\mathit{q}\mathrm{-}}$q and f(α)-α curves. A number of exactly soluable examples are investigated.

An exact solution to the Laplace equation with Dirichlet boundary conditions on a simple boundary is used in an iterative fashion to study the case of a stochastic rough surface with fractal dimension D=2+ε. For small ε, an analytic expression is derived for the spectrum of singularities f(α) of the gradient of the potential normal to the boundary, using the random multiplier approach. A binomial approximation to the multiplicative process is shown to severely underestimate f(α) for low values of α.

A Hamiltonian system of two coupled oscillators, a linear and a nonlinear, is investigated analytically and numerically in the case when the frequency of the linear oscillator is allowed to vary slowly in time. We focus mainly on the dynamic-autoresonance (DAR) effect, i.e., a persisting phase locking between the two oscillators, accompanied by a large, though slow, energy exchange between them. The two-oscillator model is motivated by the necessity, in some applications, of taking into account the energy depletion of the driving field. The theory of the internal first-order resonances in two-dimensional Hamiltonian systems is generalized to the case of a slow parameter variation. Three constraints for the existence of DAR are found, and the theory is verified numerically. Also, several routes from order to global chaos and from global chaos to order are observed numerically.

We explore the connection between the two perturbation methods that have been suggested for solving Smoluchowski-like equations with weak noise. A slight modification of the first of these methods (the size expansion of van Kampen) makes it identical to the second (the method of running coordinates). We point out that a number of modifications to both methods are possible and should be explored. We illustrate the advantages of both of these methods as applied to three solvable examples.

We study the ferromagnetic phase transition in a model consisting of one-dimensional three-state Potts spin chains with random intrachain couplings and comparatively weak ferromagnetic interchain couplings. Mean-field theory is employed to decouple approximately the interchain couplings. The transfer-matrix method is then used to study the resulting effectively one-dimensional random-bond Potts model. The free energy as a function of the ferromagnetic order is calculated numerically and a first-order ferromagnetic-paramagnetic phase transition is found over a wide range of degree of randomness.

We show a method to extract the quantum Poincaré section corresponding to an eigenstate of a two-dimensional billiard. This quantum Poincaré section is given in terms of the Birkhoff variables of the problem.

1/${\mathit{f}}^{\mathit{D}}$ noise occurs in an impressive variety of physical systems, and numerous complex theories have been proposed to explain it. We construct two relatively simple renewal processes whose power spectral densities vary as 1/${\mathit{f}}^{\mathit{D}}$: (i) a standard renewal point process, with 0<D<1, and (ii) a finite-valued alternating renewal process, with 0<D<2. The resulting event number statistics, coincidence rates, minimal coverings, and autocorrelation functions are shown also to follow power-law forms. These fractal characteristics derive from interevent-time probability density functions which themselves decay in a power-law fashion. A number of applications are considered: trapping in amorphous semiconductors, electronic burst noise, movement in systems with fractal boundaries, the digital generation of 1/${\mathit{f}}^{\mathit{D}}$ noise, and ionic currents in cell membranes.

A renormalization-group analysis is carried out of the long-time behavior of random walks in a one-dimensional lattice consisting of three kinds of bonds generated by a deterministic inflation rule. The random walker is subject to a local drift force, depending on the kind of bond it traverses. The mean-square displacement after time t is found to scale as ${\mathit{t}}^{2\nu }$ where the exponent ν is nonuniversal and depends continuously on the magnitude of the drift force. The model studied is one of a general class of systems that exhibits similar behavior.

This Rapid Communication argues that the strange attractor recently found in the context of the modulated Stokes wave [H. T. Moon, Phys. Fluids A 3, 2709 (1991)] is a universal form and indeed the asymmetric counterpart of the Lorentz attractor. Furthermore, this form appears to reveal another class of universality in that it follows the evolution pattern set by the tent map. This is in marked contrast to the Rössler attractor, which follows the universal sequence of the logistic map.

We prove that the periodic-orbit counting function, a measure of the rate of proliferation of periodic orbits, for a barrier billiard and the π/3-rhombus billiard is of the form ${\mathit{ax}}^2$+bx+c, where x is the length (equivalently, period) up to which periodic orbits are counted and a,b,c are system-specific constants. The generality of our arguments strongly suggests that the law of proliferation given here is a representation of general truth about two-dimensional plane-polygonal billiards.

We present a recursive proportional-feedback (RPF) algorithm for controlling deterministic chaos. The algorithm is an adaptation of the method of Dressler and Nitsche [Phys. Rev. Lett. 68, 1 (1992)] to highly dissipative systems with a dynamics that shows a nearly one-dimensional return map of a single variable X measured at each Poincaré cycle. The result extends the usefulness of simple proportional-feedback control algorithms. The change in control parameter prescribed for the nth Poincaré cycle by the RPF algorithm is given by δ${\mathit{p}}_{\mathit{n}}$=K(${\mathit{X}}_{\mathit{n}}$-${\mathit{X}}_{\mathit{F}}$)+Rδ${\mathit{p}}_{\mathit{n}\mathrm{-}1}$, where ${\mathit{X}}_{\mathit{F}}$ is the fixed point of the target orbit, and K and R are proportionality constants. The recursive term is shown to arise fundamentally because, in general, the Poincaré section of the attractor near ${\mathit{X}}_{\mathit{F}}$ will change position in phase space as small changes are made in the control parameter. We show how to obtain K and R from simple measurements of the return map without any prior knowledge of the system dynamics and report the successful application of the RPF algorithm to model systems from chemistry and biology where the recursive term is necessary to achieve control.

We contrast the effects of uncorrelated power-law noise and linearly correlated Gaussian noise upon the anomalous wandering of directed polymers in random media. In the first instance, we explore the role of the noise on the morphology of the ultrametric tree structure of the ensemble of locally optimal paths, and then, motivated by the work of Zhang [Phys. Rev. Lett. 59, 2125 (1987)] and more recently by Perlsman and Schwartz [Europhys. Lett. 17, 11 (1992)], provide strong evidence for the universality of the model’s river basin patterns. Lastly, we discuss those precise features of the positional and energy fluctuations that could permit a resourceful experimentalist to discern the noise distribution underlying anomalous roughening recently observed in fire fronts, bacterial growth, and fluid flow through porous media.

In an experiment on the parametric excitation of capillary waves (the Faraday instability) we have observed a stable standing-wave pattern with twelvefold orientational order. This ‘‘quasipattern’’ is analogous to a two-dimensional quasicrystal, but it occurs in a macroscopic nonequilibrium fluid-dynamical system. It is observed in containers whose side walls are of various shapes including an irregular shape, and thus it cannot be ascribed to side-wall boundary effects.

We report results from a simulated annealing study of a coarse-grained model for block copolymers. For sufficiently slow cooling rate, the system develops a well-defined lamellar structure by avoiding the formation of defects. We show that the strong segregation limit of the model is computationally accessible when the interface thickness is small enough.

The Boussinesq equation is known to be the fundamental nonlinear model, describing wave propagation in a weakly dispersive nonlinear medium. In this paper, we prove the existence of the collapse dynamics for the two basic forms of the Boussinesq equation in the case of the periodic boundary conditions. The sufficient criterion of the blow-up is found analytically.