Often it is required to make dimension estimates on data corrupted with moderate amounts of additive noise. The dimension of interest in this paper is the local intrinsic dimension of the attractor in the phase space. In this work we present a technique to estimate the local intrinsic dimension using higher-order statistical information, in particular, using statistical information beyond what is contained in methods based on the covariance. The average higher-order local intrinsic dimension (〈${\mathit{d}}_{\mathit{H}}$〉) is calculated from noisy data, the additive noise of which may be correlated and nonwhite in nature. With use of noisy data samples, the higher-order local intrinsic dimension (HOLID) is calculated from analysis of a higher-order cumulant matrix formed within local attractor regions. The local rank of the HOLID matrix is computed using the method of singular-value decomposition and the singular values so formed are then separated into signal components and noise components by applying a direct threshold. The local rank is an integer that varies with position on the attractor. To obtain 〈${\mathit{d}}_{\mathit{H}}$〉 an average of the available local HOLID’s is calculated. Results of the 〈${\mathit{d}}_{\mathit{H}}$〉 compare favorably with those of the covariance formulation 〈${\mathit{d}}_{\mathit{C}}$〉 for finding the local intrinsic dimension. Also presented, for comparison purposes, with these results are the correlation dimension for some known chaotic cases since they are so widely used. Tests performed with different signal and noise types and at various signal-to-noise ratios have indicated that 〈${\mathit{d}}_{\mathit{H}}$〉 can yield more accurate dimension information in the presence of correlated noise, but more data samples must be processed.

One-dimensional models of surface dynamics are studied analytically and by numerical simulation. In all cases the number of particles that form the deposit is conserved. We find that an initially flat surface, in general, roughens as a function of time and can be characterized by a width ξ which obeys the scaling form ξ(L,t)=${\mathit{L}}^{\mathrm{\chi }}$f(t/${\mathit{L}}^{\mathit{z}}$) for a deposit on a substrate of linear dimension L. Restricted solid-on-solid (RSOS) -type models in which the microscopic dynamics obeys detailed balance are shown to be in the ‘‘free-field’’ universality class with exponents z=4 and χ=1/2. On the other hand, if detailed balance is broken, several universality classes exist. As the ‘‘maximum-height-difference’’ constraint in a RSOS model is varied, one can observe a phase transition between a flat phase (z=2, χ=0) and a grooved phase characterized by a steady-state exponent χ=1 but with no scaling in the relaxation process. At the transition point, a nontrivially rough surface emerges with exponents (z≊3.67, χ≊0.33) close to those of the conserved Kardar-Parisi-Zhang equation. We propose a phenomenological model that may account for these observations.

Using the path-integral method, we derive the analytical solution for the following one-dimensional diffusion in random media: ∂P(x,t)/∂t=D[${\mathrm{\partial }}^2$P(x,t)/∂${\mathit{x}}^2$]+λV(x)P(x,t) , where V is a white-noise Gaussian potential. A quantity τ=(16D/9${\lambda }^4$${)}^{1/3}$ is introduced for the time scale. When the diffusion time t≪τ, the behavior of the average 〈P(x,t)〉 is essentially diffusive. When t≫τ, the random potential plays a dominant role, and the average 〈P(x,t)〉 tends to [${\lambda }^4$${\mathit{t}}^{5/2}$/8(π${\mathit{D}}^3$${)}^{1/2}$]exp[(${\lambda }^4$${\mathit{t}}^3$/48D) (1-${\mathit{x}}^2$/2Dt)]. t)].

The ideal quantum gases satisfying Bose-Einstein statistics in d dimensions (d=1,2,3,. . .) provide illuminating apparent counterexamples to the often-supposed equivalence of the Nernst-Simon and unattainability statements of the third law of thermodynamics. These gases satisfy the second law of thermodynamics and the Nernst-Simon and Planck statements of the third law but appear to violate, at least in a technical sense, the unattainability statement. They do so by virtue of the fact that their entropy vanishes at nonzero temperature and bounded pressure. Possible objections to the counterexamples are examined and additional counterexamples are produced with the same mechanism. An interpretation of this nonequivalence is proposed in terms of a statement about the kinds of Hamiltonians found in nature.

We study the crossover in the relaxation dynamics from the situation at the bifurcation point of a subcritical pitchfork bifurcation to a situation well above the bifurcation point. An approximation for the stochastic paths is given that allows the calculation of the statistical properties of the escape times. These properties can be described by a generating function that exhibits dynamical scaling of critical nature at the bifurcation point and Gaussian dynamical scaling well above the bifurcation point.

The model of a one-dimensional, diatomic gas with harmonic intramolecular forces and soft intermolecular forces, introduced by Benettin, Galgani, and Giorgilli [Phys. Lett. A 120, 23 (1987)], is studied numerically for a set of initial conditions for various values of the vibrational frequency ω and the number of molecules N. The use of many trajectories, rather than a single one, as was done before, allows us to infer conclusions about the structure of the phase space of the system using statistical analysis of simple functions of the phase. Thus, for fixed ω and small N, the average vibrational energy differs from trajectory to trajectory, implying nonergodic behavior. However, these differences become statistically insignificant for large N, implying a transition to ergodicity. While this transition is not sharp, it occurs at larger N for increasing ω, due to the increasingly difficult exchange of energy between vibrational and translational degrees of freedom. The phase-space-averaged vibrational-energy time-autocorrelation function decays on two time scales for all frequencies considered, at least for N large enough that the system behaves ergodically. A rapid, Gaussian decay on the time scale of the mean free time is followed by a much slower, approximately stretched-exponential decay, due to interactions with adiabatic islands in phase space. This second time scale increases sharply with ω and is related to the increasingly slow, possibly Arnold-like, diffusion in phase space.

We emphasize that the ordinary differential equations of a continuous dynamical system, or at least of equivalent systems, can be reconstructed from numerical scalar time series. Methods are exemplified in the case of a strange, chaotic attractor generated by a mathematical model, namely, a Rössler band. Resultant validations rely (i) qualitatively on comparisons between original and reconstructed phase portraits, and (ii) quantitatively on comparisons between generalized dimensions ${\mathit{D}}_{\mathit{q}}$ of original and reconstructed attractors. Some of the many lines of research offered by the presented results are discussed to stress potentialities of this kind of reconstruction.

Nearly ideal and very nonideal binary Yukawa mixtures are studied with Monte Carlo (MC) and mean-spherical approximation (MSA) methods. Compressibility factors, static structure factors, and local mole fractions are compared in detail. The comparison shows that MSA predicts these properties of liquid Yukawa mixtures in sufficiently good agreement. Even for very nonideal mixtures the results of MC and MSA methods agree within about 10%. The phase unstability is also well predicted by MSA. The theoretical value lies only 8% lower than the ‘‘exact’’ one and falls practically in the uncertainty range of the MC result. We conclude that the mean-spherical theory is a suitable tool for describing the thermodynamics of these types of two-component systems reliably.

In this paper, we give two independent and rigorous derivations for the quantum bound on the information transmission rate proposed independently by Bekenstein [Phys. Rev. Lett. 46, 623 (1981)] and Bremermann .ul2 [Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, edited by L. M. LeCam and J. Neyman (University of California Press, Berkeley, 1967], preceded by a heuristic argument showing why such a bound must hold. In both approaches, information carriers are quanta for some field. The first method resembles the microcanonical approach to statistical mechanics, where the strategy of overestimating the real number of states by relaxing the indistinguishability of quanta was adopted. The second is based entirely upon maximum-entropy methods. Amazingly enough, the results obtained by these physically unrelated premises turn out to be identical, namely, that the single (noiseless) channel capacity is I${\mathrm{̇}}_{\max }$=E/2πħ bits ${\mathrm{s}}^{\mathrm{-}1}$. It is further shown that, in a finite time τ, no information can ever be conveyed unless the energy threshold 2πħ/τ is reached, allowing the reinterpretation of the time-energy uncertainty in informational-theoretic language.

A self-consistent theory for vacancy entrapment in rapidly solidified metal droplets is presented. Supersaturation occurs when excess (nonequilibrium) vacancies created at the solidification front by the liquid-solid density difference are unable to diffuse back to the interface before the droplet solidifies. The model consists of heat-conduction equations for the liquid and solid phases, a vacancy-diffusion equation, and boundary conditions at the internal and external surfaces that provide coupling between regions and generate the interface dynamics and vacancy entrapment. Solutions to this system of equations are derived in the form of a set of integral equations that incorporate the boundary conditions as integral kernels. These are evaluated numerically to produce temperature and vacancy-concentration profiles for rapidly solidified droplets of various sizes, initial undercoolings, and convective cooling rates. For undercooled, micrometer-sized metal droplets, the model gives vacancy concentrations at solidification far in excess of equilibrium values.

A general formalism is developed to determine eddy viscosities for incompressible flow of arbitrary dimensionality subject to forcing periodic in space and time. The dynamics of weak large-scale perturbations is obtained by a multiscale analysis. The large-scale behavior is found to be formally diffusive (first order in time, second order in space) whenever the basic flow is parity invariant, that is, possesses a center of symmetry. The eddy viscosity is in general a fourth-order tensor, for which a compact representation is provided. Explicit expressions of the eddy-viscosity tensor are given (i) for basic flow with low Reynolds numbers, and (ii) when the basic flow is layered, i.e., depends only on one space coordinate and time. A special class of layered flow is two-dimensional, time-independent parallel periodic flow, an example of which is the Kolmogorov flow. Such parallel flow acquires a negative-viscosity instability to large-scale perturbations transverse to the basic flow when the molecular viscosity becomes less than the rms value of the stream function of the basic flow. For flows presenting less symmetry than the Kolmogorov flow, the first large-scale instability is usually found not to be transverse, thus breaking the spatial periodicity of the basic flow. Such nontransverse instabilities, observed in a lattice-gas simulation on the Connection Machine, are reported in the companion paper by Hénon and Scholl [following paper, Phys. Rev. A 43, 5365 (1991)].

In a companion paper, Dubrulle and Frisch [preceding paper, Phys. Rev. A 43, 5355 (1991)] develop a general formalism to determine eddy viscosities for incompressible flow. In particular, they predict a nontransverse instability for a modified Kolmogorov flow. We confirm quantitatively this prediction with the help of lattice-gas simulations on a Connection Machine.

Two models of reaction and consumption of porous media, such as coal and catalyst particles, are investigated. The first model is appropriate only for the kinetic regime (KR) in which the rate of reactant transport plays no role, and the consumption of the solid matrix of the medium depends only on the kinetics of the reaction. The second model incorporates the effect of the diffusion of the reactants into the pore space and, by changing the normalized reactivity of the medium, the model is capable of simulating the entire range of possible kinetic behavior, from the diffusion-limited regime (DLR) to the KR. The solid matrix of the medium is represented by a percolation cluster. In both the KR and the DLR, one observes fragmentation of the medium. However, while in the KR one can initially obtain large fragments, the fragments in the DLR are small and are mostly the result of the surface roughness of the medium (perimeter fragmentation). Moreover, in all cases the fragments obey a dynamic scaling ${\mathit{n}}_{\mathit{s}}$(t)∼${\mathit{t}}^{\mathit{w}}$${\mathit{s}}^{\mathrm{-}\mathrm{\tau }}$f(s/${\mathit{t}}^{\mathit{z}}$) where ${\mathit{n}}_{\mathit{s}}$ is the number of fragments of size s at time t, and w, τ, and z are dynamic exponents. The exponent w is positive before the maximum number of fragments has been obtained, and negative beyond that. The extension of the model for calculating other quantities of interest, e.g., the distribution of the mineral particle sizes that result from the complete consumption of the system, is also discussed. Possible similarities with other fragmentation phenomena are also discussed.

Shear flow distorts the microstructure of fluids if the Deborah number scrD becomes comparable to 1. In complex fluids, exotic hydrodynamics effects are often seen in this regime. We compute within Landau theory the structure factor S(q) of a sheared nematic liquid crystal close to the nematic to smectic-A (N–Sm-A) phase transition. As a function of increasing Deborah number, the pretransitional smectic-A fluctuation clusters become increasingly geometrically restricted, evolving from their usual three-dimensional ellipsoidal shape for scrD≪1 to an extremely anisotropic one-dimensional shape for scrD≫1. We discuss the predictions of Landau theory for x-ray diffraction experiments for various orientations of the nematic director. The suppression of pretransitional critical fluctuations by shear flow is found to raise the transition temperature ${\mathit{T}}_{\mathit{N}–\mathrm{Sm}\mathrm{-}}$A, and peculiarly, ${\mathit{T}}_{\mathit{N}–\mathrm{Sm}\mathrm{-}}$A is found to depend on the orientation of the director. The presence of the microscopic fluctuation clusters under the shear flow is also reflected on the macroscopic level. The classical theory of the hydrodynamics of nematic liquid crystals, due to Ericksen, Leslie, and Parodi (ELP), is found to be incomplete. We compute the new fluctuation-induced forces that must be added to ELP nematic hydrodynamics and we discuss their consequence, in particular for large scrD, the analog of shear-thinning for liquid crystals.

We study the linear viscoelasticity of semidilute suspensions of spherical Brownian particles. We consider a simple model in which hydrodynamic interactions and direct interactions apart from hard-sphere repulsion are neglected. We calculate the dynamic viscosity for this model system exactly to second order in the volume fraction. It turns out that the infinite-frequency shear modulus does not exist because the dynamic viscosity falls off with the inverse square root of the frequency. The relaxation-rate spectrum has corresponding inverse-square-root behavior. We find exact expressions for the spectral density and the time-dependent stress-relaxation function. Qualitatively our results resemble those found experimentally for dense hard-sphere suspensions.

We introduce effective-medium theory to describe the mechanical properties observed on sols and gels formed by addition of a cross-linker to semidilute polymer solutions. We experimentally observe that, in the entire range outside the critical domain, the variation of the viscosity (η) is well described by a linear decrease of 1/η as a function of the concentration in cross-link points. This behavior is to be related to a similar regime that we find for the resistivity of a superconductor-conductor site percolating lattice. The effective-medium calculations have been generalized to include nonsymmetrical properties of the links joining two sites of different nature, allowing us to describe the relative positions of the gelation threshold and the threshold extrapolated from the linear decrease of 1/η.

The radial distribution function g(r) provided by the solution of the Percus-Yevick (PY) equation for hard spheres is rederived in terms of the simplest Padé approximant of a function defined in the Laplace space that is consistent with the following physical requirements: g(r) is continuous for r>1, the isothermal compressibility is finite, and the zeroth- and first-order coefficients in the density expansion of g(r) must be exact. An explicit expression for the solution of the generalized mean-spherical approximation (GMSA) is obtained as a simple extension involving two new parameters, which are determined by imposing two conditions: (i) the virial and the compressibility routes to the equation of state agree consistently, and (ii) this equation of state coincides with that of Carnahan and Starling [J. Chem. Phys. 51, 635 (1969)]. The second- and third-order coefficients in the density expansion of g(r) given by the GMSA are compared with the exact ones and with those given by the PY equation.

The most notable success of the various density-functional theories of freezing is achieved for hard spheres where the density change upon freezing is relatively large, while they more or less equally fail when applied to soft inverse-power interactions where the density change upon freezing is much smaller. It is argued that this relatively large density change makes the hard-sphere freezing results insensitive to physics details in the density-functional calculational setup. As an example, it is demonstrated that the equations of state for the solid near melting as obtained by all the leading density-functional theories of freezing, in their particularly successful application to the freezing of hard spheres and hard disks, are related by a trivial parabola-shift transformation from the liquid excess free energy onto that for the solid. Conclusions regarding the relative merit of the free-energy functionals, which are based only on their performance for the hard-sphere fluid-solid transition, can be misleading.

This paper describes a light-scattering study of the glass transition in nonaqueous suspensions of sterically stabilized colloidal spheres. The observed phase behavior, fluid, crystal, and glass, is consistent with an essentially hard-sphere interaction between the particles. Metastable fluid states were obtained upon shear melting the crystalline phases by tumbling the samples. Their intermediate scattering functions, measured by dynamic light scattering, showed the emergence of a nondecaying component, implying structural arrest, at essentially the same concentration as that at which homogeneously nucleated crystallization was no longer observed. The overall forms of the intermediate scattering functions are consistent with the predictions of mode-coupling theories for the glass transition. Supplementary studies of the static structure factors indicated only short-ranged spatial order for particle concentrations ranging from the equilibrium fluid through the metastable fluid to the glass.

The glass transition studied for suspensions of colloidal particles through dynamical light-scattering experiments is analyzed within the mode-coupling theory for the supercooled liquid dynamics. These suspensions are treated as simple atomic systems with a hard-sphere interaction. The scaling properties predicted by the theory are verified. The shape of the master function for the β-relaxation region agrees well with the experimental data.