An electron in a periodic crystal potential, which is subjected to an additional periodic pulsed potential, is an example of a system which can be modeled by the kicked Harper model. The relative simplicity of this model facilitates a comparison of its quantum and classical dynamics. The model has a clearly defined semiclassical limit in which wave packets follow the classical motion. A power series in ħ yields systematic quantum corrections to the semiclassical limit. Numerical iterations of the quantum equations agree with the ħ expansion (for a limited time), provided the corresponding classical motion is not chaotic. For chaotic motion, the agreement between classical and quantum motion disappears very quickly. The quantum solutions also show (i) precursors to classical chaos, (ii) a time-dependent analog of wave-function scarring, and (iii) quantum echoes arising from quantized phase-space orbits.

Fractal dimensions and Lyapunov exponents can be estimated faster and more accurately and efficiently with the knowledge of the optimal embedding dimension and delay time. Two methods are presented to calculate the optimal embedding parameters for Takens’s delay-time coordinates. The first, called the fill factor, is a procedure that yields a global measure of phase-space utilization for (quasi) periodic and strange attractors and leads to a maximum separation of trajectories within the phase space. The second, which we call local deformation, is complementary to the fill factor. It describes the local dynamical behavior of points on the attractor and gives a measure of homogeneity of the local flow. Both methods were tested on numerical and experimental (Taylor-Couette flow) time series.

The catalyzed oxidation reaction of CO is studied by starting from macroscopic kinetic equations, which describe the evolution of the system variables. The interaction between nearest-neighbor adatoms, CO-CO, O-O, and CO-O are explicitly taken into account. In order to obtain a better ${\mathrm{CO}}_2$ production rate, the influence of the interaction is studied.

The assumption of ergodicity in Nosé’s original formulation of the constant-temperature molecular dynamics is tested for a Lennard-Jones-potential system. With the performance of very long simulations, it is shown that the extended system of a Lennard-Jones-potential system is ergodic for all values of thermostat parameters tested. It is also shown, however, that the rate of convergence to the canonical ensemble strongly depends on the value of thermostat effective mass Q. The dynamical properties of the extended system are also studied using the velocity autocorrelation function and the power spectral density. From the analysis of the simulations, it is found that the dynamical properties are not correctly represented for arbitrary values of thermostat parameters. A prescription and a set of quantitative criteria are introduced to generate physically meaningful dynamics. Thus the results of this work show that with a special choice of thermostat parameters it is possible to obtain both the correct canonical ensemble and physically meaningful dynamical behavior of the physical system.

We review the Flory-type formulas for true self-avoiding walks (TSAW’s) on fractal substrates proposed by Rammal [J. Stat. Phys. 36, 547 (1984)] and by Bouchaud and Georges [Phys. Rev. B 39, 2846 (1989)] and present a formula obtained by simple Flory approximation. We also present the Monte Carlo data for TSAW’s on a Sierpinski gasket embedded in both two and three dimensions and on a percolation backbone at the percolation threshold in two dimensions. The previous data on an infinite percolation cluster and our present data suggest that all known formulas describe fairly well the critical behavior of TSAW’s on fractals; however, that by Bouchaud and Georges and our formula show better agreement.

A scheme for detecting signatures of phase transitions associated with pure quantum states, from the knowledge of a limited set of expectation values, is introduced. An accurate prediction of critical regions in ground states of systems with a finite number of particles is obtained.

The lattice-cluster theory (LCT) is applied to describe a self-avoiding tree of specified topology on a d-dimensional simple hypercubic lattice. Equivalently, the chain can be viewed as linear and composed of structural units occupying several lattice sites. The self-interacting polymer has a nearest-neighbor contact interaction energy ɛ and occupies a fraction Φ of the lattice volume. The LCT provides a double series expansion in ɛ and ${\mathit{d}}^{\mathrm{-}1}$ for the specific free energy of the tree. Coefficients in this expansion depend on both the tree architecture and Φ. Our theory is illustrated for some tree topologies by evaluating the specific entropy, energy, and the mean number of contacts per segment. The thermodynamic properties of lightly branched trees are shown to remain identical to those of corresponding linear chains. On the other hand, dense branching (in which the number of branching points grows linearly with tree size) greatly affects the polymer properties.

We study the number of distinct sites visited by N random walkers after t steps SN(t) under the condition that all the walkers are initially at the origin. We derive asymptotic expressions for the mean number of distinct sites 〈SN(t)〉 in one, two, and three dimensions. We find that 〈SN(t)〉 passes through several growth regimes; at short times 〈SN(t)〉∼td (regime I), for tx≪t≪tx′ we find that 〈SN(t)〉∼(t ln[N S1(t)/td/2])d/2 (regime II), and for t≫tx′, 〈SN(t)〉∼NS1(t) (regime III). The crossover times are tx∼ln N for all dimensions, and tx′∼∞, exp N, and N2 for one, two, and three dimensions, respectively. We show that in regimes II and III 〈SN(t)〉 satisfies a scaling relation of the form 〈SN(t)〉∼td/2f(x), with x≡N〈S1(t)〉/td/2. We also obtain asymptotic results for the complete probability distribution of SN(t) for the one-dimensional case in the limit of large N and t.

We report the results of extensive Monte Carlo simulations of a model tethered membrane in which the particles interact through hard-core repulsion as well as a longer-range attractive potential. As the temperature is decreased, the membrane undergoes a phase transition from the usual high-temperature flat phase to a low-temperature crumpled phase. The crumpled phase, which seems to exist over a range of temperatures, is isotropic and is characterized by a fractal dimension close to the Flory estimate ${\mathit{D}}_{\mathit{f}}$=2.5, consistent with recent experimental results. At still lower temperatures the membrane is in a collapsed phase.

The problem of constructing a field theory that describes the dynamic critical behavior of semi-infinite systems whose dynamic bulk critical behavior is represented by model B and whose order parameter is conserved both in the bulk and at the surface is reconsidered. Particular attention is paid to the derivation of the boundary conditions satisfied by the order-parameter field (φ(x,t) and the associated response field φ̃(x,t). It is shown that the extremely complicated boundary conditions for φ obtained recently by Binder and Frisch [Z. Phys. B 84, 403 (1991)] simplify considerably if all irrelevant surface contributions to the action are discarded. In particular, the boundary conditions for both φ and φ̃ do not involve time derivatives. Although power counting alone admits surface terms other than those anticipated in the paper by Dietrich and Diehl [Z. Phys. B 51, 343 (1983)], the requirements of detailed balance imply that these extra terms are either absent or redundant. As an application and test, the relaxation of the order-parameter profile from a spatially homogeneous initial nonequilibrium state into thermal equilibrium is investigated, using a zero-loop approximation. The results are in conformity with those of Binder and Frisch.

The Kardar-Parisi-Zhang equation for surface growth is analyzed in the regime where the nonlinear coupling constant is small. We present detailed calculations for the mean-square surface width in terms of the bare parameters of the equation. For surface dimension d≤2, this quantity is shown to obey crossover scaling. The case d=2 is marked by an exponentially slow crossover associated with the marginally unstable character of the linear theory. For d>2 a renormalization-group analysis in the one-loop approximation yields a logarithmic scaling form at the roughening transition between smooth and rough growth phases. The crossover behavior on either side of this transition is discussed.

The roughening behavior of moving surface under a deposition and evaporation dynamics is explored within the hypercube-stacking model. One limiting case of the model is an equilibrium surface, which exhibits thermal roughening for surface dimension d≤2. Another limiting case is nonequilibrium irreversible growth, where the model is shown to map exactly to zero-temperature directed polymers on a hypercubic lattice with a random energy distribution. Results of exact calculations for d=1 and of large-scale Monte Carlo simulations [N=$2^{20}$, 11 ${520}^2$, and 2×${192}^3$ surface sites for d=1, 2, and 3, respectively] are presented that establish the Kardar-Parisi-Zhang equation as the correct continuum description of the growth process. For pure deposition (i.e., irreversible growth), careful analysis of surface width data yields the exponents β(2)=0.240±0.001 and β(3)=0.180±0.005, which violate a number of recent conjectures. By allowing for evaporation, we observe a less rapid increase of the surface roughness as a function of time. This phenomenon is consistently explained by a crossover scenario for d=1 and 2 but a nonequilibrium roughening transition for d=3, as predicted by a perturbative renormalization-group analysis of the Kardar-Parisi-Zhang equation. Detailed predictions on crossover scaling from the renormalization-group analysis are also confirmed by simulation data. In the d=1 case, some of the continuum parameters characterizing the renormalization-group flow are obtained explicitly in terms of the lattice parameters via the exact calculation of steady-state properties of the model.

The dynamic structure factor for an Enskog gas subject to thermal constraints is studied both by molecular dynamics and by fluctuating hydrodynamics. Good quantitative agreement is demonstrated.

The complex dielectric-susceptibility tensor and spectra of the molecular dipole-moment autocorrelation matrix are calculated for a system of noninteracting linear molecules located in a constant external electric field. Calculations were carried out in the context of the extended rotational-diffusion (J-diffusion) model. Sum rules for the integrated absorption are derived. Results are compared with those for magnetic relaxation of fine ferromagnetic particles in a ferrofluid and to available experimental data.

We study the stability of a solid-liquid interface in a hypercooled melt, taking into account attachment kinetics, surface energy, and surface energy in the heat balance. There is a basic-state solution with the planar interface moving at constant speed. Linear-stability theory gives a long-wave absolute-stability limit. Near this point we introduce a string model in which we use a thermal-boundary-layer approximation and obtain an evolution equation for the interface. In a limiting case this interface equation reduces to a Kuromoto-Sivashinsky equation. Comparison with experimental and numerical results are discussed and a conceptual picture of unconstrained growth for all undercoolings is addressed.

Previous work on estimating power output from heat engines is extended to estimating efficiency under finite-time constraints. A numerical method for estimating efficiencies for a class of finite-time thermodynamic models with variable-temperature reservoirs and arbitrary heat-exchange laws is proposed. The method is based on energy and entropy balances, and on the use of optimal control, Lagrange multipliers, and convex optimization. It is applicable both to lumped-parameter and distributed-parameter models. An analytical expression for the upper bound of the efficiency is obtained for constant-temperature reservoirs and Newtonian heat-exchange laws. Results of computations for two efficiency definitions are presented.

The motion of dynamically neutral Brownian particles that are influenced by a unidirectional velocity field of the form v(x,y)=${\mathit{v}}_0$‖y${\mathrm{\Vert }}^{\mathrm{\beta }}$sgn(y)x^, with β≥0, is studied. Analytic expressions for the two-dimensional probability distribution are obtained for the special cases β=0 and 1. As a function of β, the longitudinal probability distribution of displacements exhibits bimodality for β<${\mathrm{\beta }}_{\mathit{c}}$ and unimodality otherwise. A simple effective-velocity approximation is introduced that provides an integral form for the longitudinal probability distribution for general β and predicts the existence of this transition. A numerical exact enumeration of the probability distribution yields ${\mathrm{\beta }}_{\mathit{c}}$=3/4. The power-law model parallels the behavior found for tracer motion in a class of non-Newtonian fluids, where a unimodal-to-bimodal transition is also found to occur.

We introduce a shell model for the small-scale statistics of a scalar field passively carried by a three-dimensional turbulent fluid. The model mimics a cascade process for the spectral transfer of both velocity and a passive scalar. It is found to exhibit an intermittency correction to the 5/3 Obukhov-Corrsin power law for the power spectrum of a passive scalar, at wave numbers k where molecular diffusion and viscosity play a negligible role (inertial convective subrange). The structure functions and the probability distribution functions of the increments of the passive scalar are also computed. The results of our model are in excellent agreement with the experimental data.

A turbulent flow lacking homogeneity and isotropy has been studied using the technique of photon-correlation homodyne spectroscopy to measure velocity differences in different directions and at various length scales. The measured intensity correlation function G(qt,L) is found to be of the scaling form G(qtu(L)) with u(L)∼${\mathit{L}}^{\xi }$, where u(L) is the characteristic turbulent velocity at the length scale L and q is the scattering vector. The exponent ξ varies with scattering geometries and with flow cells. The scaling behavior of G(qt,L) is found to be independent of spatial positions and orientations. A scaling G(qtu(L)) indicates that the velocity-density function P(V,R) has a scaling form Q(V/u(R))/u(R). The experiment suggests that the scaling argument can still be used to describe anisotropic and inhomogeneous turbulent flows, and that the exact functional form of P(V,R) (which is of scaling form) may vary with spatial positions and orientations in turbulent flows, reflecting the inhomogeneity and anisotropy of the turbulence.

Turbulent-drag reduction in a dilute polymer solution has been studied using the technique of photon-correlation homodyne spectroscopy to measure velocity differences in a concentric cylinder cell, in which the inner cylinder rotates. A large anisotropic suppression of turbulent-velocity differences is found in the bulk region of the turbulent fluid. The suppression effect occurs at various length scales up to ≊1 mm, which is far beyond the Kolmogorov dissipation length ${\mathit{l}}_{\mathit{d}}$ (≊0.04 mm). The large-scale velocity fluctuations are suppressed, but their statistical properties over varying length scales remain unchanged. The small-scale fluctuations, on the other hand, are damped out much more strongly, resulting in a different functional form for the velocity-density function. The latter observation is consistent with the notion that the polymer-turbulence interaction causes a truncation of the turbulent-energy cascade at small scales. The measurements are also in agreement with laser Doppler velocimetry results that in turbulent polymer solutions the lifetime of large-scale vortices is increased, and the high-frequency velocity fluctuations (small-scale motions) are reduced.