The relation between the theory developed recently by Chandra [Phys. Rev. A 40, 752 (1989)] and our theory, derived previously [N. A. Cherepkov, J. Phys. B 16, 1543 (1983); 14, 2165 (1981); Adv. At. Mol. Phys. 19, 395 (1983); N. A. Cherepkov and V. V. Kuznetsov, Z. Phys. D 7, 271 (1987)], is established.

We report extensive numerical studies on the long-time behavior of a high-dimensional system of coupled symplectic maps as a function of their number N and of the nearest-neighbor coupling strength ε. The system, at a fixed value of ε, displays regular motion only in a small fraction of the phase space, whose volume vanishes exponentially with N. Regarding the chaotic motion, we find a scaling behavior of the mean-square fluctuation σ of the maximal Lyapunov exponent about its average value over initial conditions: σ≃(1/N${)}^{\mathrm{\alpha }}$ where α=O(√ε ) . Nevertheless, also for large systems, one observes a very weak Arnold diffusion, and different trajectories, with a high value of the Lyapunov exponents, maintain some of their own features for a very long time. Finally, we study the localization properties of the tangent vector. For chaotic trajectories, at small values of ε, an initially small perturbation increases only in a few directions; due to the translational invariance of the system, this behavior may be seen as a failure of ergodicity and also as a confirmation of the relevance of the Nekhoroshev scenario in high-dimensional systems.

We derive the behavior of the Thomas-Fermi equation of state near both the ‘‘hot-curve’’ (the limit as T→∞ for a fixed de Broglie length) and the ‘‘cold-curve’’ (the zero-temperature isotherm) by using perturbation methods. We derive efficient computational formulas for the coefficients in these expansions using Padé methods. We are able to meld all of these results into a structurally correct representation of the Thomas-Fermi fluid pressure. We indicate how this representational structure can be used to construct the equation of state for realistic materials over large ranges of temperature and pressure. We also discuss the computation of the degree of ionization with and without the exchange correction.

Relaxational processes in bistable potentials close to marginal conditions are studied under the combined effect of additive and multiplicative fluctuations. Characteristic time scales associated with the first-passage-time-distribution are analytically obtained. Multiplicative noise introduces large effects on the characteristic decay times, which is particularly significant when relaxations are mediated by fluctuations, i.e., below marginality and for small noise intensity. The relevance of our approach with respect to realistic chemical bistable systems experimentally operated under external noise influences is mentioned.

Localization in the quantum kicked rotator (QKR) problem leads to nontrivial dynamical correlations that are absent in the classical limit. These correlations are related to the spectral properties of the model. The effect of noise on coherence is studied. Also, its effect on the dynamical correlations is investigated, and thus one is able to determine analytically the diffusion that is induced. If the noise is white, then coherence time is the same as that for the undriven system. For colored noise it is essential to take into account the nature of the dynamical correlations. It follows that a Markovian treatment of the dynamics for a system that is coupled to a low-temperature heat bath is not valid even if the system is classically chaotic. The formalism applies also to the problem of QKR with kicks that are not strictly periodic in time, and also to the quantum kicked particle problem, where a nonperturbative mechanism for destruction of coherence has been found.

The real-time dynamics of a quantum-mechanical particle in a dissipative double-well potential at finite temperatures is solved ab initio by means of a multisite spin-hopping analysis. Both the macroscopic-quantum-coherence and macroscopic-quantum-tunneling regimes are covered. The theory gives a unified picture of tunneling transport and vibrational relaxation. The analysis is based on the Ohmic-model Hamiltonian [A. O. Caldeira and A. J. Leggett, Ann. Phys. (N.Y.) 149, 374 (1983); R. Zwanzig, J. Stat. Phys. 9, 215 (1973)] or the equivalent ‘‘particle-on-a-string’’ model [H. Dekker, Phys. Rev. A 31, 1067 (1985)]. The reduced quantum Liouville equation for the particle is derived and written in the energy representation. The usual truncation to the ground-state doublet of the vibrational ladder is not made. The interdoublet vibrational dynamics considerably simplifies upon coarse-graining and in the harmonic-well approximation. The intradoublet pseudospin dynamics is treated in the ‘‘noninteracting-blip approximation’’ [A. J. Leggett et al., Rev. Mod. Phys. 59, 1 (1987)]. The coarse-grained vibrational processes give rise to a Markovian one-step spin hopping between adjacent doublets (or sites). Inter alia ‘‘inhomogeneous broadening’’ and ‘‘motional narrowing’’ [P. E. Parris and R. Silbey, J. Chem. Phys. 83, 5619 (1985)] are discussed as special cases.

We examine numerically the self-affine scaling of time series with an imposed power-law power spectrum P(ω)=C${\mathrm{\omega }}^{\mathrm{-}\mathrm{\alpha }}$, for different exponents 1≤α≤3, and for different sequences of phases. We use two different criteria for testing self-affinity, a fractal dimension of the graph of the time series, and a more sensitive test based on the scaling of moments of probability distributions. For α≠2, our results suggest that time series with a power-law spectrum are only approximately self-affine, even in the best case of long-time series with high-dimensional, δ-function-correlated, uniformly distributed phases. Scaling curves are most sensitive to phases with long correlation times, are weakly dependent on the shape of the phase probability distribution, and are independent of the fractal dimension of the phases.

We have studied the interface-width scaling behavior of deposition models in which limited downward mobility is introduced. In all the models studied here, there is a maximum allowed slope for the interface at which the growth velocity is zero. The Kardar-Parisi-Zhang equation is resummed in order to show explicitly this slope constraint, but still incorporates parameters λ measuring the slope dependence of the growth velocity, ν measuring the surface tension, and D measuring the noise amplitude. Increasing mobility naturally smooths the interface, and is associated with a decrease in the magnitude of ${\lambda }_{\mathrm{eff}}$=λ${\mathit{D}}^{1/2}$/${\nu }^{3/2}$. A detailed study of a bridge-site deposition model, with one hop of probability p, shows that ‖λ‖ increases with p. From an independent assessment of noise-amplitude behavior, we can conclude that ν must also increase with p to ensure the required ${\lambda }_{\mathrm{eff}}$ behavior. Direct determination of ν via the Wolf-Tang procedure of imposing inhomogeneity on some length scale L supports this conclusion. (However ν depends on the inhomogeneity strength, which should be chosen small, and on L.) We comment on anticipated behavior of similar limited-mobility models in d≥2 dimensions, and compare with behavior of limited-mobility ballistic deposition and noise-reduced deposition models.

Extensive simulations of a model of directed polymers in a random potential are described. The standard deviation ΔE(t) of the lowest energy E of walks of t steps varies as ${\mathit{t}}^{\mathrm{\omega }}$, but there is a large spread in the values quoted in the literature for ω in d=2+1 and 3+1. In our model we are able to vary a parameter—the bending energy of the polymer—that affects the apparent value of ω, but, if allowance is made for corrections to scaling, then we find that ω is universal with the value 0.248±0.004 in d=2+1 and ω=0.20±0.01 in d=3+1. The probability distribution P(E,t) is found to scale as t→∞,P(E,t) dE→P(a) da, where a=(E-〈E〉)/ΔE. The scaling function P(a) is investigated and found to be universal, i.e., independent of such details as choice of distribution of the random potential, etc., and in fact identical to within our numerical accuracy to the equivalent distribution of heights in a model of ballistic aggregation.

The tunneling dynamics of the spin-boson problem has been computed using discretized path-integral simulations for temperatures T and couplings, i.e., the Kondo parameter α, spanning the entire T-α plane. The inherent problem of alternating weights has been solved using a combination of the stationary-phase Monte Carlo method and contour-distortion techniques. A transition from coherent to purely incoherent relaxation was observed for the spin correlation function. The time correlation functions and the location of the coherent-incoherent boundary on the T-α plane are well described by the noninteracting-blip approximation. In the deep-tunneling limit of large α, low T, and high bath frequency, the system relaxes exponentially, with its relaxation time constant following a power-law temperature dependence, in accord with perturbation theory. At higher T and low bath frequency, the relaxation time crosses over to a classical Arrhenius temperature dependence, reflecting the onset of activated processes. For a narrow region within 1/2<α<1, numerical results suggest that the system undergoes incoherent relaxation, with a short-time exponential decay, followed by a long-time tail of the power-law type. The short-time exponential relaxation time follows a peculiar power-law temperature dependence, with the relaxation rate increasing as a function of decreasing temperature.

A corrugated and stable ground state is found for the two-dimensional electron gas in both the normal paramagnetic and the fully polarized phases. The self-consistent Hartree-Fock method is used with a modulated set of trial wave functions within the deformable jellium model. The results for high metallic densities reproduce the usual noncorrugated ferromagnetic electron-gas behavior. A transition to a paramagnetic corrugated state for values of ${\mathit{r}}_{\mathit{s}}$≊6.5 is predicted. At lower densities ${\mathit{r}}_{\mathit{s}}$≊30, a second transition to a corrugated ferromagnetic phase is suggested.

We generalize the idea of driving a stable system to the situation when the drive signal is chaotic. This leads to the concept of conditional Lyapunov exponents and also generalizes the usual criteria of the linear stability theorem. We show that driving with chaotic signals can be done in a robust fashion, rather insensitive to changes in system parameters. The calculation of the stability criteria leads naturally to an estimate for the convergence of the driven system to its stable state. We focus on a homogeneous driving situation that leads to the construction of synchronized chaotic subsystems. We apply these ideas to the Lorenz and Rössler systems, as well as to an electronic circuit and its numerical model.

In formulating the quantum-mechanical analog of the Fermi accelerator, one encounters the problem of determination of the eigenvalues of a harmonic oscillator confined to a finite region of space with time-dependent real or imaginary frequency. Here the invariants found by Lewis [J. Math. Phys. 9, 1976 (1968)] and by Lewis and Riesenfeld [J. Math. Phys. 10, 1403 (1969)] are also constants of motion; however, they are not very useful in obtaining the eigenvalues. In their place the Heisenberg equations of motion are used to describe the time evolution of this system and show how the spacing between neighboring eigenvalues changes as a function of time.

We investigate the influence of different mobilities of A and B particles on the time evolution of the particle concentration in the diffusion-controlled reaction A+B→0. For different concentrations of A and B particles, Monte Carlo simulations display two main behaviors, depending on whether the majority species moves or is at rest. An analytical approximation based on the diffusion equation shows reasonable agreement with the Monte Carlo simulations and gives lower bounds for the concentrations’ decay.

Collective density variables ρ(k) are frequently employed in many-body physics to describe a wide variety of static and dynamic phenomena. These variables are nonlinear functions of particle positions, and consequently exhibit subtle couplings and kinematic constraints. We examine some of these features for one-dimensional systems, using both numerical exploration and analytical techniques. In particular we have considered the consequences of quenching density fluctuations [minimizing the ρ(k)’s] for sets of wave vectors surrounding the origin. This is shown, under proper circumstances, to force other sets of ρ(k)’s automatically to their minima, and even to induce perfect crystallization of the many-particle system.

The intricate connection between the distribution of exceptional points and particular fluctuation properties of level spacings and associated eigenvector statistics is shown. A procedure is given to determine the distribution of the exceptional points of the problem ${\mathit{H}}_0$+λ${\mathit{H}}_1$. Its implementation is demonstrated considering simple matrix models and the quantized chaotic quartic oscillator.

The Ruijgrok-Cohen (RC) mirror model [Phys. Lett. A 133, 415 (1988)] of a Lorentz lattice gas, in which particles are reflected by left and right diagonally oriented mirrors randomly placed on the sites of a square lattice, is further investigated. Extensive computer simulations of individual trajectories up to $2^{24}$ steps in length, on a lattice of 65 536×65 536 sites, are carried out. This model generates particle trajectories that are related to a variety of kinetic growth and ‘‘smart’’ (nontrapping) walks, and provides a kinetic interpretation of them. When all sites are covered with mirrors of both orientations with equal probability, the trajectories are equivalent to smart kinetic walks that effectively generate the hulls of bond percolation clusters at criticality. For this case, ${10}^6$ trajectories were generated, yielding with unprecedented accuracy an orbit size-distribution exponent of τ=2.1423±0.0003 and a fractal dimension of ${\mathit{d}}_{\mathit{f}}$=1.750 47±0.000 24 (without correcting for finite-size effects), compared with theoretical predictions of 15/7=2.142 857. . . and 7/4, respectively. When the total concentration of mirrors C is less than unity, so that the trajectories can cross, the size distribution of the closed orbits does not follow a power law, but appears to be described by a logarithmic function.

This function implies that all trajectories eventually close. The geometry of the trajectories does not show clear self-similar or fractal behavior in that the dependence of the mean-square displacement upon the time also appears to follow a logarithmic function. These trajectories are related to the growing self-avoiding trail (GSAT) introduced by Lyklema [J. Phys. A 18, L617 (1985)], and the present work supports the conjecture of Bradley [Phys. Rev. A 41, 914 (1990)] that the GSAT (the RC model with C=2/3) is not at a critical point. It is observed that when C<1, the trajectories behave asymptotically like an unrestricted random walk, and so for comparison the RC model in the random walk or Boltzmann approximation (BA) is also studied. In the BA, the size distribution of returning trajectories and the geometric properties of open trajectories are investigated; the time dependence of the mean-square displacement is derived explicitly and is shown to exhibit an Ornstein-Uhlenbeck type of behavior.

We carry out a detailed investigation of the spectral fluctuations for the π/3 rhombus billiard, a system known to be pseudointegrable with an invariant integral surface of genus g=2. The equilateral triangle states (which constitute half the total number) show considerable deviations from the Poisson behavior due to degeneracies in the periodic orbit actions. The ‘‘pure rhombus states’’ (the other half), however, have fluctuations closer to that of the Gaussian orthogonal ensemble of matrices (GOE) but are strongly parity dependent. The presence of localized states in the (00) mode affects the nearest-neighbor level statistics significantly. Both the Berry-Robnik and the Brody distributions, however, serve to parametrize the fluctuations of the (10) mode satisfactorily. The spectral rigidity and number variance, however, seem insensitive to the presence of localized states. Deviations from the GOE predictions become very significant for higher-order correlations.