It is shown that the multifractal properties of a period-doubling attractor and of repellers associated with symmetric unimodal maps are affected if an asymmetry is introduced in the zth derivative of the map at its maximum of order z. For a quadratic map, a universal scaling factor associated with the period-doubling attractor is calculated as a function of an asymmetry parameter. Expansions of multifractal functions in terms of an asymmetry parameter are obtained.

Three classical adiabatic ensembles of recent origin and the well-known microcanonical ensemble are combined in a unified treatment of adiabatic shell ensembles. A few ensemble-average properties are treated, and the ideal-gas law is derived in each case for comparative illustrations. Two of the ensembles represent open systems which are of current interest in molecular-dynamics studies.

In a fourfold-symmetric stochastic web generated by a two-dimensional area-preserving map, those phase-space orbits that remain forever in the central part of the chaotic channels are selected by a suitable cutoff condition. By means of a horseshoe construction, these deep orbits are classified according to (i) their paths on the square lattice of hyperbolic fixed points, (ii) their transit times between successive lattice sites, and (iii) their horseshoe branch (left or right) at each lattice site. The set of initial conditions for the deep orbits can be placed on a tree that branches with infinite multiplicity at each level: level N of the tree corresponds to paths of N+1 steps on the lattice. Empirically, one finds two scaling relations, one for the level number and the other for the number of map iterations between successive saddle points. Using a transfer-matrix technique, the scaling relations are exploited to calculate the multifractal properties [Hausdorff dimension, Hausdorff measure, the f(α) curve] of the set of deep orbits, to obtain an invariant probability measure and to derive a Green function for the diffusion process on the lattice. Numerical calculations of many of these quantities are carried out for the value 0.5 of the control parameter.

It is shown that most nonperturbative density-functional theories of classical systems can be reformulated as second-order truncations of exact perturbative expansions, thus eliminating the ad hoc nature of such theories. This reformulation is used as a basis for discussion as to why some theories work better than others when applied to hard spheres, why most theories perform poorly for continuous potentials, and in what direction they might be modified so as to improve their performance.

The quantum Fokker-Planck equation for a Gaussian-Markovian bath is deduced by applying a method proposed by Tanimura and Kubo [J. Phys. Soc. Jpn. 58, 101 (1989)]. The results are expressed in the form of simultaneous differential equations in terms of density operators and can treat strong system-bath interactions where the correlated effects of the noise play an important role. The classical Fokker-Planck equation for a Gaussian-Markovian noise is obtained by performing the Wigner transformation, and its equilibrium state is shown to be the Maxwell-Boltzmann distribution. The method is convenient for numerical studies. Calculations for quantum-system harmonic oscillators and the double-well potential problems are demonstrated for cases of Gaussian-white noise and Gaussian-Markovian noise.

Feynman path-integral simulations of a system consisting of protons and distinguishable electrons under periodic boundary conditions are presented with the aim of calculating the phase diagram of hydrogen at various temperatures and densities. For a two-electron–two-proton system, for which Fermi statistics can be ignored, the equations of state and the radial distribution functions are obtained, including the effects of temperature and pressure ionization, for temperatures down to 0.68 eV and densities up to 1 g ${\mathrm{cm}}^{\mathrm{-}3}$. These results are compared to other models. Systems with more than two distinguishable electrons are found to be thermodynamically unstable, by collapsing into a cluster at low temperature, suggesting that further simulations be used to investigate the stability of Boltzmann matter.

We consider self-avoiding Lévy walks (as a model for very stiff polymers) in different environments. We calculate the effect of screening of excluded-volume interactions by random Lévy walks themselves, i.e., a melt of such stiff polymers. We formulate the effect on Lévy walks (LW) in different environments, such as Gaussian chains, i.e., a dilute mixture of LW in ordinary linear polymers. More generally we consider the swelling due to an arbitrary solvent of fractal dimension ${\mathrm{\delta }}_{\mathit{f}}$. We find special cases where the resulting Flory exponent becomes independent of the initial configuration of the LW. The swelling and screening of flexible chains in LW are also discussed, where we find an exponent ν=(3-1/2${\nu }_0$)/(d+2) for the chain, where ${\nu }_0$ is the size exponent of the random Lévy walk in the absence of excluded-volume interactions.

It is argued that phase transitions in finite systems are likely to exhibit level statistics ascribed to quantum chaos. Apparent counterexamples become chaotic under generic perturbation. The mathematical mechanism for this to occur lies in the abundance of exceptional points in the transitional region. Exceptional points are singularities in the complex λ plane of the spectrum ${\mathit{E}}_{\mathit{n}}$(λ), where λ is the parameter that effects the transition. Examples are given for illustration. It is shown that the quality of the mean field is unaffected outside the transitional region whether or not the transitional region is chaotic. Related properties of the eigenvectors are also discussed.

We derive an exact theory of the mean first-passage time for an arbitrary one-dimensional dynamical system driven by a multiplicative external noise with finite correlation time (colored noise). Three different cases of colored noise are discussed: the random telegraph signal, the pre-Gaussian noise, and the Ornstein-Uhlenbeck diffusion process. Using random telegraph signals as a tool, we fully solve the difficult problem of non-Markovian boundary conditions associated with such a problem. The analytic solution with these boundary conditions give a complete solution of the escape time in the presence of colored noise.

The asymptotic behavior of the first-passage time on a one-dimensional random chain with 〈ln(p)〉>〈ln(q)〉 but 〈q/p〉>1 is investigated (p and q are the transition probabilities of moving to the right and left, respectively). It is shown that the average and typical first-passage times differ. Asymptotically the first-passage time to a specified site at a distance L to the right of the starting location varies as ${\mathit{L}}^{\mathit{x}}$ (x>1), with a probability of 1 even though the average first-passage time goes as exp(CL).

The relaxation time τ and the thermal correlation length ${\xi }_{\mathit{T}}$ in randomly diluted Ising magnets near the percolation threshold are related by the singular dynamic scaling lnτ=f(ln${\xi }_{\mathit{T}}$), where f(x)=${\mathit{Ax}}^2$+Bx+C, with constants A,B,C. We investigate the dimensionality dependence of A by Monte Carlo simulation and compare our observations with the theoretical predictions.

The transport of a quantum particle in a dimer is investigated for the case in which the influence of the phonons is modeled by a dichotomic stochastic process with exponentially decaying correlation functions. An exact numerical solution of the density-matrix equation of the problem is carried out, and the transition from coherent to incoherent transport is discussed.

We explore the behavior of the fractal-shot-noise-driven doubly stochastic Poisson point process (FSNDP), for which the associated impulse response functions assume a decaying power-law form. For a variety of parameters of the process, we obtain expressions for the count-number distribution and moments, Fano factor, normalized coincidence rate, power spectral density, and time probability densities. A number of these measures exhibit power-law dependencies, indicating fractal behavior. For certain parameters the power spectral density exhibits 1/f-type behavior over a substantial range of frequencies, so that the process serves as a 1/${\mathit{f}}^{\mathrm{\alpha }}$ point process for α in the range 0<α<2. We consider two physical processes that are well described by the FSNDP: Cherenkov radiation from a random stream of charged particles, and diffusion of randomly injected concentration packets.

The Siegert integral-equation approach is used to study the statistics of first passage time in a noisy linear dynamical system with a swept control parameter. Small-noise-theory results are recovered from the source terms in the integral equations. Numerical results obtained using a simple algorithm are found to agree well with simulation results. An interesting case in which the first-passage-time probability density has the two most probable values is pointed out.

A generalization is presented of Kramers’s Brownian motion model in order to include nonisothermal activation events in noisy ‘‘macroscopic’’ systems (e.g., Josephson-junction devices). The theory is based on the thermodynamical significance of the effective potential for the pertinent degree of freedom (e.g., the magnetic flux) and heavily leans on notions from nonequilibrium thermodynamics. The thermodynamic potential generally depends on the specimen’s local temperature (which may differ from the reservoir temperature), such that the entropy varies along the reaction coordinate. For thermally isolated activation events, the escape rate from the metastable state is determined by the entropy barrier, which is generically higher than the usual isothermal Gibbs (or Helmholtz) free-energy barrier. For systems with a small heat capacity, the theory predicts a substantial entropic barrier enhancement.

An example of a strongly nonhyperbolic repeller with a continuous measure along the unstable direction is demonstrated.

From the viewpoint of eigenvalue level statistics, harmonic-oscillator systems are unusual. Although integrable, these systems are nongeneric, and a spacing distribution does not exist even as the number of levels N→∞. The origins of this pathological behavior are explored using methods of number theory and ergodic analysis. However, such nongenericity is extremely fragile, and the smallest nonlinearity asymptotically restores generic behavior. These results are of relevance to the study of molecular spectra, as well as to the quasienergy spectra of integrable quantum maps.

Specific features of a quantum map that can be connected to periodic orbits of the corresponding classical system are investigated. The model we studied is the kicked top for which it is known that many statistical properties of the quantum system, such as distributions of eigenvalues and eigenvectors, are well described by the random matrix theory. It is shown that other statistical measures of eigenvector statistics (e.g., Shannon entropy) reveal deviations from predictions based on ensembles of Gaussian matrices. These differences may be associated with periodic orbits. A formula, valid for an arbitrary quantum map, which can be helpful in associating scarred wave functions with a particular periodic orbit, is given. A comparison is made with the semiclassical results.

Some novel critical phenomena are discovered in a class of nonanalytic twist maps. It is found that the degree of inflection z plays a role reminiscent of that of dimensionality in phase transitions with z=2 and 3 corresponding to the lower and upper critical dimensions, respectively. Moreover, recurrence of invariant circles has also been observed. An ‘‘inverse residue criterion,’’ complementary to the ‘‘residue criterion’’ for the determination of the disappearance point, is introduced to determine the reappearance point of invariant circles.

Previously published Monte Carlo molecular-dynamics data on the velocity autocorrelation function of the hard-sphere fluid are supplemented by data for the time-dependent transport coefficient to yield estimates for the self-diffusion coefficient throughout the fluid regime, from 25 to 1.6 times the close-packed volume. The dependence on system size is explored through results for 108–4000 particles. It is found essential to include long-time contributions to the Green-Kubo integral based on the previously validated mode-coupling theory. It is shown that the calculations of Easteal, Woolf, and Jolly [Physica 121A, 286 (1983)] fall significantly below the current estimates, which are in qualitative agreement with the Alder, Gass, and Wainwright [J. Chem. Phys. 53, 3813 (1970)] values.