A phase transition associated with dynamical properties of a type-I intermittency is analytically studied on the basis of the statistical-mechanical formalism previously proposed. At the saddle-node bifurcation point the continuous phase transition is found to occur at ${\mathit{q}}_{\mathit{c}}$ less than unity, due to the coexistence of different types of strange attractors with different measures. For nonvanishing shift parameter, crossover phenomena are discussed to obtain scaling exponents for the generalized entropies and the singularity spectrum. The exponents including a logarithmic correction are shown to depend on ${\mathit{q}}_{\mathit{c}}$, which means an absence of universality, in contrast to the usual critical phenomena.

We study the dynamical properties of the electric microfield at a fixed neutral point immersed in a one-component plasma. We introduce an effective-field approach for describing the correlations between the microfield densities at two different times. In this approach, the essential features of the dynamics of the charges that produce the microfield are incorporated via suitable choices of the effective field. We present two versions of the theory which, in the static limits, reduce to the mean-force and adjustable-parameter exponential (APEX) approximations for the equilibrium distribution of the microfield. Both versions rely on a few ingredients determined through existent theories for the dynamics and the statics of the particles. The comparison to the molecular-dynamics data shows that the dynamical extension of APEX is the most reliable theory. The predictions of the Brissaud and Frisch model [J. Quant. Spectrosc. Radiat. Transfer 11, 1767 (1971)] for the microfield dynamics are also tested against the simulation results. This model turns out to be rather reasonable. However, it is not as accurate as APEX, and it misses oscillatory behaviors (originating from the plasmon modes) which, on the contrary, are qualitatively reproduced by the latter theory.

We show how to obtain expressions for the probability density of structures defined by d degrees of freedom in a Poissonian distribution of points. Methods are given for the determination of the coefficients that appear in these expressions. The connection, in the high-density limit, with Gaussian random fields is pursued.

Using renormalization theory it is shown that systems of coupled maps can be characterized with a few universality groups. Identical maps with quadratic maxima, linearly coupled, exhibit α and δ scaling in the parameter space of coupling strength and amplitude of the uncoupled maps. Nonlinear coupling leads to 2 and δ scaling in the same parameter space. Systems with degenerate coupling, where the coupling terms have the same form as the uncoupled maps, show special features. Examples for two coupled maps, and many maps with nearest-neighbor coupling as well as mean-field coupling are worked out and scaling is demonstrated.

The concept of reduced Renyi information is extended for fractal measures possessing a natural nonuniform partition. The spectra of the generalized dimensions and the reduced information of the ‘‘devil’s staircase’’ arising in the critical sine circle map is determined by an effective numerical method. The interpretation of the reduced information as quantities characterizing the complexity of a fractal measure is proposed.

We study analytically the evolution of the concentrations of reactants initially contained in a one-dimensional array of lamellae. The model corresponds to the reaction of two mechanically mixed liquids. The asymptotic behavior of the reactant concentrations c(t) depends on the initial distribution p(x) of striation thicknesses. If the first two moments of p(x) are finite, then in general c(t)∼${\mathit{t}}^{\mathrm{-}1/4}$; the exception is the case of a strictly periodical system, for which c(t) decays exponentially. If the second moment of p(x) is infinite, c(t) depends in a sensitive way on p(x). Thus for p(x)∼${\mathit{x}}^{\mathrm{-}1\mathrm{-}\mathrm{\alpha }}$, as x→∞, with 1<α<2, one has c(t)∼${\mathit{t}}^{\left(1\mathrm{}\mathrm{-}\mathrm{\alpha }\right)/4\mathrm{}}$. This case corresponds to initially poorly mixed liquids.

Glass-sphere avalanches have been studied experimentally using a drum partly filled with beads and rotating slowly (Ω) around its horizontal axis. The statistics of the avalanche characteristics (i.e., duration D and size δΘ) have been determined as a function of the rotation speed Ω, the sphere diameter d, and the drum length l. The widths of these statistics are broad, but avalanches do not exhibit either periodicity or 1/f noise. We conclude also that avalanches are governed by inertia and gravity. We recall then classical results of soil mechanics; we will see that triaxial test results, together with the so-called ‘‘critical state’’ of granular material and the Granta gravel model of sandpiles will make evident and quantify the well-known effects of friction, caging, and dilatancy in granular samples. Using these established results, especially those on the ‘‘critical’’ state of soil, we will demonstrate that the maximum angle of repose of a pile may exceed the angle of friction for initially dense enough materials, but that this leads to a catastrophic event (avalanche). This critical-state approach also allows us to relate the mean angle per avalanche to the mean avalanche duration, using an inertial process, and to predict the avalanche duration. According to our model, the avalanche size is controlled by the difference between the real pile specific volume v and that of the ‘‘critical’’ state ${\mathit{v}}_{\mathit{c}}$; macroscopic avalanches are obtained for v<${\mathit{v}}_{\mathit{c}}$ (i.e., a first-order process), but we expect critical fluctuations (and probably 1/f noise) when v=${\mathit{v}}_{\mathit{c}}$ (i.e., a second-order transition). This theory makes a link between the theory of self-organized criticality of sandpile avalanches and experimental data; it links also the Coulomb approach of the stability of a free surface and the dilatancy effect discovered by Reynolds.

Assuming a scalar wave, the Bethe-Salpeter (BS) equation for a system of random medium with rough boundaries is first reviewed, together with the scattering matrices involved. Emphasis is placed on the optical condition of each scattering matrix, as well as that of a random layer with pronounced boundary effect as one scatterer. Their optical expressions are obtained in terms of the cross sections along with the respective optical conditions. The enhanced backscattering can be understood as a natural consequence of requiring coordinate-interchange invariance of the second-order Green’s function, and the BS equation is rewritten as an equation for the function of the four coordinates involved, so that the invariance is immediately clear. With the solution, specific expressions of cross sections are obtained for a random layer to the approximation of using the boundary-value solution of the diffusion equation. Nevertheless, the angle distribution in the enhanced backscattering holds sufficient accuracy as long as the optical width of the layer is long enough, although not quite for the background term. Another method of using asymptotic evaluation of the cross sections under the diffusion condition is also discussed. A numerical example is shown for the enhanced backscattering.

The principles of statistical physics are applied to the study of cellular systems. Cells that fill a flat surface are characterized by area, perimeter, and number of sides. A set of constraints is established, taking into account the main features of the structure, from both geometrical and physical points of view. The distribution functions of number of sides and area are calculated as well as the thermodynamic parameters. The results are in agreement with available experimental and computational data.

The velocity drift of a Brownian particle in a random environment under the influence of a constant external force is considered. It is established that the singularities leading to the anomalous diffusion are controlled not by the time but by the external force. The consequence is that (1) the diffusion becomes normal and (2) the mobility and the diffusion coefficient become nonlinear functions of the force.

A generic mechanism for intermittency in two-dimensional bounded Hamiltonian dynamics (with two different time scales) is analyzed. This analysis sheds some light on the importance of tori jumps as a source of chaos. We show how intermittency arises in the systems considered, and by using ergodic considerations we estimate the distribution of laminar lengths. The phenomenology found shares features corresponding to both type-I and type-II intermittency. The intermittent behavior is then used to model the dynamics as an incoherent superposition of laminar periods. We derive a formula for the frequency spectrum based on this model and on ergodic theory. All the results predicted are tested on a concrete system.

We examine the question of accurately determining, from an observed time series, the Lyapunov exponents for the dynamical system generating the data. This includes positive, zero, and some or all of the negative exponents. We show that even with very large data sets, it is clearly advantageous to use local neighborhood-to-neighborhood mappings with higher-order Taylor series, rather than just local linear maps as has been done previously. We give examples using up to fifth-order polynomials. We demonstrate this procedure on two familiar maps and two familiar flows: the Hénon and Ikeda maps of the plane to itself, the Lorenz system of three ordinary differential equations, and the Mackey-Glass delay differential equation. We stress the importance of maintaining two dimensions for converting the scalar data into time delay vectors: one is a global dimension to ensure proper unfolding of the attractor as a whole, and the other is a local dimension for capturing the local dynamics on the attractor. We show the effects of changing the local and global dimensions, changing the order of the mapping polynomial, and additive (measurement) noise. There will always be some limit to the number of exponents that can be accurately determined from a given finite data set. We discuss a method of determining this limit by numerically obtaining the singularity spectra of the data set and also show how it is often appropriate to make this choice based on the fractal dimension of the attractor. If excessively large dimensions are used, spurious exponents will be generated, and in some cases the accuracy of the true exponents will be affected. We present methods of identifying these spurious exponents by determining the Lyapunov direction vectors at particular points in the data set. We can then use these to identify numerical problems and to associate data-set singularities with particular exponents. The behavior of spurious exponents in the presence of noise is also investigated, and found to be different from that of the true exponents. These provide methods for identifying spurious exponents in the analysis of experimental data where the system dynamics may not be known a priori.

In order to treat the effects of local nonequilibrium in fluids the thermodynamic metric, previously used for measuring dissipation, is extended from a space of extensive state variables into a space of the field variables involving nonequilibrium quantities such as heat flux. The metric is investigated in the context of the field description of transport and flow phenomena off but close to equilibrium. Thermodynamic potentials and corresponding tensors are constructed on the basis of the extended Gibbs equation involving heat and macroscopic velocity terms. An extension of the theory is suggested to the problem of multicomponent heat and mass diffusion.

The time-convolution as well as the time-convolutionless generalized master equations (GME) are rewritten in a simple form permitting an exact formal solution. Using this form, it is argued that possible approximations must be performed consistently on both sides of the GME in order to control deviations of the emerging approximate solution from the exact one. Arguments are given that after such consistent approximations, the initial-condition term can no longer in general be taken as proportional to (scrI-${\mathit{scrD}}_0$)‖${\mathrm{\rho }}_0$) (with ${\mathit{scrD}}_0$ being the projector used in GME). These conclusions are illustrated for the asymptotic-time symmetry-breaking problem of the spin-boson model. Although standard approximate GME approaches do indicate a possible symmetry breaking, we argue that these results are strongly influenced by the inconsistency of the approximations used. To support this statement we prove the absence of any asymptotic-time symmetry breaking in the zero-bias spin-boson model with a partial site-occupation relaxation.

Recent experiments on one-dimensional convection in binary fluids at moderate negative values of the separation ratio ψ have revealed the existence of a spatially localized ‘‘stationary pulse’’ of traveling waves whose spatial shape and length are fixed. In a long system, this pulse is destroyed slightly above onset by convectively amplified fluctuations. In a short system, however, these fluctuations are suppressed, allowing the observation of the unstable extension of the pulse state. Unlike the stationary pulse, this ‘‘unstable pulse’’ can have any length.

Non-Boussinesq effects in a Rayleigh-Bénard convection system lead to a symmetry breaking between the top and the bottom boundary layers. We have found that the two layers adjust their temperature drops and the thicknesses λ such that their temperature scales νκ/gα${\lambda }^3$ are equal, where α is the thermal expansion coefficient, g is the gravitational acceleration, and ν and κ are the kinematic viscosity and thermal diffusivity, respectively.

A model is presented for the behavior of amphiphilic platelets at an interface between immiscible liquids. In the model Hamiltonian, the edges and faces of a cylindrical platelet have different strengths of interaction with a given liquid. For almost all of their parameter space, the model platelets prefer to sit at the interface rather than in either liquid. However, the ground-state orientation of the platelets (flat or upright) is found to be a function of cylinder geometry and strengths of interaction. The equation of state of a platelet system is investigated using a simplified version of the model. At low temperatures, there is a rapid change in platelet orientation with pressure, but this does not appear to be a phase transition. At high pressure, the platelet system takes on the appearance of a nematic phase.

The Fréedericksz transition and the dynamic response of homeotropical liquid-crystal films with a free surface (FS) induced by a perpendicular dc magnetic field have been studied theoretically and experimentally for two nematic substances: 4’-n-pentyl-4-cyanobiphenyl (5CB) and 4’-n-octyl-4-cyanobiphenyl (8CB). For either one of these two materials, it is found that the critical magnetic field of these two nematic liquid-crystal (NLC) films with a free surface is equal to that of the NLC film with hard boundaries (HB) of the same thickness. This thus demonstrates a very strong orientational anchoring for 5CB and 8CB at the free surface. The intrinsic structures of FS samples are homeotropic, just as the HB ones are. For the dynamic response, the turn-off rates of FS samples are found to be about two times faster than those of the HB ones. This is interpreted with the backflow theory and the lack of positional anchoring on the free surface. Finally, a sensitive approach to determine the viscosity coefficients, ${\gamma }_1$ and ${\mathrm{\eta }}_{\mathit{c}}$, of the NLC film is suggested.

Detailed experiments are reported of the growth of droplets when water vapor condenses from a saturated carrier gas onto a hydrophobic plane substrate. We have investigated the effects of the carrier-gas flow velocity, the nature of the gas, the experimental geometry, and heat transfer through the substrate. Individual drops grow according to a power law with exponent μ=1/3. At high flow velocities, the temperature of the substrate can rise significantly, which lowers the condensation rate and leads to lower apparent growth-law exponents. A self-similar regime is reached when droplets interact by coalescences. The coalescences continuously rescale the pattern, produce spatial correlations between the droplets, and accelerate the growth, leading to a power law with an exponent ${\mathrm{\mu }}_0$=3μ. The experiments are compared to predictions of scaling laws and to simulations.

A microscopic theory of planar crystal surface melting is described and applied to rare gases with Lennard-Jones potential. Explicit results for the solid-vapor density profiles are presented that exhibit surface melting in the vicinity of the triple point. Our approach is based on density-functional techniques combined with a van der Waals approximation and a hard-sphere fluid as a reference system.