For systems that undergo period-doubling cascades there also exists an ‘‘inverse cascade’’ of chaotic band mergings. The frequency spectrum associated with a chaotic orbit of $2^k$ constituent bands has $2^k$ δ-function spectral components superposed on a broadband continuous component. At merging, pairs of the $2^k$ bands join to produce $2^{k\mathrm{-}1}$ bands. Associated with this, the number of δ-function spectral peaks halves. This happens via the acquisition of a finite spectral broadened width by every other δ function (hence making them no longer δ functions). This paper investigates this transition in detail with emphasis on its scaling properties.

This study concerns ultrasonic velocity and damping in the smectic-A phases of four liquid crystals, using a resonance technique. The compounds studied are terephthal-bis-p-p’-butylaniline (TBBA), octylcyanobiphenyl (8CB), octyloxycyanobiphenyl (8OCB), and p-cyanobenzylidene-p-octyloxyaniline (CBOOA). The results obtained for TBBA reveal abnormal damping behavior which shows that conventional hydrodynamics does not apply in the smectic-A phases of liquid crystals. Analysis of these results shows that anharmonic effects related to fluctuations in the smectic layers have an order of magnitude which is compatible with the lowest-order predictions of the theory of Mazenko, Ramaswamy, and Toner (MRT), when δ${\eta }_5$<0. Measurements taken on 8CB 8OCB, and CBOOA show that anharmonic effects are negligible in these three compounds at ultrasonic frequencies. This result is consistent with the lowest-order predictions of MRT theory. The difference between the behavior of TBBA and that of the other three compounds shows that anharmonic effects only appear at ultrasonic frequencies in compounds with a sufficiently high B/$K_1$ value (B and $K_1$ are the layer compressibility and splay constants, respectively).

We study the fluctuating nonlinear hydrodynamics of compressible fluids. Development of the appropriate field-theoretical description for this problem requires treatment of nonlinearities which arise through the relationship g=ρV, where g is the momentum density, ρ is the mass density, and V is the velocity field. We show how this constraint can be naturally included in a field theory of the Martin-Siggia-Rose type. We analyze the structure of the resulting field theory using the available fluctuation-dissipation theorem. We also develop the perturbation-theory expansion in powers of the temperature and evaluate the contributions from the nonlinearities to one-loop order. We show that the theory is renormalizable in the hydrodynamic limit. This field-theoretical model is used to systematically investigate the origins and viability of the nonlinear density feedback mechanism first identified by Leutheusser as a source of the liquid-glass transition. While we find that the nonlinear couplings driving this mechanism are present, we also find contributions, arising from the nonlinear constraint relating g, ρ, and V, which cut off the mechanism. The cutoff arises from a nonhydrodynamic correction not treated in previous work. While we find that there is no sharp transition, we do find evidence for a rounded version of the transition.

The system of coupled linearized Boltzmann equations for a multicomponent gas mixture of particles interacting via a Maxwellian-type potential is solved for the spatially homogeneous case. The αth component of the gas mixture is described by its concentration $c_{\alpha }$, mass $m_{\alpha }$, and the initial distribution of velocities $h_{\alpha }^0$(v); the collisions with particles of species β are characterized by the interaction parameter ${\sigma }_{\alpha \beta }^0$. The eigenvalue spectrum of the linearized collision operator for arbitrary concentrations, masses, and interaction parameters is studied and the time-dependent velocity distribution function for each component is derived. As examples the binary and ternary systems are discussed briefly. The time-dependent velocity distribution function for a binary mixture is calculated for several mass ratios showing the development of two branches within the first few collisions. Furthermore the mass dependence of Boltzmann’s H function is studied in detail. Finally we discuss the time dependence of the mean velocity and mean-square velocity for each species in a ternary mixture. While in a binary mixture these quantities vary monotonically in time, there are mass and concentration ratios in ternary systems, where one of the species—starting with less than its equilibrium share of energy—can acquire more energy than its equilibrium value for a limited period of time.

Practically all the known Monte Carlo algorithms for identical fermions require the subtraction of separate contributions arising from, respectively, even and odd permutations of the particles. At low temperatures this becomes a serious problem since the two contributions nearly cancel. A new method is proposed for obtaining this difference directly. This amounts to a constraint that limits configuration space to a region bounded by the nodal surface of the fermionic ground-state wave function. A detailed description of the method is given for the general real Hamiltonian in the continuum where the partition function paths are approximated by ‘‘polymers’’ of finite length. The only departure from the usual bosonic treatment is the use of a nonlocal constraint which prohibits the polymer from simultaneously occupying regions of configuration space related by an odd permutation. With this constraint, the respective even- and odd-permutation images of the polymer are viewed as the two components in a phase-separated binary mixture. The condition for equilibrium of the interface formed between the two phases can be shown to be consistent with the boundary condition of the fermionic wave function at the nodal surface. Extension of the method to lattice systems and nonzero temperature is not straightforward and is not discussed.

The nature of statistical cross correlation between center-of-mass molecular translation and rotation of the same molecule is investigated for carbon tetrachloride liquid by molecular dynamics computer simulation. It is shown that rotation and translation in this $T_d$-symmetry spherical top are indeed correlated through the existence of the molecular Coriolis, centripetal, and nonuniform accelerations of the rigid molecule, thus invalidating the Debye theory of rotational diffusion under the conditions for which it was originally proposed. The simple cross-correlation function 〉ω(t${)\mathrm{v}}^T$ (0)〈/[〈${\omega }^2$(0)${〉}^{1/2}$ 〈$v^2$(0)${〉}^{1/2}$] vanishes, however, for all t and in all frames of reference for the spherical top diffusing in three dimensions in a bath of other spherical tops.

We consider the effect of periodic perturbations on open reactive systems far from the linear thermodynamics domain. The systems present a center-manifold contraction of the phase space achieved by sustaining a hard-mode instability. The critical regime is governed by a Ginzburg-Landau potential defined on the locally attractive, locally invariant center manifold. Within the framework of this reduction scheme, scaling properties of the Green’s function sensitivity matrix are obtained. It is demonstrated that the entrainment is produced by a projection of the perturbing time-dependent field on the center manifold. Thus, the reduced equations for entrainment in the order parameter space are derived. It is demonstrated that two inherent properties of the system favor the entrainment: (a) a small sensitivity of the amplitude of the bifurcating limit cycle with respect to changes in the control parameter, (b) the departure of the system from the region of marginal stability. The results are applied in two different contexts: When there exists a separation of relaxation-time scales (in a truncation of Hopf’s model for hydrodynamic turbulence) and when there is only one time scale involved (Brusselator). Agreement with previous derivations of the entrainment regions is found. Finally, a realistic experiment coupling two oscillatory reactors is suggested in order to test the theoretical findings. In this case, the frequency of the perturbation is a function of the bifurcation control parameter (the residence time) which measures the departure of the entrained system from marginal stability. The results are applicable in the case of convection in a rotating layer and convection driven by the Soret-Dufour effect since the oscillatory convection corresponds to a center manifold contraction of the phase space. This manifold contains the dominant velocity modes when the frequency of oscillation is small.

A simple model of quasirandomness is presented. The model is a self-similar space division: A d-dimensional space is divided successively into smaller subspaces with randomness. Positions of subspaces are fixed. Then centers of subspaces distribute quasirandomly. Two typical cases of the model are discussed. In one case (repulsive case) the power spectrum of centers of subspaces exhibits a $k^{\mathrm{-}\alpha }$ (α≊d) singularity in a broad range of a randomness parameter, where k is the wave number. This spectrum corresponds to a so-called 1/f noise spectrum for a one-dimensional time sequence. The $k^{\mathrm{-}d}$ singularity will be observed in the scattering intensity from a phase-separating system if droplets cannot move freely due to the strong correlation. This singularity does not seem to depend on details of the model, and is universal. We can show the same singularity analytically for a modified model. In the other case (attractive case) the power spectrum exhibits a $k^{\mathrm{-}\alpha }$ (α≊0.4d) singularity in a broad range of the randomness parameter. This singularity is equivalent to the typical singularity observed in the universe. The physical reason for this agreement is discussed.

Using a structured approach to the Hopfield model of associative memory, we address the question of maximal information packing. We show that input binary vectors which are orthogonal to one another are not guaranteed to form a faithful set, i.e., a set which generates the same output. We point out the importance of the binary product operation in sorting out the faithful sets. We prove that any binary basis has a total binary product proportional to the unit vector. We construct examples of faithful sets by requiring that certain subsets be forbidden to have a total binary product proportional to the unit vector.

The electron mobility μ was measured in liquid Kr at 95×${10}^{26}$≤n≤172×${10}^{26}$ molecules/${\mathrm{m}}^3$, and 121≤T≤206 K. At low fields a field-independent μ=${\mu }_0$ was obtained at each density. The density dependence of ${\mu }_0$ is compared to that predicted by the deformation potential theory of Basak and Cohen; ${\mu }_0$ is maximum at (13.7±0.4)×${10}^{27}$ molecules/${\mathrm{m}}^3$. Scattering lengths were calculated for both liquid Kr and Xe, and their density dependence was examined using the recent model of Baird [Phys. Rev. A 32, 1235 (1985)]. At higher fields, μ is field dependent. The effect of density on the field dependence is examined.

A simple renormalization procedure is constructed directly from map equations for the sine circle map and the dissipative standard map. Spectra of the maps are obtained. The relation between the two maps is shown.

A new Monte Carlo method, the incomplete-enumeration method, is used to calculate both the square radii of gyration 〈$R_N^2$〉 and the number of animals $a_N$ of size N up to N=30 in the case d=3 and N=20 in the cases d=4–7. We obtain about 2% accuracy for $a_N$ and 0.2% accuracy for 〈$R_N^2$〉 for the largest sizes N. For d=9 longer runs yield values of $a_N$ and 〈$R_N^2$〉 to better than 0.25% and 0.04% accuracy, respectively. Our data are consistent with upper critical dimension $d_c$=8 and with the Flory formula for all d<$d_c$.

A theory is given for the momentum-transfer dependence of the quasielastic-scattering linewidth and relaxation time for solutions of surface and mass fractals. The linewidth and relaxation time are found to be fractional powers of the momentum transfer if the distribution of fractal sizes is a power law, in agreement with recent experimental findings.

Our previously developed stochastic trajectory analysis technique has been applied to the calculation of first-passage time statistics of bound processes. Explicit results are obtained for linearly bound processes driven by dichotomous fluctuations having exponential and rectangular temporal distributions.

The spectrum of a spectroscopically active particle which makes a random walk on a finite one-dimensional lattice, upon each of whose n sites the particle radiates or absorbs radiation with a different frequency, is discussed. The correlation function of the radiating dipole is found to decay asymptotically as a Gaussian times $t^{\mathrm{-}\left(n\mathrm{-}1\right)/2}$ in the case when the site frequencies are independent Gaussian random variables. Numerical simulations of the random walk confirm the asymptotic result. The difference between these results for a finite system and previous results for an infinite system are discussed.

A ‘‘class’’-c orbit is one that rotates around a periodic orbit of class c-1 with some definite frequency. This contrasts with the ‘‘level’’ of a periodic orbit which is the number of elements in the continued-fraction expansion of its frequency. Level renormalization is conventionally used to study the structure of quasiperiodic orbits. The scaling structure of periodic orbits encircling other periodic orbits in area-preserving maps is discussed here. Renormalization fixed points of p/q bifurcations are found and scaling exponents determined. Fixed points for q>2 correspond to self-similar islands around islands. Frequencies of the island boundary circles at the fixed points are obtained. Importance of this scaling for the motion of particles in stochastic regions is emphasized.

The Lorenz equations are cast in the form of a single stochastic differential equation in which a ‘‘deterministic’’ part representing a bistable dynamical system is forced by a ‘‘noise’’ process. The properties of this effective noise are analyzed numerically. An analytically derived fluctuation-dissipation-like relationship linking the variance of the noise to the system’s parameters provides a satisfactory fitting of the numerical results. The connection between the onset of chaotic dynamics and the breakdown of the separation between the characteristic time scales of the variables of the original system is discussed.

The equilibrium polymerization of chains and rings together is a ($n=1\mathrm{}$) critical phenomenon with aspects of bicriticality, governed by exponents not found in any symmetric $\mathrm{O}\mathrm{}\left(n\right)\mathrm{}$ model. The bicritical nature of the critical point is the result of a competition between a transition to form long-chain polymers and a transition to form an infinite-rings condensate. As a result the fraction of monomers incorporated in chains and in rings varies with temperature according to a different power law than that for the total fraction of polymerized material and is governed by an exponent that depends on the crossover exponent $\phi$ for quadratic anisotropy for the $\mathrm{O}\mathrm{}\left(n\right)\mathrm{}$ vector model of magnetism in the limit $n\to 1$. Moreover, the fraction of monomers in polymeric rings is found to decrease with temperature with infinite slope just beyond the transition in the polymerized phase. Similarly, the geometry of large polymers is described by a power law $R\sim N^{\frac{\nu }{\phi }}$, where $R$ is the distance spanned by a polymer segment of $N$ monomers and where the exponent is not the de Gennes result $\nu \mathrm{}(n=0\mathrm{})\mathrm{}$ for chains alone, but rather also depends on the crossover exponent $\phi$ for $n=1\mathrm{}$. We study the equilibrium polymerization of chains and rings together using a variety of techniques, including simple equilibrium theory, a lattice model, field-theoretic correspondence to magnetism and a direct renormalization-group calculation on a polymer model. An explicit parametric form for the equation of state is presented to lowest order in $\epsilon$. Results are obtained for the fraction of polymeric material in chains and rings and for various correlation functions of interest in considering the geometry and distribution of chains and rings.

The binary-mixture phase diagram of n-propyl-4’-n-decyloxybiphenyl-4-carboxylate [3(10)OBC] and n-butyl-4’-n-decyloxybiphenyl-4-carboxylate [4(10)OBC] has been investigated by optical microscopy, x-ray diffraction, and heat-capacity studies. Here 3(10)OBC and 4(10)OBC have the isotropic (I)–smectic-A (SmA)–hexatic-B and I–SmA–smectic-I (SmI) transition sequence, respectively. This binary-mixture study reveals two new features, namely, the existence of the hexatic-B–SmI transition and a SmA–hexatic-B–SmI point where the relevant three phases coexist.

An algorithm devised for measuring the dimension of a strange attractor from a time series is applied both to autocorrelated Gaussian noise and to a dynamical system. It is analytically shown that a finite sequence of stochastic data—where by ‘‘finite’’ it is meant that N<2${\tau }^{m/2\mathrm{}}$, where N is the number of points in the sequence, τ is the autocorrelation time (in units of sampling period), and m is the embedding dimension—exhibits anomalous structure in its correlation integral. The anomaly is seen numerically in both stochastic and dynamical data. Unrecognized, it can lead to unnecessarily inaccurate and possibly spurious estimates of dimension. We propose a slight modification of the standard algorithm which eliminates this difficulty.