A stochastic version of the lattice nonlinear Schrödinger equation, allowing treatment by means of the inverse-scattering technique and having an exact one-soliton solution, is introduced. It is shown that such a model is a useful tool for investigation of a wide class of nonlinear lattices affected by spatiotemporally random forces. A number of the most important statistical characteristics of soliton dynamics governed by such models can be evaluated without any assumption about the smallness of random perturbations. The problem is studied in detail in two limiting cases: small and large intensities of fluctuations of a stochastic term in the integrable equation.

Considering the random walk of the vacancies in a one-dimensional (single-file) system, the moments of the probability distribution curves of molecular displacements are calculated yielding a Gaussian distribution curve.

Visualization studies of the phase separation of the critical binary liquid mixture, aniline-cyclohexane, in the late stage of the gravity-dominated regime are reported. The sedimentation behavior, following the formation of the bulk interface, shows that the residual domains grow in size according to a(t)=${\mathit{At}}^{\mathrm{\phi }}$, with φ=0.27±0.06 and A=(7±5)×${10}^{\mathrm{-}7}$ m ${\mathrm{s}}^{\mathrm{-}\mathrm{\phi }}$. Domain growth in the earlier stages of spinodal decomposition and nucleation mechanisms also yield similar growth exponents and prefactors. The differences are probably due to some hydrodynamic interactions during the sedimentation.

The collisions of m=0.3–2.0 g, v=5–50 cm/s mercury drops are studied experimentally. A transition between a fusion and a nonfusion outcome is observed, and found to be influenced by angular momentum. The mass dependence measured for the limiting angular momentum ${\mathit{L}}_{\mathit{c}}$(expt) is compared with predictions ${\mathit{L}}_{\mathit{c}}$(theor) of a surface potential model widely used in nuclear physics. A systematic ${\mathit{L}}_{\mathit{c}}$(expt)≤${\mathit{L}}_{\mathit{c}}$(theor) discrepancy is found in these and in other drop-collision data. Dynamical considerations and the use of more elaborate surface shapes than those assumed by the model are found to reduce the disagreement.

Model calculations are made of surface segregation profiles for polymer and oligomer blends, both for the case in which surface segregation is driven purely by short-range forces, and for the case in which long-range (dispersion) forces are important. As expected, the inclusion of long-range forces affects the profiles as complete wetting is approached, but the effect on the shape of the surface segregation profile for a high polymer blend in the one phase region of the phase diagram is small; recent discrepancies between theory and experiment cannot be accounted for by this explanation.

We present an exact determination of the correlation-length exponents as well as an accurate numerical estimation of the bulk and surface entropic exponents at the ordinary and special Θ points for a directed-polymer model. The numerical results were obtained using an exact enumeration of walks up to 29 steps.

Presented are results of experimental studies of the critical amplitude and exponent describing the behavior of the nonlinear dielectric effect on approaching the critical consolute point. Plots of the critical amplitude as a function of dielectric properties of components of binary solutions of limited miscibility show good agreement with the Goulon-Greffe-Oxtoby droplet model [J. Chem. Phys. 70, 4742 (1979)]. An analysis of temperature dependence of the critical exponent ψ suggests that an observed discrepancy between the experimental value of ψ and that predicted by the theoretical models is due to the relaxation of critical fluctuations.

The results of the numerical calculations of the growth probabilities of diffusion-limited-aggregation clusters based on the Spitzer theorem are presented. They support the recent results of Schwarzer et al. [Phys. Rev. Lett. 65, 603 (1990)] on the non-power-like size dependence of minimum-growth probabilities and the resulting phase transition in the multifractal spectrum.

The dispersion relation of the long-wavelength fluctuations of an interface exhibited by the Cahn-Hilliard equation is studied analytically and numerically. The expected asymptotic dispersion relation ω∼${\mathit{k}}^3$ is demonstrated. Further, using a well-defined microscopic length scale ξ, the dispersion relation is numerically found to have a nearly universal form ω/${\mathit{k}}^3$=(1/ξ)Ω(kξ) for a wide variety of potentials.

Evolving random cellular structures are observed to reach a universal scaling regime. A mean-field approach to finding fixed-point distributions in cell-side number is extended to distributions for the average area of cells with a given number of sides. This approach leads to simplified equations that can be analyzed analytically and numerically. The theory’s results are compared to experimental results on dynamics and distributions in soap froths and good agreement is achieved.

Frictional stick-slip dynamics is studied experimentally in an elastic continuum by means of a stretched latex membrane in contact with a translating glass rod. In contrast to other laboratory-scale experiments, the characteristic stiffness length here is small compared to the size of the system but large compared to the spatial resolution of the measurements. The internal displacement field u(s,t) is measured in detail with imaging techniques, which are shown to be more sensitive than total-force measurements. The magnitudes μ of slipping events extend over a wide range and include both small localized events and large spatially extended ones. The shape of the distribution P(μ) depends somewhat on the measurement threshold and also varies with the parameters describing the frictional interaction, which are affected by wear. The frictional force increases with velocity and an instability is due to waves of detachment. The temporal statistics of slipping events are also discussed. The experimental results are compared qualitatively to simplified models of earthquake faults.

In this paper it is shown that elementary tools of Riemannian differential geometry can be successfully used to explain the origin of Hamiltonian chaos beyond the usual picture of homoclinic intersections. This approach stems out of first principles of mechanics and fundamental tools of Riemannian geometry. Natural motions of Hamiltonian systems can be viewed as geodesics of the configuration-space manifold M equipped with a suitable metric ${\mathit{g}}_{\mathit{J}}$, and the stability properties of such geodesics can be investigated by means of the Jacobi–Levi-Civita equation for geodesic spread. The study of the relationship between chaos and the curvature properties of the configuration-space manifold is the main concern of the present paper and is carried out by numerical simulations. Two different mechanisms for chaotic instability are found: (i) the trajectories are ‘‘scattered’’ by random encounters of regions of negative curvature (either scalar or Ricci curvature—it depends on the averaging procedure adopted); (ii) the ‘‘bumpiness’’ of (M,${\mathit{g}}_{\mathit{J}}$) yields oscillations of the Ricci curvature along the geodesics so that parametric resonance makes them unstable also in regions of positive curvature.

The geometric approach is intrinsically nonperturbative because everything is well defined at any energy, i.e., quasi-integrability is not required as in the case of classical perturbation theory. Therefore this approach is fit to describe the existence of the strong-stochasticity threshold (SST) in high-dimensional Hamiltonian flows. This threshold refers to a transition between weak and strong chaoticity of the dynamics and, correspondingly, between slow and fast mixing in phase space. In view of applications to equilibrium and nonequilibrium statistical mechanics, the SST appears as the transition feature of high dimensional Hamiltonian flows with the greatest physical significance. As the SST concerns chaotic dynamical behaviors, its existence cannot be understood within the framework of classical perturbation theory, whereas it is shown that the SST can be related to some major geometrical change of the constant-energy surfaces of phase space. A clear-cut distinction can be made between integrable and nonintegrable systems. Finally, a new meaning is given to the standard algorithm to compute numerical Lyapunov exponents, and it is shown that Oseledets multiplicative theorem is not necessary to justify it.

We investigate the dynamics generated from iterated maps and analyze the motion in terms of the probabilistic continuous-time random-walk (CTRW) approach. Two different CTRW models are considered: (i) Particles jump between sites (turning points) or (ii) particles move at a constant velocity between sites and choose a new direction at random. For both models we study the mean-squared displacement 〈${\mathit{r}}^2$(t)〉 and the propagator P(r,t), the probability to be at location r at time t having started at the origin at t=0. Iterated maps are used to generate both dispersive and enhanced diffusion and the results are analyzed using the CTRW framework and scaling arguments. For the case of dispersive motion we discuss the problem of the stationary state and point out its relevance.

Various modes in three coupled chemical oscillators in a triangular arrangement were observed. As a well-defined nonlinear oscillator, the Belousov-Zhabotinsky reaction was studied in a continuous-flow stirred tank reactor (CSTR). Coupling among CSTR’s was performed by mass exchange. The coupling strength was quantitatively controlled by changing the flow rate of reacting solutions among the three CSTR’s using peristaltic pumps between each pair of the reactors. As a key parameter to control the model of coupling, we changed the symmetry of the interaction between the oscillators. In the case of the symmetric coupling, a quasiperiodic state or a biperiodic mode, an all-death mode and two kinds of synchronized modes appeared, depending on the coupling strength. On the other hand, under the asymmetric coupling, a quasiperiodic state or a biperiodic mode, an all death mode and four kinds of synchronized modes appeared. Those modes have been discussed in relation to the idea of ‘‘frustration’’ in the Ising spin system, where the three-phase mode appears as a transition from the Ising spin system to the XY spin system.

Several attempts have been made recently to generalize the multifractal formalism, originally introduced for singular measures, to fractal signals. We report on a systematic comparison between the structure-function approach, pioneered by Parisi and Frisch [in 2 Proceedings of the International School on Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, edited by M. Ghil, R. Benzi, and G. Parisi (North-Holland, Amsterdam, 1985), p. 84] to account for the multifractal nature of fully developed turbulent signals, and an alternative method we have developed within the framework of the wavelet-transform analysis. We comment on the intrinsic limitations of the structure-function approach; this technique has fundamental drawbacks and does not provide a full characterization of the singularities of a signal in many cases. We demonstrate that our method, based on the wavelet-transform modulus-maxima representation, works in most situations and is likely to be the ground of a unified multifractal description of self-affine distributions. Our theoretical considerations are both illustrated on pedagogical examples and supported by numerical simulations.

We report on the study of nonequilibrium ordering in the reaction-diffusion lattice gas. It is a kinetic model that relaxes towards steady states under the simultaneous competition of a thermally activated creation-annihilation $(—reaction$)— process at temperature T, and a diffusion process driven by a heat bath at temperature T’≠T. The phase diagram as one varies T and T’, the system dimension d, the relative priori probabilities for the two processes, and their dynamical rates is investigated. We compare mean-field theory, new Monte Carlo data, and known exact results for some limiting cases. In particular, no evidence of Landau critical behavior is found numerically when d=2 for Metropolis rates but Onsager critical points and a variety of first-order phase transitions.

The time evolution of river patterns and earth’s relief is simulated on lattice by modeling the process of water erosion. Starting from a randomly perturbed surface, the river pattern and earth’s relief develop simultaneously. The river pattern becomes stationary after all lakes have vanished. In the stationary state the river pattern shows some fractal properties such as a power-law size distribution of the drainage basin area and Horton’s laws. The fractalities are shown to be not exactly self-similar but self-affine. A mean-field theory for the river pattern is discussed.

We study the long-wavelength properties of the one-dimensional Kuramoto-Sivashinsky equation. We determine all the parameters in the effective long-wavelength equation, interpret the phenomenological coefficients in terms of microscopic quantities, and estimate the time and length scales where the behavior crosses over from linear diffusive to that of the full nonlinear equation. We corroborate our analysis by studying variants of the model with more general linear terms.

We study unbiased random walks in discrete time n on a square lattice, in the form of a strip of finite width N in the y direction, with a family of boundary conditions parametrized by a stay probability Γ per time step at the edge sites. The diffusion coefficient K=${\lim }_{\mathit{n}\to \mathrm{\infty }}$〈${\mathit{X}}_{\mathit{n}}^2$〉/n is computed analytically to exhibit its dependence on N and Γ. The result is generalized to the case of a strip with side branches attached to the boundary sites to simulate the effect of rough edges. A further generalization is made to obtain K for a random walk in d dimensions on a lattice bounded in one of the directions. Thus, K serves as a probe of both the transverse size of the region in which diffusion takes place and the nature of the bounding surfaces.

A dynamically corrected variational transition-state theory is formulated for the thermally activated escape of a particle trapped in a potential well separated from a different well or continuum by a barrier and coupled to a heat bath. The theory is based on the Hamiltonian-equivalent formulation of the generalized Langevin equation. The dynamical corrections are obtained by utilizing the reactive-flux method in which the choice of dividing surface is guided by minimization of the transition-state flux. Analytic correction formulas, valid for memory friction, are obtained for the Kramers-Grote-Hynes estimate of the rate in the range from moderate friction to the large-friction limit. The analytic expansion is in terms of the inverse barrier height (1/β${\mathit{V}}^{\mathrm{‡}}$). For the special case of an extended Smoluchowski equation containing finite damping corrections, the exact expansion is also obtained using the mean-first-passage-time formulation. The dynamically corrected variational transition-state-theory expansion is shown to be identical to the mean-first-passage-time result.