Stochastic resonance has been examined in a number of systems that have steady states. We examine a dynamical system that allows for the escape of a particle from a potential well, and that is subjected to the combination of a periodic forcing term and noise. We show that in such a system the addition of a small amount of noise can actually increase the average time for the particle to escape from the well. This effect depends on the frequency of the external field and in this sense is a form of stochastic resonance.

We consider the effect of quantum-nondemolition monitoring of the photon number on the quantum suppression of classical chaos in a recently proposed quantum optical model, the parametrically kicked nonlinear oscillator [G. J. Milburn, Phys. Rev. A 41, 6567 (1990)]. Classically the effect of the quantum-nondemolition measurement is equivalent to a phase diffusion in the phase plane of the oscillator. A similar result holds in the quantum description, but in addition the measurement rapidly diagonalizes the system density operator in the photon-number basis. This has the effect of causing the evolution of the quantum moments to approach the corresponding classical moments and thus restores the classical dynamics for practical purposes.

Dynamical zeta functions, defined as Euler products over classical periodic orbits, have recently received enhanced attention as an important tool for the quantization of chaos. Their representation as a Dirichlet series over pseudo-orbits has proven to be particularly useful, since these series seem to possess in the general case much better convergence properties than the original Euler product. The convergence of the Dirichlet series depends crucially on the asymptotic distribution of the pseudo-orbits and thus on the ergodicity of the underlying dynamical system. It is shown that the lengths ${\mathit{l}}_{\mathit{n}}$ (or rather exp${\mathit{l}}_{\mathit{n}}$) of the classical periodic orbits play mathematically the role of generalized prime numbers. Based on the theory of Beurling’s generalized prime numbers, we derive an exact law for the proliferation of psuedo-orbits for the Hadamard-Gutzwiller model, which is one of the main testing grounds of our ideas about quantum chaos. The strength of growth of the pseudo-orbits is determined by the ratio ZETA(2)/ZETA’(1), where ZETA(s) denotes the Selberg zeta function. Two explicit, complementary representations are given that allow the computation of this ratio solely from the length spectrum {${\mathit{l}}_{\mathit{n}}$} of the classical periodic orbits, or from the quantal energy spectrum {${\mathit{E}}_{\mathit{n}}$}. One of these relations depends exponentially on the generalized Euler constant ${\gamma }_{\mathrm{\Delta }}$, which is therefore also studied. The formulas are applied to two strongly chaotic systems. It turns out that our asymptotic law describes the mean proliferation of pseudo-orbits very well not only in the asymptotic region, but also surprisingly well down to the shortest pseudo-orbit.

We generalize open random aggregating systems to the case of exponential growth of each aggregating unit. A statistically stationary state is confirmed under the uniform injection both numerically and theoretically. The steady-state mass distribution shows two different characteristic ranges: a small-mass range where the random aggregating process dominates, and a large-mass range where exponential growth rules the system. In both ranges, mass distributions follow power laws. In the large-mass range, the distribution converging asymptotically to P(m)∝${\mathit{m}}^{\mathrm{-}1}$ is observed universally.

We consider the effect of periodic perturbations on a dynamical system tuned near the onset of a period-doubling bifurcation, by studying a two-dimensional iterative map. In contrast to earlier work, we focus on the response at the idler frequency. We derive simple scaling laws, valid in the weakly damped limit, which can be readily tested by experiments. The results agree with numerical simulations of a Josephson-junction circuit.

A constant (N,P,T) statistical method is developed for a finite-size system. A scaling variable a is introduced to describe the size of the system following the method used for the bulk system. The histograms of the Boltzmann distribution function f(${\mathit{E}}_{\mathit{c}}$,V;T,P) (${\mathit{E}}_{\mathit{c}}$ and V being the configuration energy and the volume) for a 55-atom cluster bound by Lennard-Jones pair potentials are calculated at several (T,P) values by Monte Carlo (MC) simulations. From the density of states Ω(${\mathit{E}}_{\mathit{c}}$,V) constructed from the MC results, important thermodynamical quantities are then obtained. In the phase-transition region, f(${\mathit{E}}_{\mathit{c}}$,V;T,P) shows a bimodal distribution on the (${\mathit{E}}_{\mathit{c}}$,V) plane indicated by a ‘‘twin-peak’’ structure. The full phase equilibrium, including volume or pressure changes, of a cluster is explored in a systematic manner, and thus a complete picture of the phase diagram of a cluster is presented.

Upper and lower bounds are derived for the Rényi entropies in terms of the properties of periodic orbits. The bounds are valid for a broad class of chaotic systems and are applied as criteria for the existence of dynamical phase transitions associated with an anomalous scaling of the probability of observing orbits with certain symbolic sequences. The invariance of these criteria under singular coordinate transformations is emphasized and demonstrated in one-dimensional examples.

By obtaining analytically the length spectrum of periodic orbits, a semiclassical analysis is carried out for the 1/3π-rhombus billiard system, which is the simplest possible example of a class of the almost-integrable billiard system. A complete set of length spectra enables us to examine the difference of the semiclassical density of states between the almost-integrable (even-parity states) and completely integrable (odd-parity states) case. Within the framework of the ordinary periodic orbit expansion including only families of periodic orbits, the universality of the quadratic long-range level statistics for the present almost-integrable billiard system cannot be discriminated qualitatively from that for the corresponding integrable billiard system. Nevertheless, an extensive numerical study for the nearest-neighbor level-spacing distribution and the spectral rigidity demonstrates that there exists level repulsion in the almost-integrable case.

The three-particle (ppp) propagator, which is that component of the three-particle Green’s function describing triple ionization, is investigated. In particular, a second-order approximation scheme for this propagator is presented, suitable for the calculation of triply charged electronic states. Two different methods are used to derive the explicit working equations. The first method is purely algebraic and is based on the connection of propagators and effective Hamiltonians. The second is the well-established algebraic-diagrammatic-construction method, which makes use of Feynman diagrams. These methods lead to equivalent working equations represented in different forms.

A description of noise-induced reaction rates in a fluctuating medium is presented. The environment is assumed to exist in a number of distinct configurations and transitions between these configurations are governed by a Markov process. The environment fluctuations drive transitions between reactant and product regions described by a set of generalized coordinates. General expressions for the rate coefficients are presented and illustrated for a reaction driven by dichotomous switching between fixed-point and limit-cycle states.

The failure of the one-dimensional gravitational system to relax to equilibrium on predicted time scales has raised questions concerning the ergodic properties of the dynamics. A failure to approach equilibrium could be caused by the segmentation of phase space into isolated regions from which the system cannot escape. In general, each region may have distinct ergodic properties. By numerically investigating the stability of two classes of periodic orbits for the N-body system, we have unequivocally demonstrated that stable regions in the phase space exist for N≤10. For populations 11≤N≤20 we find numerical evidence for multiple, chaotic, invariant regions. Thus the failure of large systems (say, N≥100) to equilibrate may be a result of microscopic dynamical restrictions, rather than imposed macroscopic contraints.

We approximate the noisy Burgers equation by the single-step model, alias the asymmetric simple exclusion process. The generator of the corresponding master equation is identical to the ferromagnetic Heisenberg spin chain with a purely imaginary XY coupling. We Bethe diagonalize this nonsymmetric Hamiltonian. We show that the gap between the ground state and first excited state scales as ${\mathit{N}}^{\mathrm{-}3/2}$ for large system size N. The gap between the largest and next-largest eigenvalue scales as ${\mathit{N}}^{\mathrm{-}1}$. This property hints at conformal invariance. We also explain the connection to the six-vertex model.

We study both analytically and numerically the A+B→C reaction-diffusion system for the case of initially separated reactants. We find the functional forms of the reaction front and also of the concentration profiles of the reactants. The profile of the production rate R(x,t) at site x and time t scales as ${}_{}{}^{\mathit{t}\mathrm{-}2/3}{}_{}{}^{}$(x/${\mathit{t}}^{1/6}$${)}^{3/4}$exp[-2/3 (λx/${\mathit{t}}^{1/6}$${)}^{3/2}$] for the spatial range ${\mathit{t}}^{1/6}$≪x≪${\mathit{t}}^{1/2}$. Our results are supported by numerical solutions of the reaction-diffusion equation, and are in agreement with Monte Carlo simulations in dimension d=2.

Particles released from a target surface by laser-pulse bombardment normally collide with each other and, as a result, exhibit three limiting categories of gas-dynamic behavior. One possibility, (a), is that the particles pass directly into unsteady adiabatic expansion (UAE), a problem for which an analytical solution already exists. Alternatively, (b), the particles first form a Knudsen layer (KL), then pass into UAE, and are not subject to recondensing at the target surface if the release pulse terminates, or else, (c), the particles go through the same sequence of KL formation and UAE but are subject to recondensing. Closely related to (b) is the behavior of particles that escape into vacuum from a nozzle. We here present an analytical, one-dimensional, continuum solution for case (b) when the release is pulsed. The KL (like the throat of the nozzle) is treated as a boundary condition with a nonzero flow velocity. During the release pulse the solutions take on our previously derived forms for a planar UAE from an infinite reservoir. When the pulse ends (or the nozzle closes), there is an abrupt change of boundary condition from finite to zero flow velocity and from high to intermediate sound speed (thence density); at the same time the flow pattern breaks up into three regions. These analytical results are finally compared with the numerical solution of the flow equations by Knight [AIAA J. 20, 950 (1982)], with Monte Carlo solution of the Boltzmann equation by Sibold and Urbassek [Phys. Rev. A 43, 6722 (1991)], and with the explicit nanosecond-time-scale photography of Braren, Casey, and Kelly [Nucl. Instrum. Methods B 58, 463 (1991)]. The comparison with the Boltzmann equation is particularly important, as the generally good agreement with the flow equations suggests the latter to be useful in spite of being founded on the assumption of persistent local equilibrium.

The kinetic theory for tagged-particle problems in lattice-gas cellular automata is extended beyond Boltzmann’s mean-field approximation by including correlated ring-type collisions. This theory provides explicit expressions for the velocity autocorrelation function (VACF) for all times in terms of the ring-collision integral, as well as corrections to the Boltzmann values of the transport coefficients. For times long compared to the mean free time, the ring integral equation yields the phenomenological mode-coupling theory and the long-time tails. For intermediate times it describes a slow transition from initial exponential decay to the long-time tails. At short times the ring kinetic theory is exact. In particular, deviations from the Boltzmann result in the VACF of three-dimensional systems after two time steps are calculated explicitly and compared with computer simulations.

We show that the nonlinear phase equation that applies to propagating patterns allows for a large range of parameter values for propagating confined states for which a spatially localized region with wavelengths different from that of the background travels on this background. This phenomenon is the generalization of the stationary confined states predicted a few years ago by the authors, which have since been seen experimentally in various systems. We suggest that the propagating confined states found here could arise in spirals in Taylor vortex flow or in convective systems showing traveling waves far above onset. We find that the propagating confined states can be replaced by a pattern that is irregular in space and time as the control parameter in the nonlinear phase equation is varied.

We derive a Green-Kubo relation for the mutual diffusion tensor (MDT) of a binary fluid mixture subject to a strong Couette shear field. We calculate the MDT of a Lennard-Jones (LJ) mixture and the self-diffusion tensor (SDT) of a pure LJ fluid as functions of state point and strain rate. The qualitative dependence of the diffusion tensors upon these variables is explained in greater detail by analyzing the velocity and mass current autocorrelation functions. Finally, we compare the zero-color field color conductivity with the SDT. For nonzero shear rates we find that the color conductivity and the SDT are different, confirming the prediction by Evans, Baranyai, and Sarman [Mol. Phys. (to be published)], according to which these two transport coefficients should only be related at zero shear rate.

Rayleigh-Bénard flow for Rayleigh numbers Ra=${10}^8$–${10}^{11}$ (hard turbulence regime) is studied solving the Boussinesq equations with the Fourier-Weierstrass ansatz introduced recently [Eggers and Grossmann, Phys. Fluids A 3, 1958 (1991)]. The plumes and swirls detaching from the boundary layers are mimicked by volume stirring on all scales down to the Rayleigh-number-dependent scale of these thermals. Wave-number spectra, frequency spectra, and structure functions are presented both for velocity and temperature fluctuations. As the scale decreases the velocity-temperature cross correlation decreases much faster than both velocity and temperature autocorrelations. In the viscous subrange all wave-number spectra decay exponentially. Based on the experimental Ra dependence of the mean temperature fluctuations we can calculate the Ra dependence of the mean velocity fluctuations as well as of the mean temperature and velocity time derivatives. The inner length scale η is found to scale ∝${\mathit{Ra}}^{\mathrm{-}0.32\pm 0.01}$.

The stochastic Swift-Hohenberg equation is studied as a model of Rayleigh-Bénard convection in a simple fluid. The equation has been solved numerically in two spatial dimensions to obtain the convective heat current and the roll pattern when either a bulk stochastic forcing field or different models of thermal-diffusivity mismatch at the sidewalls of the convective cell are considered. The parameters that enter the equation have been chosen to match the ramping experiments on Rayleigh-Bénard convection by Meyer, Ahlers, and Cannell [Phys. Rev. Lett. 59, 1577 (1987)]. For any combination of forcing mechanisms, we are able to find values of their various amplitudes that lead to excellent fits to the experimentally measured convective current. In the case of a bulk random forcing field, we find an amplitude of ${\mathit{F}}_1$=5×${10}^{\mathrm{-}5}$, compared to ${\mathit{F}}_{\mathrm{th}}$=1.92×${10}^{\mathrm{-}9}$, the value obtained from fluctuation theory. A random, cellular pattern of rolls is observed, in agreement with experiments involving a gel sidewall designed to eliminate the influence of the sidewalls on the onset of convection. A thermal-conductivity mismatch at the sidewall has also been modeled by a variety of forcing fields. In all cases a roll-like pattern that reflects the geometry of the sidewalls is observed. Different combinations of both types of forcing fields have also been studied and found to yield patterns intermediate between cellular and roll-like, while yielding a very reasonable fit to the convective heat current measured experimentally.

A small-angle x-ray scattering (SAXS) study of dense AOT-water-decane microemulsions [AOT denotes sodium bis(2-ethylhexyl) sulfosuccinate] was undertaken in order to delineate clearly the phase behavior and corresponding structural transitions for AOT-plus-water volume fractions ranging from φ=0.60 to 0.95. Spectra were collected for temperatures between T=3 and 65 °C. The resulting T-vs-φ phase diagram indicates three distinct structural domains when the water-to-AOT molar ratio is fixed at W=40.8, namely, the previously investigated ${\mathit{L}}_2$ droplet phase, a high-temperature ${\mathit{L}}_{\mathrm{\alpha }}$ lamellar phase, and a low-temperature ${\mathit{L}}_3$ phase consisting of randomly connected lamellar sheets. A significantly wide coexistence region accompanies the droplet-to-lamellar phase transition, which is demonstrated to be first order. For W between 15 and 40, an analysis of the lamellar structure using a one-dimensional paracrystal model produces a Hosemann g factor indicative of an approximately constant variation in the lamellar spacing of about 8%. The SAXS study was supplemented by dielectric-relaxation, shear-viscosity, and quasielastic light-scattering measurements in order to substantiate the observed phase transitions and further our understanding of the structural and dynamical properties of the ${\mathit{L}}_3$ phase. It was found that the ${\mathit{L}}_3$ phase exhibits Newtonian behavior up to a shear rate of 790 ${\mathrm{s}}^{\mathrm{-}1}$, in contradiction to previous theoretical considerations. The phase exhibits two distinct relaxation modes. A relaxation time of ∼1 ms characterizes the Brownian motion of a single lamellar sheet, while the motion of the entire interconnected sheet assembly has a relaxation time on the order of 1 s.