A field is represented by its Fourier modes u(k,t) on the wave-number interval 0≤k≤${\mathit{k}}_{\max }$. We seek to reduce the number of degrees of freedom needed to describe the system by eliminating those modes in the band of wave numbers ${\mathit{k}}_{\mathit{c}}$≤k≤${\mathit{k}}_{\max }$, the ${\mathbf{u}}^{+}$, while retaining their average effect on the remaining modes, the ${\mathbf{u}}^{\mathrm{-}}$. Because of mode-mode coupling, this requires, in principle, a conditional average over the ${\mathbf{u}}^{+}$, in a subensemble in which the ${\mathbf{u}}^{\mathrm{-}}$ are held (approximately) constant. The conditional average can be related to the full ensemble average by means of the decomposition ${\mathbf{u}}^{+}$=${\mathbf{v}}^{+}$+${\mathrm{\Delta }}^{+}$, where ${\mathbf{v}}^{+}$ is independent of ${\mathbf{u}}^{\mathrm{-}}$ but has the same global mean properties as ${\mathbf{u}}^{+}$, while ${\mathrm{\Delta }}^{+}$ represents the effect of mode coupling on the conditional average. Implementation of the formalism for any particular physical system requires an ansatz for the relationship between ${\mathbf{v}}^{+}$ and ${\mathbf{u}}^{+}$ such that the conditional average of ${\mathrm{\Delta }}^{+}$ is small. This procedure is illustrated by its application to the Navier-Stokes equation, where ${\mathbf{v}}^{+}$ is taken to be the first-order expansion of ${\mathbf{u}}^{+}$ in a Taylor series about k=${\mathit{k}}_{\max }$, thus permitting a renormalization-group calculation of the turbulent effective viscosity, as reported previously [Phys. Rev. Lett. 65, 3281 (1990)]. For systems like this, with deterministic time evolution, the conditional average must be based on the weak, or imprecise, condition that ${\mathbf{u}}^{\mathrm{-}}$+${\mathrm{\Phi }}^{\mathrm{-}}$ is held constant, where ${\mathrm{\Phi }}^{\mathrm{-}}$ is small (but not zero) and has zero mean under the conditional average, in order to ensure chaotic ${\mathbf{u}}^{+}$. We show that our formalism, relating conditional and full ensemble averages, is independent of ${\mathrm{\Phi }}^{\mathrm{-}}$ to second order in small quantities.

A model of many globally coupled phase oscillators is studied by analytical and numerical methods. Each oscillator is coupled to all the other oscillators via a global driving force that takes the form ${\mathit{tsum}}_{\mathit{j}}$ g(${\mathrm{\phi }}_{\mathit{j}}$), where g(${\mathrm{\phi }}_{\mathit{j}}$) is a periodic function of the jth phase. The spatiotemporal properties of the attractors in various regions of parameter space are analyzed. In addition to simple spatially uniform fixed points and limit cycles, the system also exhibits spatially nonuniform attractors of three kinds. First, there are cluster states in which the system breaks into a few macroscopically big clusters, each of which is fully synchronized. Second, there is a stationary state with full frequency locking but no phase locking. The distribution of phases is stationary in time. Third, in an extremely narrow regime of parameters, a nonperiodic attractor exists. It is found that the cluster state is stable to the addition of weak stochastic noise. Increasing the level of noise beyond a critical value generates a continuous transition to a stationary ergodic state. In the special case where the nonlinearities in the dynamics involve only first harmonics, marginal states are observed, characterized by a continuum of marginally stable limit trajectories. These states are unstable under the introduction of noise.

We discuss an extension of Gutzwiller’s [J. Math. Phys. 8, 1979 (1967); 10, 1004 (1969); 11, 1791 (1970); 12, 343 (1971); Chaos in Classical and Quantum Mechanics (Springer, New York, 1990)] semiclassical theory for chaotic systems that allows a determination of matrix elements in terms of classical periodic orbits. Associated zeta functions are derived. The semiclassical predictions are found to be in good agreement with Fourier transforms of quantum spectra of hydrogen in a magnetic field. Expressions for off-diagonal matrix elements are derived that are extensions of the Bohr correspondence relations for integrable systems.

A quantum condition on phase-space distribution functions is introduced based on the probability for the quasimomentum k and quasicoordinate q. This condition is complementary to the conditions on the momentum p and the coordinate x. It is shown that the Wigner distribution does not satisfy this condition, and it is the Rihaczek function that obeys it. An equality is established for sums on the Wigner and the Rihaczek distributions. Linear dependences of phase-space distribution functions are derived based on the zeros of the kq functions.

A problem of correlated fluctuations in multielement systems is considered in the framework of the theory of stochastic branching processes and is formulated in terms of a population model. Correlations have been taken into account by means of the heredity function. An explicit mathematical method is proposed to solve the problem without referring to the complicated theory of general branching processes. Using this method the global statistical properties of the system (e.g., the ensemble average and the probability of extinction) have been found. Possible applications of the present model are discussed.

An alternative forecasting technique for chaotic time series, based on the optimal association concept, is presented. The method is applied on series generated by the logistic and Hénon maps, and on experimental data corresponding to rainfall of a storm event. In all three cases the quality of the forecasts is analyzed in terms of the prediction interval, the length of the historic data available on the time series, and the dimension of the embedding space. It is shown that the method is capable of producing very satisfactory short-term forecasts for data sequences of small lengths as they often occur in real experiments. Our results also show that the present technique can be used to discriminate complex signals associated with deterministic chaos from those of random origin.

The complete rule of the Feynman-diagram expansion of a nonequilibrium process is given by using the path-integral technique. At an initial time ${\mathit{t}}_{\mathit{I}}$, the system is in equilibrium and is brought into the nonequilibrium state by a time-dependent Hamiltonian. All the interactions, including the initial correlation, are taken into account by a diagrammatic expansion. Besides the well-known equilibrium propagators and the conventional 2×2 nonequilibrium propagator matrix, the propagator becomes a 3×3 matrix that contains extra elements depending explicitly on ${\mathit{t}}_{\mathit{I}}$. Both the Feynman rule of the coordinate representation and the coherent-state representation are presented. It is shown that the propagator matrix is expressed compactly by the contour representation. The classical limit is also discussed.

We have extended our earlier analysis of a periodically driven anharmonic oscillator with higher-order nonlinearity. In the present paper, to start with, we present a brief review of the stationary situation with the help of a bifurcation diagram, confirming the period doubling. Next we pass over to the nonstationary case, when the amplitude and frequency of the driving force are both time dependent. Here we have considered two types of time dependence for the frequency, linear and cyclic. We have also made a detailed analysis of sub- and superharmonic-resonance formation. An interesting phenomenon occurs in the transition from the primary to the superharmonic-resonance region, which paves the way to chaos. Finally, the case of quasiperiodic driving force is considered for the analysis of the combination resonances. The whole situation is analyzed on the basis of a phase-space plot and response curve.

The use of cycle expansions for spin systems with long-range interactions is explored numerically. It is found that the thermodynamic ζ function is an effective tool, both in practice and in principle, for the study of Ising models with power-law interactions. To deal with phase transitions the cycle expansion is factorized, and accurate phase-transition points for several power-law models are obtained, together with other thermodynamic quantities.

A numerical study of propagation of a particle through a one-dimensional dissipative medium is presented. The medium is composed of several dissipative sections, which are characterized by their friction coefficients η. In particular, we have considered two types of friction coefficients distributed orderly or disorderly along the chain. For the same relative proportion of the coefficients, we have found that 2 transport can be enhanced in the disordered distribution in comparison with the ordered one. We also show how this can be considered an approximated way to treat the propagation in a dissipative medium with a position-dependent friction coefficient.

A continuous-time approach for stochastic renormalization of fractal random processes is suggested. The jump clustering is described in terms of two waiting time densities: the probability density ${\mathrm{\eta }}_{\mathit{c}}$(τ) for the time interval τ within which the clustering takes place and the density ${\mathrm{\eta }}_{\mathit{r}}$(τ) for the relaxation time attached to the probability that a clustering event occurs. The condition of scale invariance requires that ${\mathrm{\eta }}_{\mathit{c}}$(τ) and ${\mathrm{\eta }}_{\mathit{r}}$(τ) are exponential functions of time. The statistical properties of the number n of jumps from a cluster are determined by a single parameter H=τ${\mathrm{¯}}_{\mathit{r}}$/τ${\mathrm{¯}}_{\mathit{c}}$, where τ${\mathrm{¯}}_{\mathit{r}}$ and τ${\mathrm{¯}}_{\mathit{c}}$ are the mean times attached to the distributions ${\mathrm{\eta }}_{\mathit{r}}$(τ) and ${\mathrm{\eta }}_{\mathit{c}}$(τ). For 0<H≤1 all moments of n are infinite. In this case the renormalization leads to a Lévy flight whose characteristic exponent is equal to 2H. The occurrence of lacunary series is not a necessary condition for the nonanalytic behavior of renormalized structure functions. The fractal behavior of the renormalized random walk is due only to the fractal behavior of the number n of jumps from a cluster.

A stochastic analog of renormalization-group transformations for fractal random processes is suggested. We start from a translationally invariant continuous-time and -space random walk. The jump events are grouped into hierarchical clusters. The process of jump clustering is described by means of a set of random aggregation probabilities selected from certain probability laws. The number of jumps from a cluster is a random variable. The renormalization consists in replacing the jump events from a cluster by a single renormalized (i.e., overall) jump. Explicit conditions for the occurrence of statistical fractal jump distributions are derived. If the probability distribution for the number of the jump events from a cluster is fractal, the same is true for the renormalized probability densities of the waiting time and displacement vector. The scaling of the jump events, space, and time is characterized by means of a unique fractal exponent. The renormalized stochastic process is compared with an alternative approach based on the use of random multiplicative processes. The comparison between the two formalisms illustrates Lévy’s theory of probability limit distributions with infinite moments. Although the rates of convergence and the corresponding parameters may be different, both models lead asymptotically to the same universal Lévy distribution. The possible applications and generalizations of this method are also considered.

The influence of homoclinic tangles and separatrices on the eigenstates and quasienergies of a quantized, nonintegrable, piecewise-linear standard map is investigated both analytically and numerically. The strength of scarring on hyperbolic fixed points is found to depend strongly on the phase-space area covered by the corresponding homoclinic tangles. It is particularly pronounced when the tangle degenerates to an unbroken separatrix and is seen to decrease with decreasing ħ. In the case of the unbroken separatrix, an interesting resonance phenomenon, which occurs for certain values of ħ, is also discussed. This effect is related to a degeneracy in the quasienergy differences for a triplet of states which lie energetically below, on, and above the saddle. This degeneracy is shown to give rise to periodic recurrences for a coherent state initially placed at the hyperbolic fixed point. The relevant time scale T is calculated semiclassically and exhibits the asymptotic behavior T∼lnħ. The implications of these results to other systems is also discussed.

This paper analyzes the size and shape of a closed, two-dimensional random walk with a pressure difference p between the inside and outside, which couples to an algebraic (signed) area. This pressurized-random-walk (PRW) model is, in some respects, closely related to a computer model studied by Leibler, Singh, and Fisher [Phys. Rev. Lett. 59, 1989 (1987)]. Since all terms in the Hamiltonian are quadratic in the position-vector field r, the partition function and its derivatives can be evaluated exactly. The most notable feature of the PRW model is an instability, which occurs at ‖p‖=${\mathit{p}}_{\mathit{c}}$. For ‖p‖<${\mathit{p}}_{\mathit{c}}$, the system has a finite algebraic area and an anisotropic shape; for ‖p‖≥${\mathit{p}}_{\mathit{c}}$, the algebraic area diverges and the shape is circular. The asphericity is also calculated. A form of bending rigidity, also quadratic in r, is introduced into the model; however, the resulting macroscopic properties are quite different from those one might ordinarily expect. This difference can be traced to the absence of a fixed link size in the model.

The quantum dynamics of a particle in a double-well potential under a monochromatic external driving force is computed by using a minimum-uncertainty Gaussian wave packet as an initial state. Irregular or regular time development of dynamical variables is obtained, depending on whether the initial wave packet is located in the classical chaotic sea or the regular region in phase space, respectively. Tunnelings between nonresonant islands and between resonant islands are obtained. Tunneling times between nonresonant islands decrease smoothly, in general, with the increase of the amplitude of the driving force, but tunneling times between resonant islands are more erratic.

We study the relationship between classical and quantum structures for a map on the sphere whose behavior can be chaotic in the classical limit. On the classical side we implement an efficient method to locate periodic points on symmetry lines. On the quantum side we show how matrix elements of the propagator in the coherent-state representation are connected to classical structures. Diagonal and off-diagonal matrix elements are related to periodic points, symmetry lines, and other invariant structures in phase space, both in the time and in the energy domains. The scarring phenomena related to the short periodic orbits and their homoclinic neighborhoods are discussed.

The hypernetted-chain closure is modified to include three-body potentials. The latter is reduced to a state-dependent effective pair potential. Explicit expressions for the Helmholtz free energy and for the chemical potential are derived for the modified closure. The approximation is shown to be relatively self-consistent and in qualitative agreement with perturbation results. An argonlike fluid is discussed using the Lennard-Jones pair potential and the Axilrod-Teller triple-dipole potential. In agreement with earlier work, it is found that the effective three-body contribution is positive, increasing the pressure and the internal energy by about 10% in the liquid state.

The nonequilibrium canonical distribution function of the generalized Boltzmann equation entails a nonequilibrium Kirkwood hierarchy of integral equations for dynamic correlation functions. A sheared simple fluid at constant density and temperature is considered. Shear effects appear in the nonequilibrum potential as additional contributions to the intermolecular potential. Approximations are introduced into the hierarchy to derive a closed integral equation for the dynamic pair-correlation function that reduces identically to the Percus-Yevick integral equation as the shear rate vanishes. Two exact results are obtained: the dynamic pair-correlation function is proven to possess the symmetry of the nonequilibrium potential, and the corresponding structure factor is shown to be positive definite for all wave-number vectors. The dynamic pair-correlation function is expanded in terms of the spherical harmonics, and the equations for coupled radial components are solved numerically. The shear rate and angular dependences of the dynamic pair-correlation function and structure factor are shown. In the low-shear-rate limit, the shear stress ${\mathit{P}}_{\mathit{x}\mathit{y}}$, pressure difference p(γ¯)-p(0), and normal-stress differences ${\mathit{P}}_{\mathit{y}\mathit{y}}$-${\mathit{P}}_{\mathit{z}\mathit{z}}$ and ${\mathit{P}}_{\mathit{y}\mathit{y}}$-${\mathit{P}}_{\mathit{z}\mathit{z}}$ obey power laws with exponents 1, 2, and 2 respectively; ${\mathit{P}}_{\mathit{x}\mathit{x}}$-${\mathit{P}}_{\mathit{y}\mathit{y}}$ is zero identically.

The transport properties of an ensemble of charged test particles diffusing from infinity through a second charged species that fills the positive half-space are studied. It is assumed that there are no incoming particles at the boundary between the medium in the positive half-space and the vacuum that fills the negative half-space. This problem has been referred to as the Milne problem and has previously been considered for neutral species [M. J. Lindenfold and B. Shizgal, Phys. Rev. A 27, 1657 (1983)]. The present paper examines the rarefied-gas dynamical effects for the Coulomb interaction between the two species. The distribution function of the background gas is taken to be a Maxwellian and the test-particle distribution is determined from a solution of the Boltzmann equation with its expansion in Burnett functions, products of Laguerre polynomials, and spherical harmonics. The extrapolation length is calculated together with the density and temperature profiles and compared with the Chapman-Enskog results valid far from the boundary. A comparison with the previous results for the hard-sphere cross section describing collisions between two neutral species is also presented.

Measurements of dye transport by /Idisordered capillary waves (generated by the Faraday instability) are reported as a function of the wave amplitude. The transport is diffusive, with effective diffusion coefficients that are typically at least a factor of ${10}^3$ larger than the molecular values. A model based on fluctuations in the Stokes drift caused by the disordered surface waves is tested and found to predict diffusion coefficients of the right order of magnitude, but roughly a factor of 3 too small. Residual correlations in the wave field are suggested to account for this discrepancy.