The equations of motion which govern the evolution of collective variables introduced into a nonlinear Klein-Gordon system are obtained by projecting the nonlinear field equation onto vectors in Hilbert space. For systems in the zero-kink sector, which admit breather and kink-antikink dynamics, the magnitude of some vectors onto which the field equation is projected become zero at particular instants in time. At these instants in time, the projections are not defined, and consequently the collective-variable equations of motion themselves are not defined: the transformation from the original field variables to the collective variables becomes singular. Although the singularity problem has already appeared in various contexts in the literature, it has not been resolved in the context of a collective-variable theory which contains collective variables in addition to that of the center of mass. For such cases, we show the singularities are removed by correctly taking into account the relativistic invariance of the underlying theory and give an example Ansatz which admits a well-defined collective-variable theory which would otherwise have been singular. We comment on the increased complexity of the present Ansatz relative to the utility of collective-variable theory.

The dynamics of a one-dimensional Brownian particle are treated theoretically over a wide time scale in the presence of random reflecting barriers. Green functions for particle motion with two kinds of barriers are formulated, one being diffusion with fixed random barriers and the other with stochastic random barriers. The stochastic barriers are expected to appear randomly on the time axis as well as with positional disorder and then, after a while, to disappear. To treat these problems, we first develop a mean-field Green-function MFG theory. The effects of many barriers, regarded as a multiple perturbation to the free diffusion of the particle, are decomposed into many elements. Their average effects (the first cumulant of the random perturbation elements) are sequentially incorporated into the unperturbed Green function up to the number of the elements. The first-cumulant MFG is thus obtained and covers the whole range of the perturbation intensity of the mean element. To improve the approximation, the second cumulants of the perturbation elements are incorporated one by one into the first-cumulant MFG, yielding the second-cumulant MFG. Taking into account a higher-order correlation among the perturbation elements yields a higher-order cumulant MFG. By applying the first-cumulant and the second-cumulant MFG theories to the particle diffusion with fixed and with stochastic barriers, we calculated the mean-square displacement as a function of time. It was found that the first-cumulant MFG describes qualitatively the dynamic behavior of the particle on all the time scales, whereas the second-cumulant MFG gives a more accurate numerical coefficient in the estimation of the time-dependent mean-square displacement.

Bistable systems subject to a stochastic force that is singular at the deterministic unstable point are investigated. Various noise-induced critical phenomena, such as pitchfork bifurcations and saddle-node bifurcations, are revealed. A particular codimension-two critical point is found. In a certain parameter region the systems can be trapped in the deterministically unstable state. The relaxation and the mean first passage times of the systems are analyzed.

We have studied the critical behavior of a generalized Frenkel-Kontorova model in which the external potential is nonanalytic and has a variable degree of inflection z. For z>3 a sequence of pinning and depinning transitions has been observed. The critical exponents of the gap in the phonon spectrum, the coherence length, and the Peierls-Nabarro barrier of the ground state are found to vary with z when 2≤z<3, but are independent of z when z≥3. It suggests that the degree of inflection plays a role quite similar to that of dimensionality, with z=2 and 3 corresponding, respectively, to the lower and upper critical dimensions.

We show, with precision numerical experiments, that far enough from the origin, the phase space of a Hamiltonian oscillator, kicked at a frequency that is incommensurate with its natural frequency, is featureless at some level of precision; that is, the periodicities that lead to island chains in resonantly forced oscillators are strongly suppressed. Therefore this system is suitable for the study of high-dimensional analogies of chaos, such as Fokker-Planck dynamics and Einstein diffusion. We determine the diffusion constants and the probability densities numerically and compare them to previous theoretical predictions. We observe a crossover behavior of the diffusion constant between regions of weak and strong forcing.

We study globally coupled chaotic maps modeling an optical system, and find clear evidence of nonstatistical behavior: The mean-square deviation (MSD) of the mean field saturates with respect to an increase in the number of elements coupled, after a critical value, and its distribution is clearly non-Gaussian. We also find that the power spectrum of the mean field displays well-defined peaks, indicating a subtle coherence among different elements, even in the ‘‘turbulent’’ phase. This system is a physically realistic model that may be experimentally realizable. It is also a higher-dimensional example (as each individual element is given by a complex map). Its study confirms that the phenomena observed in a wide class of coupled one-dimensional maps are present here as well. This gives more evidence to believe that such nonstatistical behavior is probably generic in globally coupled systems. We also investigate the influence of parametric fluctuations on the MSD.

We consider the effect of a high-frequency pumping E cosωt on the escape rate of a classical under-damped Brownian particle out of a deep potential well. The pumping frequency ω is assumed to be close to the frequency ω(0) of small oscillations. The energy dependence of the oscillation frequency ω(E) is also assumed to be weak on the scale of thermal energy, ωE′(0)T∼ω(0)T/U0≪ω(0) [ωE′(0) is the derivative of ω(E) at E=0, U0 is the barrier height, U0≫T]. The quadratic (in E) contribution to the decay rate is calculated exactly as a function of two reduced parameters, the anharmonicity of the potential ν≡ωE′(0)T/γ and the resonance detuning δ≡[ω-ω(0)]/γ, where γ is the friction coefficient. It is shown that the effect of the pumping increases with diminishing |ν| and at small ν is proportional to ν-1. In this regime dependence on δ is stepwise: the pumping contribution is large for νδ>0 and small for νδ<0. In the framework of our theory, the decay rate is invariant against the simultaneous change of the signs of δ and ν.

We study the transition to chaos caused by overlap of resonances in a dissipative standardlike map exhibiting recurrence of invariant circles. In particular, the effects of the recurrence on the structure of a critical line at which the motion is always mode locked are investigated. When there is no recurrence of invariant circles, a smooth critical line exists. In the recurrence case, however, the critical line is folded back and has discontinuous parts. At this line the mode-locked intervals trace a folded devil’s staircase with isolated parts. Scaling behavior of the complementary set to the folded staircase is also discussed.

Some investigations are made on the distribution of periodic-orbit lengths in a two-degrees-of-freedom pseudointegrable Hamiltonian system. Recently, it has been conjectured that N(T), the number of orbits having their length ≤T, scales as a fractal for such systems. We provide an explicit numerical test of this result using the simplest possible pseudointegrable barrier-billiard system proposed by Hannay and McCraw [J. Phys. A 23, 887 (1990)].

The one-dimensional single-species diffusion-limited-coagulation process with irreversible random particle input (A⇆A+A reversibly and B→A irreversibly), under the influence of external fluctuations in the system parameters, is formulated in terms of a closed and linear partial-differential equation. Our theoretical treatment includes both internal fluctuations and external noise simultaneously and without approximation, allowing investigation of the interplay of their effects on the macroscopic behavior of this diffusion-reaction system. For the reversible model with the rate of the A→A+A reaction fluctuating between two values as a Markov stochastic process, we solve the system exactly. We observe that spatially homogeneous macroscopic fluctuations in the system parameters can induce microscopic spatial correlations in the nonequilibrium steady state. Direct Monte Carlo simulations of the microscopic dynamics are presented, confirming the theoretical analysis and directly illustrating the external-noise-induced spatial correlations.

In a previous paper [T. Ohta, A. Ito, and A. Tetsuka, Phys. Rev. A 42, 3225 (1990)] we have shown that the excited domains arrayed periodically in space in a Bonhoeffer–van der Pol–type model system exhibit an in-phase oscillation when the time evolution of the inhibitor is slow compared to that of the activator. This was obtained by the linear-stability analysis for deviation of domain boundaries around the equilibrium position. In this paper, we study the postthreshold behavior. Our main concern is to derive a coupled set of equations for the phase and the amplitude of the oscillating domains. We start with an isolated oscillating domain and take into account the interaction among the domains perturbatively. It is shown that the nearest-neighbor-interaction strength changes the sign as the domain width is changed. As a result, when the number of the interacting domains is finite, the system exhibits bistability where in-phase and antiphase oscillations can coexist.

We investigate classical chaotic scattering between two atoms interacting via a Morse potential in the presence of a laser field. We find that stable resonance islands appear in phase space above the field-free dissociation energy in an impulsively driven model. A local expansion relating this model to the Hénon map predicts the existence of such resonance islands within some finite domain in parameter space for any impulsively driven potential model. By switching on the external field during collision we show how stable field-dressed molecules can be formed with energy above the field-free dissociation threshold. The experimental and quantum relevance of such phenomenon is discussed.

We address the problem of estimating parameters in systems of ordinary differential equations which give rise to chaotic time series. We claim that the problem is naturally tackled by boundary value problem methods. The power of this approach is demonstrated by various examples with ideal as well as noisy data. In particular, Lyapunov exponents can be computed accurately from time series much shorter than those required by previous methods.

Nearest-neighbor distribution functions characterize the probability of finding a nearest neighbor at some given distance from a reference point in systems of interacting particles and are of fundamental importance in a variety of problems in the physical and biological sciences. We extend the formalism of Torquato, Lu, and Rubinstein [Phys. Rev. A 41, 2059 (1990)] for identical spheres to obtain exact series representation of nearest-neighbor functions (void and particle probability densities) and closely related quantities for systems of interacting D-dimensional spheres with a polydispersivity in size. Polydispersivity constitutes a basic feature of the structure of random systems of particles and leads to a wider choice of possible definitions for nearest-neighbor functions. The most relevant definition for a polydispersed system of particles involves the ‘‘nearest particle surface’’ rather than the ‘‘nearest particle center’’ and thus we refer to them as ‘‘nearest-surface distribution functions.’’ For the special cases of D-dimensional hard and overlapping spheres, we obtain analytical expressions for the nearest-surface functions that are accurate for a wide range of sphere concentrations. Employing these results, we are able to compute the corresponding mean nearest-surface distances for polydispersed hard spheres. Finally, we determine the nearest-surface functions for bidispersed systems from Monte Carlo computer simulations and find that our theoretical results are in very good agreement with the data.

The covariant formulation of shallow-water theories on a rotating sphere is derived, taking into account the non-Euclidean geometry of the spherical surface. Geometric and dynamic consistency of the inertial and gravitational forces require particular consideration. Covariant diagnostics and equations of motion uniquely determine the energy and angular-momentum budgets. It is shown that the absence of pressure and frictional torques is insufficient to conserve the angular-momentum vector with respect to the rotation axis. On the other hand, it will be seen that in the framework of shallow-water theory topography does not affect the torque. The specific form of the inertial forces on a rotating sphere gives rise to a modified momentum vector, which is conservable under appropriate circumstances. Baroclinic effects are discussed in the framework of covariant bishallow-water theory.

A theory of incoherent inelastic neutron scattering from liquid water is presented, and recent experimental data obtained by employing the spallation sources are interpreted. This theory is a statistical theory, similar to those describing optical spectra of liquids. The experimental spectra of liquid ${\mathrm{H}}_2$O and ${\mathrm{D}}_2$O are reproduced theoretically. The incoherent neutron- and light-scattering spectra of water are compared. The suitability of this type of experiment to study high-frequency modes of translational-rotational motion of molecules in liquids is emphasized.

We study a three-mode truncation of the equation ${\mathit{u}}_{\mathit{t}}$+${\mathit{uu}}_{\mathit{x}}$+δ${\mathit{u}}_{\mathit{x}\mathit{x}\mathit{x}}$+${\mathit{u}}_{\mathit{x}\mathit{x}}$ +${\mathit{u}}_{\mathit{x}\mathit{x}\mathit{x}\mathit{x}}$=0. This simple model allows us to understand analytically the role played by dispersion in the appearance of well-defined traveling pulses. In the absence of dispersion, that is, for δ=0, the solutions obtained from the truncated model are in quantitative agreement with the known results for the Kuramoto-Sivashinsky equation for small horizontal periodicity.

We consider the propagation of a plane front separating a turbulent region of fluid from a quiescent region. Initially, the turbulent-energy distribution as a function of z, the displacement normal to the front, is assumed to be localized, and after a time t, general renormalization-group arguments show that there is a similarity solution of the form q(z,t)∼${\mathit{t}}^{\mathrm{-}(2/3+2\mathrm{}\mathrm{\alpha }\mathrm{̃})}$f (${\mathit{zt}}^{\mathrm{-}(2/3+\mathrm{\beta })}$, ε), where α̃ and β are ε-dependent anomalous dimensions, satisfying the scaling law α̃+β=0 and ε is a measure of the dissipation. Using perturbation theory, we calculate values of α̃ and β to O(ε), which are in good agreement with numerical calculations, and we explicitly verify the above scaling law and find the form of the scaling function f.

A model for randomly stirred or homogeneous turbulent fluids is analyzed using renormalization-group methods on a path-integral representation of the Navier-Stokes equations containing a spatially and temporally colored noise source. For moderate Reynolds numbers and certain values of the dynamic exponent governing the noise correlation, an additional scaling regime is found at wave vectors k beyond those where the Kolmogorov 5/3 law holds. In this case, the energy spectrum decays as ${\mathit{k}}^{\mathrm{-}1\mathrm{-}\mathit{z}}$, where 1<z<2, the fluid homodyne-scattering function decays as (time${)}^{\mathrm{-}2/\mathit{z}}$, and the velocity-distribution function (as characterized by its skewness) deviates from a Gaussian. The additional scaling region disappears, and the Kolmogorov constant and Prandtl number become universal in the limit of infinite Reynolds number. In three spatial dimensions, the latter two equal 3/2( 5) / 3 ${)}^{1/3}$ and √0.8 , respectively. The recent homodyne scattering experiments of Tong and co-workers [Phys. Rev. Lett. 65, 2780 (1990)] are analyzed, and the connection of the new scaling region with intermittency is discussed.

We show experimental evidence of traveling hole defects, which are reminiscent of analytical solutions to complex Ginzburg-Landau equations. Also, these objects seem to play an important role in the development of phase instability of oscillatory patterns.