A simple irregular scattering system is investigated classically and quantum mechanically. The existence of sharp resonances due to the quantum tunneling between the chaotic and regular parts of the classical phase space is demonstrated.

We test the reliability of the different generalized fractal dimension estimators, when applied to point distributions with a priori known scaling properties. We consider the effects of varying the amount of available data and the dimensionality of the distribution. The present work is motivated by the growing interest in cosmological context to safely analyze the scale-invariant properties of the observed galaxy distribution; these results may also be of value in all physical situations where the statistical analysis of a fractal ‘‘dust’’ is required. We consider (a) a monofractal structure with dimension D=1, (b) a multifractal structure, and (c) a scale-dependent structure, behaving like a D=1 monofractal at small scales and an homogeneous dust at large scales. For this structure, the clustering strength and the point number density have been chosen as to be similar to those observed for the galaxy distribution. Although the different methods display different advantages and pitfalls, we find that the presently available galaxy samples can be usefully employed to trace the scaling properties generated by nonlinear clustering.

Two chaotic orbits can be synchronized by driving one of them by the other. Some of the variables of the driven orbit are set continuously to the corresponding variables of the drive orbit. It has been seen that synchronization can be achieved if the subsystem Lyapunov exponents corresponding to the remaining or response variables are all negative. We find that a procedure where the drive variable is set at discrete times can also achieve synchronization. However, the synchronization criterion is altered by the effect of the drive being set at finite time steps. An important consequence of this is found in the Lorenz system where synchronization can be achieved with z as the drive variable despite the existence of a marginal subsystem Lyapunov exponent. We also find that synchronization can be achieved for the Rössler attractor with z as the drive, even though the largest subsystem Lyapunov exponent is positive. In addition, we find that there is an optimal time step corresponding to the fastest rate of convergence for both cases above. Our synchronization criterion reduces to the usual subsystem-Lyapunov-exponent criterion in the limit of the time step tending to zero.

A crossover behavior is investigated in Scheidegger’s river-network model [Bull. Int. Acco. Sci. Hydrol. 12, 1 (1967); 12, 15 (1967)] where a river meanders left with probability p and right with probability 1-p. Near p=1 (or p=0), the crossover phenomenon occurs from linear rivers at smaller length scales than the crossover length ${\mathit{t}}_{\mathit{c}}$ to the river network of a self-affine fractal at larger length scales than ${\mathit{t}}_{\mathit{c}}$. For 0<p<1, the river network always crosses over the self-organized critical state. The mean river size 〈S〉 scales as 〈S〉≊t for t<${\mathit{t}}_{\mathit{c}}$ and 〈S〉≊${\mathit{t}}_{\mathit{b}}^{\mathit{d}}$ (${\mathit{d}}_{\mathit{b}}$=1.50) for t>${\mathit{t}}_{\mathit{c}}$ where ${\mathit{d}}_{\mathit{b}}$ is the scaling exponent of the drainage basin area. The crossover length ${\mathit{t}}_{\mathit{c}}$ scales as ${\mathit{t}}_{\mathit{c}}$≊(Δp${)}^{\mathrm{-}1/\mathrm{\phi }}$ (1/φ=1.033±0.050) where Δp=1-p near p=1 (or Δp=p near p=0). The mean river size is described by the scaling form 〈S〉=tf(t/${\mathit{t}}_{\mathit{c}}$) where f(x)≊1 for x≪1 and f(x)≊${\mathit{x}}_{\mathit{b}}^{\mathit{d}}$-1 for x≫1. For a sufficiently small Δp, the mean river size 〈S〉 also scales as 〈S〉≊Δ${\mathit{p}}^{\gamma }$ (γ=0.484±0.020). The cumulative river size distribution ${\mathit{N}}_{\mathit{S}}$ scales as ${\mathit{N}}_{\mathit{S}}$≊(Δp${)}^{\mathrm{-}2\gamma /3\mathrm{}}$${\mathit{S}}^{\mathrm{-}1/3}$.

A characteristic-functional approach is introduced to study the space and time stochastic fluctuations of the electron population in a simple one-dimensional model of ionization growth. Two electron sources are considered: (a) ionization by direct collisions and (b) photoemission at the cathode due to de-excitation of atoms. The motion of the ions is neglected and the electrons are assumed to move with constant drift velocity. An equation for the characteristic functional G[θ(x),t]=〈exp[i${\mathcal{F}}_0^{\mathit{L}}$dx θ(x)n(x,t)]〉 is obtained, where n(x,t) is the electron density, and θ(x) is a conjugate function; from this, equations for the moments, e.g., the average density and the density-density correlation function, can easily be derived. Similarly to using the method of compounding moments, this technique avoids the use of a probability in function space; however, it has the benefit that the motion in configuration space may be incorporated self-consistently, whether the motion is free streaming or diffusion. Numerical examples are used to illustrate the time behavior of the mean total population in the gap, which shows a good agreement with previous results; in addition, we analyze the time evolution of the associated electron mean density, the density-density correlation function, and the fluctuations around the mean population.

We study the nonlinear relaxation in the presence of multiplicative noise by means of a simple approximation scheme valid outside the critical region and exact asymptotic expansion at the critical point. The theory is developed in the Malthus-Verhulst stochastic model case. We find nonmonotonic growth of fluctuations during the transient. At the critical point we study the statistical properties of the finite time average of the original process. We obtain an exact result for the generating function exhibiting scaling asymptotic behavior at the critical point. We deduce also an asymptotic sum rule for the n-times correlation function of the original process and the asymptotic expression of the two-times correlation function. Our theoretical results are compared with numerical simulations and steady-state known properties.

In this contribution, the generalized thermodynamic formalism is applied to a nonhyperbolic dynamical system in two comparable situations. The change from one situation to the other is small in the sense that the grammar and the singularities of the system are preserved. For the discussion of the effects generated by this change, the generalized entropy functions are calculated and the sets of the specific scaling functions which reflect the phase transition of the system are investigated. It is found that even under mild variations, this set is not invariant.

The small-angle Krein collision (SAKC) in four-dimensional reversible maps refers to the codimension-2 bifurcation, where the eigenvalues of the Jacobian of the map at a symmetric fixed point collide close to +1 as some relevant parameters are varied. SAKC is always associated with the bifurcation of nearby fixed points. We investigate the existence and stability of invariant curves around these fixed points for a particular family of reversible maps, and find a rich structure of the phase space.

Stochastic resonance, a phenomenon in which random transitions between states in a multistate system are modulated by an external periodic signal, is usually studied statistically. In an experiment with a period-doubled circuit, we observe transitions between the two phases of the response induced by noise or chaos. Adding a weak periodic modulation allows us to observe stochastic resonance. It is also noted that transitions between phases are caused by a crisis which may be observed in the circuit. As a result, we are able to describe stochastic resonance with the techniques of deterministic dynamics, providing a link between the fields of deterministic and stochastic mechanics.

I consider the dynamics of a soft-spin Ising magnet, subject to the time-varying external magnetic field h(t)=h cosΩt. The system is modeled with a time-dependent Ginzburg-Landau equation and is studied at a mean-field level. The time-averaged magnetization (scrM) acts as the order parameter and divides the temperature-h plane into two phases. In contrast to a previous result that predicted a dynamical tricritical point separating a line of continuous and discontinuous transitions in scrM, I find that the transition is always continuous. The previous work utilized Glauber dynamics which, it is here argued, becomes inapplicable near the phase boundary line due to a critical slowing down.

With a simple rigid four-center interaction model we computed the thermodynamics and transport properties of liquid n-butane by equilibrium molecular dynamics. Unlike previous works, which investigated only one state near the boiling point with semiflexible models, this work considers six states ranging from the boiling point to the triple point. In contrast to the studies based on semiflexible potential models, our calculations give good agreement with experiment for states near the boiling-point state. Reasonable agreement with experiment was achieved for the pressure and the shear viscosity at denser states near the triple point. The equilibrium time-correlation functions were computed for different shapes of the model molecules. Their time dependence leads to interesting conclusions concerning the anisotropic behavior of the molecules in the liquid state. Unlike the correlation functions, the transport coefficients were found to be nearly independent of the molecule shape. This explains partially why our rigid potential model is successful.

We present a technique for calculating the Lyapunov spectrum from a scalar time series. The technique is particularly useful when the data set is short and/or noisy. The method is based on an orthogonal polynomial expansion of the dynamics. For comparison purposes we test the new technique to two previous methods. We find that the global method performs as well as, or better than, either of the previous techniques.

In order to determine the mean first-passage time for systems driven by a superposition of suitably scaled independent dichotomous Markovian processes (pre-Gaussian noise), the well-known absorbing boundary conditions must be complemented in the generic case by a novel type of natural boundary conditions. We treat explicitly a linear stochastic flow for the superposition of up to five dichotomous processes and compare the analytic results with a digital simulation for these processes and an Ornstein-Uhlenbeck process.

Local icosahedral order is found to increase as random hard-sphere packings (one- and two-component) generated on the computer are densified from the (recently established) ‘‘loose random-packing’’ limit to the ‘‘dense random-packing’’ limit. While icosahedral ordering in ‘‘atomic’’ systems is frequently ascribed to the energetic stability of icosahedral clusters, the present results show that icosahedral ordering can arise from packing constraints alone. However, the icosahedra are often distorted, partly due to the lack of preferred distance between hard spheres. At high density one-third to one-half of the pairs in the first peak of the radial distribution function (RDF) are icosahedral fragments. The splitting of the second peak, which is characteristic of packings of spherical particles, was studied by decomposing the RDF into components according to the local environment of the pairs. Linear trimers of spheres are responsible for the second subpeak while the first subpeak arises roughly equally from tetrahedra sharing a face and triangles with adjacent sides. The hard-sphere packings were compared with packings of soft, attracting spheres by relaxing the configurations under a Lennard-Jones potential. The fraction of pairs characteristic of local crystalline order in the first peak of the RDF was found to increase. The reversal of the relative height of the two parts of the split second peak results from a broadening of the distribution of distances within the linear trimers, while the distribution sharpens for the face sharing tetrahedra and adjacent triangles.

We construct a soluble model to describe the retrieval properties of the quantum transverse Hopfield model in neural networks. A set of equations for the order parameters describing retrieval and spin-glass phases of the Hopfield model in a transverse field are obtained. Based on these equations, phase diagrams are examined and the memory storage capacity of the networks is analyzed as a function of the temperature and transverse field.

The first-passage-time distribution function (FPTDF) is derived for a relativistic random walk (RW) in one-dimensional (1D) space limited by one boundary. Thereby two configurations are considered with the start of the RW directed towards or away from the boundary. For each configuration the distribution function (DF) of the RW path ends is evaluated subjected to either absorption or reflection occurring at the boundary. The DF’s are identical with the solutions of the telegrapher equation in 1D space under the same initial and boundary conditions. The calculus can be extended to two boundaries, where all combinations of reflecting and absorbing boundaries are possible, for which the Laplace transforms of the path-end DF’s and the FPTDF’s are presented. The derivation is demonstrated and cross-checked for the case of two absorbing boundaries.

The motion of an overdamped particle in a bistable potential U(x), driven by quasimonochromatic noise (high-frequency, narrow-band noise), has been investigated by means of electronic analog simulation. The escape rate from one potential well to another was found to be exponentially small compared to the reciprocal mean first-passage time to the top of the potential barrier. The logarithm of the quasi- stationary probability distribution was observed to fall extremely sharply at a particular value of x, quite close to the equilibrium position. Theory describing the nonanalytic dependence of this logarithm on the bandwidth of the noise is presented and shown to be in good agreement with experiment. Data are also presented for a symmetric monostable potential. In a certain parameter range, the quasistationary distribution is demonstrated to be independent of the form of such a potential.

The Landauer cost for erasing information demands that information about a physical system be included in the total entropy, as proposed by Zurek [Nature 341, 119 (1989); Phys. Rev. A 40, 4731 (1989)]. A consequence is that most system states—either classical phase-space distributions or quantum pure states—have total entropy much larger than thermal equilibrium. If total entropy is to be a useful concept, this must imply that work can be extracted in going from equilibrium to a typical system state. The work comes from randomization of a ‘‘memory’’ that holds a record of the system state.

We investigate the dynamics of models involving deposition and evaporation of dimers, trimers, . . .k-mers, using analytical methods and numerical simulations. Autocorrelation functions show power-law decays in time, which are related to broken continuous symmetries in associated spin-1/2 Hamiltonians. These include the Heisenberg ferromagnet as the first nontrivial example. For k≥3 the models exhibit strongly nonergodic behavior. The numbers of both partially and fully jammed subspaces, within which the evolution takes place, increase exponentially with the size of the system. Evidence of finite-size scaling and universality over k is presented which supports a phenomenological diffusive picture for the dynamics in many subspaces.

The coarsening process of froths and foams is described statistically in two dimensions by a ‘‘random-neighbor model.’’ This nonlinear dynamical model contains no free parameters; all rates are determined dynamically. It describes normal growth, agrees with experimental results, throws light on old data, and, hopefully, inspires new simulations and measurements. Lewis’s law [Anat. Rec. 38, 341 (1928); Am. J. Bot. 30, 74 (1943)], and some other properties, can be derived analytically from the model.