One technique of estimating the local intrinsic dimensionality of attractors is to apply the nearest-neighbor (NN) approach to local regions of attractor data in the phase-space domain. In noise-free, infinite signal-to-noise ratio data, the NN method is shown to produce results in reasonable agreement with the correlation dimensions for some known examples. All NN results are compared to the correlation dimension, as computed by the standard Grassberger-Procaccia algorithm. However, it is shown here that even a small amount of corrupting noise will severely degrade the performance of NN-type dimension estimators. NN methods are therefore of limited use for these calculations if data are noisy.

We consider the relaxation dynamics of a spin-1/2 particle in a fluctuating transverse magnetic field. With a cumulant expansion technique (up to sixth order) we calculate the time dependence of the approach to equilibrium of the population difference of the two spin levels and of the phase coherence. We find that for intermediate strength coupling to the fluctuations, the cumulant expansion converges, and the relaxation to equilibrium can have a pronounced non-Markovian character. We compare these analytic results with numerically exact data from stochastic simulations.

Chaotic dynamical systems can be organized around an underlying strange set, which is comprised of all the unstable periodic orbits. In this paper, we quantify the complexity of such an organization; this complexity addresses the difficulty of predicting the structure of the strange set from low-order data and is independent of the entropy and the algorithmic complexity. We refer to the new measure as the grammatical complexity. The notion is introduced, discussed, and illustrated in the context of simple dynamical systems. In addition, the grammatical complexity is generalized to include metric properties arising due to the nonuniform distribution of the invariant measure on the strange set.

Multifractal properties of repelling sets generated by hyperbolic maps are studied as a function of a parameter describing a transition to an attracting interval. Critical indices in the scaling behavior of multifractal functions are found when a uniform probability density is assumed. A constant probability is also considered, and the resulting thermodynamiclike functions are investigated close to the critical value of the parameter.

Ordering of a one-dimensional stochastic Ising model evolving by a combination of spin flips and spin exchanges is investigated. The spin-flip rates satisfy detailed balance for the equilibrium state of the Ising model at temperature T, while the spin exchanges are random and of arbitrary range. Analytical methods and Monte Carlo simulations are used to show that, depending on the details of the spin-flip rate, finite-temperature phase transition may or may not occur in the system. When ordering occurs, it is of mean-field type and the scaling function describing the finite-size scaling of the magnetization fluctuations is found to be indistinguishable from that of an equilibrium Ising model with infinite-range interaction.

We study a spin-1/2 entity interacting with a set of quantum-mechanical oscillators using the Heisenberg picture. In the weak-coupling limit, the Heisenberg equation is given an analytical solution, suggesting that upon increase of the coupling the system should exhibit a critical transition to an overdamped regime, like that of a heavily damped oscillator. Thus the lifetime of the trapped state increases with increasing temperature. Under the assumption of very small tunneling frequency, another analytical solution is found, which predicts the occurrence of trapped states, whose lifetime decreases with an increase of temperature. As a result of the competition between the two trapping processes, the lifetime of the localized state is shown to be a nonmonotonic function of temperature with a minimum lifetime at a certain critical value of temperature. The predictions of this theoretical treatment are carefully checked by numerical solution of the interaction between a 1/2-spin entity and a dissipative oscillator, and a very satisfactory agreement between theory and numerical solution is found over the whole range of coupling strength.

Statistical properties of deterministic chaos are investigated in systems with a variational structure, driven by a delayed force. The delayed force is chosen to be periodic with respect to the amplitude of the motion. When the period is much smaller than the response time of the system, ${\mathit{T}}_1$, the chaotic solutions are shown to be noiselike, i.e., they have statistical properties very similar to those of random chaos resulting from nonlinear Langevin equations. As the period increases to ${\mathit{T}}_1$, the delay term is shown to act in a manner similar to a colored noise in all cases of large delay, but with different statistics. However, when the delay is about equal to ${\mathit{T}}_1$, very special effects like phase transitions appear, giving rise to new peaks in the probability distribution. The ‘‘phase transition’’ is shown to be well described by an approximate Fokker-Planck equation. An analytical expression is proposed for the drift term in this equation, which agrees with numerical computations.

We propose an approximation to the density-functional theory of classical nonuniform systems that reproduces all the formal properties of the free energy and requires only the direct correlation function of the uniform system as input. By introducing additional assumptions into this theory a direct relation can be established with most of the existing nonperturbative theories. When the theory is worked out for the case of the hard-sphere solid, very good agreement is found with the computer simulations. The free energies, pressures, and fluid-solid coexistence data are reproduced to within the error bars of the simulations. The theory also predicts stable bcc and sc phases that could play a role in the final nucleation of the equilibrium fcc phase.

The excited states and the one-particle energy spectra are evaluated for the electron gas in the deformable jellium model. A screened Coulombic interaction is used. The Hartree-Fock approximation is carried out with linear combinations of periodic functions as the trial wave functions. The behavior of the energy spectra with the density of the system is discussed. The existence of overlapping bands is established. This result suggests that finite conductivity at low densities can be obtained within the Hartree-Fock approximation.

The quantum-mechanical problem of tunneling of a particle coupled to a heat bath that consists of a large number of harmonic oscillators is studied in this paper. Here the equation of motion for the particle which is a nonlinear integro-differential equation is found by eliminating the degrees of freedom of the oscillators. The nonlinear operator equation may be solved by noting that the position operator at the time t can be expanded as a power series in terms of the initial position and momentum with time-dependent c-number coefficients. These coefficients also satisfy nonlinear integro-differential equations. For the problem of dissipative tunneling of a particle in a double-well potential, the first few terms of expansion have been calculated numerically. The expansion parameter in this formulation is a small dimensionless parameter which is inversely proportional to the product of the width and the square root of the height of the barrier.

A method to determine the resolvent of a quantum system coupled to a harmonic-oscillator bath is derived by extending the continued-fraction theory of a Gaussian-Markovian bath that has been presented by Tanimura and Kubo [J. Phys. Soc. Jpn. 58, 101 (1989)]. The results are expressed in terms of continued fractions and apply to an oscillator bath with a general spectral density, corresponding to colored noise, at various temperatures. Exact values of the resolvent can be calculated for arbitrary strength of the system-bath interaction by making use of the convergence properties of the continued fractions. For the weak-interaction case these results agree with the quantum master equation. The physical meaning of the results is also discussed by a diagrammatic method. As an application, the result of the Gaussian-Markovian system is extended to the case of the low-temperature bath. Correlated (unfactorized) initial conditions are also discussed.

The amount of regularity of classical trajectories and quantum states of a two-dimensional anharmonic oscillator far from the semiclassical limit are compared. It is found for this model system that the degree of regularity is considerably higher for the quantum system and that the amount of regularity of the quantum system increases with increasing energy in the high-energy region while the classical system shows the opposite trend.

Invariant sets arising in chaotic dynamics can be organized around their underlying skeleton of unstable periodic orbits. In this paper a scaling function for the eigenvalues of the unstable periodic orbits of strange sets embedded in two dimensions is introduced. For the piecewise-linear Lozi mapping, the scaling function is obtained analytically, as a convergent series in b, the inverse dissipation strength. In general, the scaling function is shown to converge only for uniformly hyperbolic systems, while the inadequacy of this description for other systems is demonstrated and discussed. The converged periodic-orbit scaling function is applied to yield accelerated convergence of the multifractal spectrum of a strange set.

We present the calculation of Ruelle’s rotation number (frequency) Ω for a chain of damped and driven coupled Duffing oscillators using the recently discovered symplectic structure of the system. A relation between the eigenvalues of periodic orbits and the winding number ‖Ω‖/ω is conjectured by making use of their Krein signatures. The occurrence and nonoccurrence of plateaus with respect to the driving frequency ω can be explained via this relation.

We consider a two-dimensional (2D) square billiard with one particle where the interaction of the particle with an elementary wall segment is the Lennard-Jones (LJ) potential. We name the resulting potential the LJ stadium (LJS). It has three qualitatively different regions and here we study the region with energy E>0 (Sinai’s region). We also find an integrable potential close to that of the LJS and discuss the motion in the LJS in terms of this potential. We draw the Poincaré surface of section for the LJS at various energies and the frequency-frequency curves (FFC) and find that, the lower the energy, the more probable a chaotic motion exists. We identify the corners as the probable source of chaos and discuss the mechanisms of resonance interaction and jumps of tori. We evaluate the Lyapunov exponent and discuss it in terms of qualitative characteristics of the potential. By deleting the attractive terms in the LJ potential we study the square billiard system with a softened repulsion. We find that the motion in this billiard is regular. Therefore we conclude that by softening the hard-sphere potential there are no qualitatively new phenomena, while with an attractive interaction, stochastic motion appears at low energies and global stochastic behavior can be seen.

A Green’s function is given for the canonical form of the thermodiffusion problem. The analytical and numerical solutions obtained for an initially homogeneous mixture reveal that the process at the boundaries is enhanced during the initial phase. This has to be related to the observations made by Tanner [Trans. Faraday Soc. 23, 75 (1927)] that the concentration change is ‘‘extremely rapid’’ at the boundaries. Impact on liquid-alloy crystallization is briefly analyzed.

This paper is concerned with the fluctuations of the liquid-gas surface. We study the approach to the strong fluctuation regime as the strength of the external field pinning an interface is decreased. By means of a variational approach we treat self-consistently the order parameter and its correlations in three dimensions. Our numerical results show that the width of the profile increases as the external field decreases according to a logarithmic law. We relate this behavior to the capillary-wave-like excitations of the interface and discuss the limitations of the present approach.

We develop an effective interface formalism to perform the linear stability analysis of long-wavelength perturbations of a periodic array of fingerlike cells in the small-Péclet-number limit of directional solidification. The growth rate Ω for perturbations of wave number q is found to have two branches. The first branch ${\mathrm{\Omega }}_1$(q) is stable for all wave numbers. The second branch is of the diffusive form ${\mathrm{\Omega }}_2$(q)=-${\mathit{Bq}}^2$, where B is proportional to the derivative dΔT/dΛ of the thermal undercooling ΔT with respect to the cell spacing Λ. The point of marginal array stability in this system therefore coincides with the point of minimum undercooling in an analogous way to lamellar eutectics.

In 1974, Halperin, Lubensky, and Ma (HLM) [Phys. Rev. Lett. 32, 292 (1974)] predicted that the nematic–smectic-A transition of pure compounds and their mixtures should be at least weakly first order. One way to obtain such a prediction is to treat the smectic order parameter as a constant and integrate out the director fluctuations. The coupling between the director fluctuations and the smectic order parameter then generates a cubic term in the effective free energy for the nematic–smectic-A(N–Sm-A) transition, which tends to drive the transition first order. So far, however, there has not been clear experimental evidence in support of this prediction: Some materials appear to exhibit a first-order transition but others a second-order transition. In this paper we introduce two new approaches to test the predictions of HLM. First, we note that if a cubic term in the effective free energy for the smectic order parameter is present, its effect is dominant near the Landau tricritical point (LTP), where the quartic term in the free energy vanishes.

In a mean-field approximation, a universal scaling form of the latent heat can then be derived close to the LTP. Its form depends sensitively on the presence of the cubic term. By reanalyzing earlier calorimetric measurements near the LTP, we find that these data yield evidence for the presence of the cubic term predicted by HLM. The second new approach to experimentally determine whether a transition is weakly first order or second order is a dynamical method. This general method is based on the observation that when a transition is (weakly) first order, the dynamics of interfaces are symmetric about ${\mathit{T}}_{\mathit{c}}$, so that an interface can propagate into both phases, depending on whether the sample is undercooled or overheated (corresponding to ‘‘melting’’ and ‘‘freezing’’). For a weakly first-order transition, a simple scaling relation for the interface speed can be derived. In contrast, the dynamics of propagating fronts close to a second-order transition are very asymmetric. Results of moving interfaces close to ${\mathit{T}}_{\mathit{c}}$ in 8CB-10CB (where CB represents cyanobiphenyl) and 9CB-10CB mixtures are presented and shown to support both qualitatively as well as quantitatively the prediction that the transition is always at least weakly first order. For the N–Sm-A transition in these compounds, our comparison finds that the dynamic experiments are more sensitive than the adiabatic calorimetry experiments by about one order of magnitude and more sensitive than the x-ray-diffraction experiments by about two orders in detecting the phase-transition order.

A cell-dynamical-system (CDS) model of the microphase separation of block copolymers is described in detail. The model suggests a partial-differential-equation (PDE) model that allows us to find a close relation between the block copolymer and spinodal decomposition problems. A detailed numerical study of the CDS model is given, which indicates that hydrodynamic effects are crucial for very late time-ordering processes. The strong segregation regime is studied analytically with the aid of an exactly solvable PDE model, which is suggested by the universality supported by the CDS simulations. The results obtained show that dimensional analysis gives the correct exponent for the block copolymer lamellar thickness.