Average switching time and its variance in a generic bistable system are studied in the transition regime from dominantly stochastic to dominantly deterministic behavior. Our experimental and numerical results are in excellent agreement with recent theoretical predictions, and a coherent picture of the physical mechanism emerges.

We describe one-dimensional discrete diffusive motion that is driven by a purely deterministic system, the logistic map. We find normal diffusive motion above a critical value ${\lambda }_1^{\mathrm{*}}$ of the chaotic parameter. At ${\lambda }_1^{\mathrm{*}}$, the diffusion constant D plays the role of an order parameter, with a critical exponent of 1/2. In the presence of external noise, D is expressed in terms of a universal scaling function and the critical exponent associated with the noise is found to be 1. We also consider biased diffusive motion driven by the logistic map. We find that the diffusion process thus generated is different from random biased diffusion. In contrast to the case of the biased ‘‘random walker’’ where all configurations (‘‘walks’’) are still possible, here certain configurations are not allowed. However, the statistical properties approach those for random biased diffusion in the infinite time limit. We define a measure of the effective randomness.

The stochastic differential equation ${\mathrm{\tau }}_0$${\mathrm{\partial }}_{\mathit{t}}$A=ε(t)A-${\mathit{g}}_3$${\mathit{A}}^3$+f¯(t), where f¯(t) is Gaussian white noise, is studied for arbitrary time dependence of ε(t). In particular, cases are considered where ε(t) goes through the bifurcation of the deterministic system, which occurs at ε=0. In the limit of weak noise an approximate analytic expression generalizing earlier work of Suzuki [Phys. Lett. A 67, 339 (1978); Prog. Theor. Phys. (Kyoto) Suppl. 64, 402 (1978)] is obtained for the time-dependent distribution function P(A,t). The results compare favorably with a numerical simulation of the stochastic equation for the case of a linear ramp (both increasing and decreasing) and for a periodic time dependence of ε(t). The procedure can be generalized to an arbitrary deterministic part ${\mathrm{\partial }}_{\mathit{t}}$A=D(A,t)+f¯(t), but the deterministic equation may then have to be solved numerically.

By using a path-integral approach we have studied the asymptotic probability distribution of a periodically swept supercritical bifurcation. The steepest-descent approximation has been used with the corresponding time-dependent Onsager-Machlup-Lagrangian of the Fokker-Planck-Equation. We prove by using the Lyapunov function the uniqueness of the asymptotic time-periodic probability distribution for periodically forced Markov processes; then the mixing property for these types of stochastic processes is proved. An iterative matrix procedure is introduced to calculate the long-time behavior of the probability distribution. Monte Carlo simulations were performed in order to show the agreement between the path-integral approach and the numerical solution of the corresponding periodically forced stochastic differential equation. A discussion on the problem of calculating the weak-noise Graham-Tel invariant measure is presented.

The distance between the nearest diffusing particle to a single trap has been shown to be a useful characterization of self-segregation in low-dimensional reaction kinetics. Recent studies in two dimensions show that the average of this distance increases asymptotically as (ln t${)}^{1/2}$. In this paper we study a two-dimensional system in which the trap is an imperfect one, modeled in terms of the radiation boundary condition. Our exact solution shows that there exists a spatial and temporal dependence on the trap absorptivity, but the asymptotic time dependence remains unchanged. These results enable us to follow the crossover between the two limiting cases, namely, perfect trapping and total reflection. Analytical expressions are also given for the concentration profile and the reaction rate at the trap.

The problem of chaotic behavior in quantum mechanics is investigated against the background of the theory of quantum-nondemolition (QND) measurements. The analysis is based on two relevant features: The outcomes of a sequence of QND measurements are unambiguously predictable, and these measurements actually can be performed on one single system without perturbing its time evolution. Consequently, QND measurements represent an appropriate framework to analyze the conditions for the occurrence of ‘‘deterministic randomness’’ in quantum systems. The general arguments are illustrated by a discussion of a quantum system with a time evolution that possesses nonvanishing algorithmic complexity.

We present a systematic derivation of the percolation thresholds in anisotropic systems composed of permeable elongated boxes. The analytic calculation is based on an order-by-order diagrammatic expansion of the pair-connectedness function. A comparison of the results with those of Monte Carlo simulations shows excellent quantitative agreement. We conclude that, of the analytic methods suggested thus far, the present approach is the most suitable one for quantitative derivation of system properties in continuum percolation.

Statistical properties are studied numerically for a cellular-automaton version of the dynamical stick-slip model for earthquakes. The celluar-automaton model studied is deterministic and spatially uniform and is designed to reproduce most of the properties of the original dynamical model studied recently by Carlson and Langer [Phys. Rev. Lett. 62, 2632 (1989); Phys. Rev. A 40, 6470 (1989)]. Behavior of the large system with many blocks changes distinctively as parameters of the model change, and the changes seem to correspond to those of the small system with three blocks, but the deviation by small perturbations suggests that there are differences in behavior between the large and small systems.

In this paper we show how an extension of the nonequilibrium-statistical-operator method, relying upon the maximum-entropy principle set up by Jaynes [Am. J. Phys. 33, 391 (1965)], may be used to describe the time evolution of an arbitrary many-body system. The Gibbs space of the observables describing the macrostates of the system is extended to include not only the conserved variables, but additional ones whose origin is directly related to the microscopic nature of the system manifested in its Hamiltonian. This allows us to go beyond linear irreversible thermodynamics and enter into the domain of what is now known as extended irreversible thermodynamics (EIT). Transport equations for the extended basic set of macrovariables are derived, showing that the Maxwell-Cattaneo-Vernotte equations of EIT are obtained. The relaxation times and transport coefficients contained therein can be calculated from the microscopic dynamics of the system averaged over an appropriate nonequilibrium coarse-grained probability density. Other outstanding features of the methods are emphasized and related to already-established results for nonequilibrium systems.

The general theory described in the preceding article [Phys. Rev. A 43, 6622 (1991)] based on the nonequilibrium-statistical-operator method, which provides mechano-statistical foundations for phenomenological irreversible thermodynamics, is applied to a specific problem. This is the case of a highly excited plasma in a semiconductor, where fluxes of mass and energy naturally appear, as well as other higher-order fluxes, as basic variables necessary for the description of the macroscopic state of the system. A criterion for the truncation of the basic set of variables is presented. The equations of motion for the macrovariables are derived for the case of a simple model. They have the structure of nonlinear and nonlocal transport equations, which fit into a natural extension of those of linear irreversible thermodynamics. In particular, Maxwell-Cattaneo-Vernotte-type equations of extended irreversible thermodynamics are recovered, having relaxation times and transport coefficients that may be calculated from the microscopic dynamics of the system composed of averages over the nonequilibrium ensemble.

A two-dimensional Monte Carlo simulation method based on the NpT ensemble and the Voronoi tesselation, which was previously developed for single-species hard-disk systems, is extended, along with a version of scaled-particle theory, to many-component mixtures. These systems are unusual in the sense that their composition is not fixed, but rather determined by a set of internal degeneracies assigned to the differently sized hard disks, where the larger disks have the higher degeneracies. Such systems are models of monolayers of molecules with internal degrees of freedom. The combined set of translational and internal degrees of freedom leads to a rich phase structure that includes solid-liquid transitions (governed by the translational variables) as well as transitions involving changes in average disk size (governed by the internal variables). The relationship between these two types of transitions is studied by the method in the case of a binary mixture, and results are presented for varying disk-size ratios and degeneracies. The results are also compared with the predictions of the extended scaled-particle theory. Applications of the model are discussed in relation to lipid monolayers spread on air-water interfaces, and it is concluded, by comparison with experiments, that the hard-disk mixture is an excellent candidate for a minimal model of lipid-monolayer phase behavior.

In this paper we study three fundamental questions: the characterization of the nonequilibrium states of a macroscopic system using exclusively the information that is controllable by the experimenter through the constraints imposed on the system, the construction of a variational method generating a probability distribution function in which the information available to the experimenter is contained, and finally the study of the time behavior of a functional, here referred to as the Shannon-Jaynes entropy, in order to examine the existence of a criterion for irreversibility. It is shown that the answer to the first two questions is provided by a theorem establishing the equivalence between the coarse-graining operation in phase space as suggested by Ehrenfest and the action of a projection operator, in fact, Zwanzig’s projector, acting on the full probability distribution function containing all the information available when the system is initially prepared. The third question is partially answered by deriving an inequality characterizing the fact that every time the system is observed, information is lost. Yet the full proof establishing the equivalent of Boltzmann’s H theorem is only qualitatively analyzed. The connection between this work and a similar one carried out from a macroscopic point of view is also established.

A closed, macroscopic description of the dynamics of an assembly of electrons under the influence of a space-time-varying field is presented. This is accomplished by approximating the phase-space distribution function in an equivalent space whose coordinates are the infinite set of moments, {${\mathit{m}}_{\mathit{j}}$;j=1,...,∞}. An attempt has been made to formulate a rigorous procedure for obtaining approximate distributions with varying degrees of space-time resolution and for establishing their range of validity. Expressions are presented for the approximate distribution function and the transport parameters with resolution corresponding to the scale for energy relaxation.

The motion of the interface between two fluids in a quasi-two-dimensional geometry is studied via simulations. We consider the case in which a zero-viscosity fluid displaces one with finite viscosity and compare the interfaces that arise with zero surface tension with those that occur when the surface tension is not zero. The interface dynamics can be analyzed in terms of a complex analytic function that maps the unit circle into the interface between the fluids. The physical region of the domain is the exterior of the circle, which then maps into the region occupied by the more viscous fluid. In this physical region, the mapping is analytic and its derivative is never zero. This paper focuses upon the determination of the nature of the interface and the positions of the singularities of the derivative of the mapping function g. Two kinds of initial conditions are considered: case A, in which the singularities closest to the unit circle are poles; and case B, in which the t=0 interface is described by a function g with only zeros inside the unit circle. In either case, different behaviors are found for relatively smaller and larger surface tensions. In case A, when the surface tension is relatively small, the problem is qualitatively similar with and without surface tension: the singularities move outward and asymptotically approach the unit circle. For relatively large surface tension, the singularities, still polelike, move towards the center of the unit circle instead. In case B, for zero surface tension, the zeros move outward and hit the unit circle after a finite time, whereupon the solution breaks down. For finite but relatively small surface tension, each initial zero disappears and is replaced by a pair of polelike excitations that seem to approach the unit circle asymptotically, while for a relatively large surface tension, each initial zero is replaced by a polelike singularity that then moves towards the unit circle.

We experimentally examine the hypothesis that the scaling of the Nusselt number Nu (nondimensional heat flux) with Rayleigh number Ra in turbulent thermal convection is affected by large-scale flows near and within the boundary layers. The natural flows are enhanced artificially by a moving layer of mercury beneath the convecting fluid. Recirculating large-scale flows are thereby induced near the lower thermal boundary layer, and Nu is enhanced by as much as 70%. The response to forcing allows a determination of the large-scale flows needed to account for the observed scaling of Nu with Ra in the unperturbed situation. The required flows are found to be comparable to those present naturally, about 1.5 mm/s at Ra=${10}^8$ and a Prandtl number near 5. The experiments thus indicate that Nu is at least largely determined by a balance between diffusion of heat into the boundary layers and advection from them by large-scale flows. In an appendix, the form of the temperature probability distribution in the interior is shown to be related to the coherence of the thermal plumes.

Linear hydrodynamics yields the onset of convection for ascending autocatalytic reaction fronts in laterally bounded geometries. The system is studied in the limit of infinite and zero thermal diffusivity. For convection in a vertical slab of thickness a or a long vertical cylinder of radius a, the appropriate dimensionless driving parameter scrS=δ${\mathit{ga}}^3$/ν${\mathit{D}}_{\mathit{c}}$ involves the fractional density difference δ between the unreacted and reacted fluids, the acceleration of gravity g, the kinematic viscosity ν, and the catalyst molecular diffusivity ${\mathit{D}}_{\mathit{c}}$. Calculated critical values ${\mathit{scrS}}_{\mathit{c}}$ for onset of convection agree with recent experiments on iodate–arsenous acid systems. We also show that, compared with the unbounded system, the sidewalls tend to suppress convection.

In a variety of one-dimensional nonequilibrium systems, there exist uniform states that may undergo bifurcations to spatially periodic states. The long-wavelength dynamics of these spatial patterns, such as an array of convective rolls or a driven interface between two thermodynamic phases, can often be derived on the basis of the symmetries of the physical system. Secondary bifurcations of these patterns may be associated with the subsequent loss of remaining symmetries. Here, we study the transition that results from the loss of reflection symmetry (parity) and show that observations made in several recent experiments appear to be signatures of this bifurcation. Most of the common features seen in the disparate experiments follow from the simplest Ginzburg-Landau equations covariant under the remaining symmetries. It is shown that nucleated localized regions of broken parity travel in a direction determined by the sense of the asymmetry, and the passage of a localized inclusion of broken parity leads to a change in the wavelength of the underlying modulated state, and leaves the system closer to an invariant wavelength. Such behavior is in close correspondence with the properties of ‘‘solitary modes’’ seen in experiment. When a system supports extended regions of broken parity, a boundary between those of opposite parity can be considered as a source or sink of asymmetric cells and a ‘‘spatiotemporal grain boundary.’’ The creation or destruction of cells at the interface is reminiscent of ‘‘phase slip centers’’ in one-dimensional superconductors. The simplest dynamics consistent with the symmetries are identical to those of the time-dependent Ginzburg-Landau equation for a superconductor in an applied electric field. The experimental observation of approximate length subtraction of colliding regions of broken parity follows from this analogy.

The one-dimensional flow of particles thermally desorbed from a plane surface into vacuum is studied on the basis of a Monte Carlo simulation of the Boltzmann equation. It is assumed that particles desorb during a finite period of time with a fixed temperature. It is shown then that the flow is determined by a single parameter, which is essentially the number of monolayers Θ desorbed, and is inverse to the Knudsen number of the problem. The cases of negligible (Θ≪1) and in part also of very intense (Θ≫1) desorption fluxes can be treated analytically using collision-free flow and ideal gas dynamics, respectively. Here the simulation yields very good agreement with theory, indicating that the code can be used on a wide range of Knudsen numbers. For the gas-dynamical case, a Knudsen layer is formed by the equilibration of the flow in the vicinity of the surface; it is studied in some detail. Simulation of flows with Θ≲1 shows that the particle distribution is far from thermal equilibrium everywhere in the flow, and deviates strongly from the analytically accessible cases. For stronger desorption fluxes, the flow is partially equilibrated, such that the velocity component in the direction of the flow reaches thermal equilibrium, although with a smaller temperature than the velocity component perpendicular to the flow. The time integrated velocity spectra of particles measured far away from the surface are discussed for moderate desorption fluxes. They are surprisingly well described by a thermal distribution, even though no Knudsen layer forms in this case.

A method of modeling isotropic turbulence with a passive scalar is presented that can derive the high-order temperature structure functions with reasonable asymptotic behavior. As a necessary tool for this purpose, a theory of joint intermittency exponents of the dissipation of turbulent energy and scalar variance is developed, which is led to be equivalent to the joint multifractal formalism dealing with these two quantities simultaneously.

We examine sound propagation in two-phase states of one- and two-component fluids by taking into account heat and mass transport between the two phases. For the sake of simplicity, detailed calculations are performed on near-critical fluids undergoing nucleation or spinodal decomposition, which exhibit very large acoustic anomalies at relatively low frequencies. As a universal relation the zero-frequency sound speed is reduced to 82% of the sound speed without domains in near-critical pure fluids. However, our predictions can be applied even to fluids far from criticality. One of our main findings is that, when droplets are sparsely distributed, sounds can induce latent-heat generation or absorption at the interfaces and produce long-range temperature gradients extending far from the droplets. The sounds are then anomalously attenuated at low frequencies, and the effect may be used to detect onset of nucleation. We also calculate a frequency-dependent adiabatic compressibility in two-phase states, which is valid even far from criticality and is applicable to bubbly fluids. It reproduces the effective-medium theory at relatively high frequencies and a Landau-Lifshitz result in the zero-frequency limit. The mechanism investigated is general and is not limited to fluids.