We derive, for a classical fluid, a new and exact hierarchy of integro-differential equations for the free energy and direct correlation functions of successively higher order. This hierarchy gives the evolution of these quantities as the attractive part w of the interaction is turned on by the successive inclusion of the different Fourier components of w, starting from large momenta. The full hierarchy is also written as a unique functional differential equation for the free energy of a nonuniform fluid. This hierarchy is the basis of a unified theory of fluids. Near the critical point and at small momenta, our equations become equivalent to renormalization-group equations and we recover the usual ε expansion. A very simple approximation gives, in three dimensions, γ=2ν≊1.38. Suitable truncation of the hierarchy reproduces the correct low-density (virial coefficients) and high-density (optimized random-phase) limit.

Energy transport and photon emission in a thin Al foil bombarded by an energetic proton beam is theoretically investigated with use of a one-dimensional numerical model which self-consistently treats energy deposition by the beam, the hydrodynamic response and ionization dynamics of the target, and the detailed radiation emission and transport. An extensive atomic-level structure with collisional-radiative-equilibrium (CRE) ionization dynamics is employed in conjunction with a hybrid radiation transport scheme consisting of a multifrequency ray tracing formalism for continuum transport and a probabilistic treatment for line transport. The interaction of a 1-MeV-proton beam with a flux of ${10}^{26}$ protons/${\mathrm{cm}}^2$ sec on a 15-μm-thick Al slab is investigated. It is found that half or more of the incident energy can be converted to radiation, suggesting that radiation can play a dominant role in the subsequent plasma evolution. Temperature and density profiles, energy histories, emission spectra, and other results are presented.

The nonlinear charge storage property of driven Si p-n junction passive resonators gives rise to chaotic dynamics: period doubling, chaos, periodic windows, and an extended period-adding sequence corresponding to entrainment of the resonator by successive subharmonics of the driving frequency. The physical system is described; equations of motion and iterative maps are reviewed. Computed behavior is compared to data, with reasonable agreement for Poincaré sections, bifurcation diagrams, and phase diagrams in parameter space (drive voltage, drive frequency). N=2 symmetrically coupled resonators are found to display period doubling, Hopf bifurcations, entrainment horns (‘‘Arnol’d tongues’’), breakup of the torus, and chaos. This behavior is in reasonable agreement with theoretical models based on the characteristics of single-junction resonators. The breakup of the torus is studied in detail, by Poincaré sections and by power spectra. Also studied are oscillations of the torus and cyclic crises. A phase diagram of the coupled resonators can be understood from the model. Poincaré sections show self-similarity and fractal structure, with measured values of fractal dimension d=2.03 and d=2.23 for N=1 and N=2 resonators, respectively. Two line-coupled resonators display first a Hopf bifurcation as the drive parameter is increased, in agreement with the model. For N=4 and N=12 line-coupled resonators complex quasiperiodic behavior is observed with up to 3 and 4 incommensurate frequencies, respectively.

The transmission of acoustic waves through a sequence of alternating layers with random thicknesses but otherwise fixed characteristics is studied by means of the transfer-matrix formalism of one-dimensional disordered chains. The law ${\lim }_{\mathrm{N}\to \mathrm{\infty }}$ln(‖$T_N$‖/N)≡-λ(ω) of the exponential decay of the transmission coefficient $T_N$ as a function of the number (2N) of layers is determined in a weak- (strong-) disorder regime for an arbitrary (uniform) distribution of layer thicknesses. The localization constant λ(ω) has a particularly simple form at extreme low and high frequencies ω. Namely λ(ω→0)=const×${\omega }^2$ with a slope given in terms of physical characteristics of the layers and λ(ω→∞)=const defined by a transmission coefficient of a single interface. The predictions are tested by Monte Carlo simulations of a simple model with characteristics of certain rocks. For all frequencies beyond the weak-strong disorder turnover region discrepancies between theoretical and numerical results are merely a few percent.

Stochastic dynamical models described by Fokker-Planck equations, in the limit of weak noise, can be formally associated with Hamiltonian dynamical systems. Of special interest are ‘‘Fokker-Planck Hamiltonians’’ with a certain smooth separatrix at zero energy, since a differentiable macroscopic potential was shown to exist in this case. In the present paper, integrability of Fokker-Planck Hamiltonians with two degrees of freedom is investigated, with the aim of identifying cases in which a smooth potential exists and cases in which the eigenvalue problem of the Fokker-Planck operator becomes separable. Additional first integrals of polynomial and nonpolynomial form and also explicitly time-dependent conserved quantities are obtained. The singular-point analysis testing for the Painlevé or the weak Painlevé property and polynomial conserved quantities in dynamical systems is applied. This method is found to be useful for the purpose of identifying solvable special cases in classes of Fokker-Planck models with free parameters. However, the utility of this method in a search for systems with a smooth potential is found to be severely limited because it turns out that complete integrability with polynomial conserved quantities is a much stronger requirement than the existence of a smooth separatrix at zero energy.

The concept of pattern formation, which to current researchers is a synonym for self-organization, carries the connotation of deductive logic together with the process of spontaneous inference. Defining a pattern as an equivalence relation on a set of thermodynamic objects, we establish that a large class of irreversible pattern-forming systems, evolving along idealized quasisteady paths, approaches the stable steady state as a mapping upon the formal deductive imperatives of a propositional function calculus. In the preamble the classical reversible thermodynamics of composite systems is analyzed as an externally manipulated system of space partitioning and classification based on ideal enclosures and diaphragms. The diaphragms have discrete classification

capabilities which are designated in relation to conserved quantities by descriptors such as impervious, diathermal, and adiabatic. Differentiability in the continuum thermodynamic calculus is invoked as equivalent to analyticity and consistency in the underlying class or sentential calculus. The seat of inference, however, rests with the thermodynamicist. In the transition to an irreversible pattern-forming system the defined nature of the composite reservoirs remains, but a given diaphragm is replaced by a pattern-forming system which by its nature is a spontaneously evolving volume partitioner and classifier of invariants. The seat of volition or inference for the classification system is thus transferred from the experimenter or theoretician to the diaphragm, and with it the full deductive facility. The equivalence relations or partitions associated with the emerging patterns may thus be associated with theorems of the natural pattern-forming calculus. The entropy

function, together with its derivatives, is the vehicle which relates the logistics of reservoirs and diaphragms to the analog logistics of the continuum. Maximum path probability or second-order differentiability of the entropy in isolation are sufficiently strong interpretations of the second law of thermodynamics to define the approach to and the nature of patterned stable steady states. For many pattern-forming systems these principles define quantifiable stable states as maxima or minima (or both) in the dissipation. An elementary statistical-mechanical proof is offered. To turn the argument full circle, the transformations of the partitions and classes which are predicated upon such minimax entropic paths can through digital modeling be directly identified with the syntactic and inferential elements of deductive logic. It follows therefore that all self-organizing or pattern-forming systems which possess stable steady states approach these states according to the imperatives of formal logic, the optimum pattern with its rich endowment of

equivalence relations representing the central theorem of the associated calculus. Logic is thus ‘‘the stuff of the universe,’’ and biological evolution with its culmination in the human brain is the most significant example of all the irreversible pattern-forming processes. We thus conclude with a few remarks on the relevance of the contribution to the theory of evolution and to research on artificial intelligence.

Coherent dynamic structure factors S(k,ω) obtained by means of thermal neutron inelastic scattering are presented. The experiments were performed on liquid ${}_{}{}^{36}{}_{}{}^{}\mathrm{Ar}$ at four densities along the 120-K isotherm covering a range of wave numbers k from 4.2 to 39.0 ${\mathrm{nm}}^{\mathrm{-}1}$. The neutron time-of-flight spectra are corrected for all known experimental effects with an improved data-reduction system. Special attention is paid to corrections for multiple scattering, duty-cycle overlap, and instrumental resolution. The importance of various correction steps is shown. The reliability of the corrected data is assessed by means of two independent consistency checks, viz., the detailed-balance condition and the first frequency moment of S(k,ω). The S(k,ω) data are presented both as a function of k at fixed ω and as a function of ω at fixed k. The peak height and full width at half maximum of S(k,ω) at fixed k are shown for all densities together with the small-k (hydrodynamic) and large-k (free-gas) asymptotes. The frequency moments of S(k,ω), evaluated up to the fourth moment, are consistent with results from computer simulations and from theoretical calculations. The longitudinal current correlation function $C_l$(k,ω), derived from the experimental S(k,ω), is examined and both its peak position (yielding the dispersion curve for longitudinal current fluctuations) and its peak height (a measure of the lifetime of the fluctuations) are discussed.

Recently, by exploiting the Riemannian structure of the thermodynamic state space, a system of equations was given for the distribution of the fluctuations of arbitrary thermodynamic state variables. Here we present an alternative version of the theory, which incorporates explicitly the balance equations for the extensive parameters. For small fluctuations the usual Gaussian law is reproduced.

The far-infrared spectra of diatomic polar molecules, immersed in a rare-gas liquid, are calculated using two non-Markovian formalisms: the total-time-ordering-cumulant (TTOC) and the partial-time-ordering-cumulant (PTOC) formalisms. Both TTOC and PTOC spectra coincide in the Markovian limit. The interaction between the diatomic molecule and the liquid is described by means of a directing intermolecular field with known statistical properties. In the secular approximation both TTOC and PTOC spectra contain a sum of resonances which become Lorentzian in the Markovian limit. The interference effect among resonances is taken into account using iterative methods in which the secular profile is the first term in the iterative process. We apply the theory to calculate the far-infrared spectra (0–200 ${\mathrm{cm}}^{\mathrm{-}1}$) of a HCl molecule in Ar, Kr, and Xe liquids. There is good agreement between experimental and theoretical absorption spectra.

Late-stage diffusion-controlled coarsening during the process of phase separation is considered. We derive a reduced theory in terms of ordinary differential equations for the cluster density, mean cluster size, width of the cluster size distribution, and supersaturation. Parameters introduced by the approximations are fixed such as to yield agreement with the asymptotic Lifshitz-Slyozov-Wagner theory. The theory improves upon the work of Langer and Schwartz because it includes subcritical clusters as part of the minority phase and explicitly considers the width of the size distribution which controls the rate of coarsening. The theory is constructed in a way which allows treating open systems. For a two-cell model with diffusional coupling we numerically solve the full equation for the size distribution and prove the quality of our theory through the comparison. In addition, we propose an approximate procedure to include the effect of a finite volume fraction of the minority phase and the process of coagulation.

Numerical calculations [D. Mattis and S. Rudin, Phys. Rev. Lett. 52, 755 (1984)] indicate the absence of bound states of three identical spin-(1/2) fermions moving on a two-dimensional lattice and interacting with attractive two-body zero-range potentials. Here, we give a proof of this result in the strong-coupling limit, and we also prove that, in the weak-coupling limit, the ground state of six fermions consists of three bound pairs. We discuss the relevance of this result to superconductivity.

We present a pseudopotential local-density calculation for ${\mathrm{Be}}_2$ using a molecular supercell scheme. The results for the binding energy (-0.43 eV), equilibrium bond length (4.53 a.u.), and ground-state vibrational frequency (373 ${\mathrm{cm}}^{\mathrm{-}1}$) are in reasonable agreement with recent measurements and other quantum-chemical calculations.

The transmission of coherent laser light through the semiconductor CuCl is studied and consideration of the backward-propagating wave’s contribution, produced by the mismatch of the boundary dielectric constants, is included in the treatment. The interference of the backward wave with the forward wave, the étalon effects, is absent from experimental data. We explain this discrepancy by assuming rough surfaces and treating an ensemble average of CuCl samples of different length. We conclude that surface roughness is an important experimental factor and that it may lead to an apparent attenuation which would be interpreted as an increase in the damping constants, especially for long samples.

It is shown that the optical-field-induced first-order molecular reorientation and intrinsic optical bistability in a nemiatic liquid crystal can always be enhanced or suppressed by a low magneitc field (typically <1 kG) and hence can be seen in all existing nematics at a low optical power (typically <1 kW/${\mathrm{cm}}^2$).

We discuss the long-wavelength, low-frequency excitations of non- equilibrium systems with a helical structure as observed in spiral vortex flow in the Taylor instability in long cylinders. We find that a single-phase equation can give rise to a propagating mode, a situation unknown from hydrodynamics in local thermodynamic equilibrium.

The transition to a wavy vortex state from the Taylor vortex state has been studied under a quasistatic ramping condition. Six different values of the radius ratio, η ranging between 0.7204 to 0.7399, were used. For η≤0.7248, the flow always evolves into an m=2 state, while for η≥0.7260, the flow evolves into an m=3 state. (From an earlier study, it is known that the latter transition type—i.e., to an m=3 state—extends to values of η at least as great as 0.7767.) Therefore, the neutral stability curves for the m=2 and m=3 states must intersect somewhere within the narrow range 0.7248 <η<0.7260. The flow states near the intersection should exhibit behavior characteristics of a multiple bifurcation of codimension 2. No unusual transition behavior has yet been observed. Our results compare favorably with recent stability calculations by C. A. Jones.

An experiment, performed on a circular liquid metal flow subjected to an external magnetic field, shows that the fractal dimension of the attractors describing turbulence can discontinuously increase with the control parameter.

The results of an experimental investigation into the role of ‘‘end effects’’ in Taylor-Couette flow with moderate aspect ratio up to 47 are presented. The investigation has concentrated on the lower and upper stability limits for the steady ‘‘anomalous’’ 40-cell state. It is found that the finite length of the cylinders plays a crucial role in determining the overall stability of the flow.

A subset of pseudoregular eigenfunctions of the classically chaotic stadium billiard is shown to participate strongly in vertically directed motion, supporting the conjectures of McDonald and of Heller regarding periodic orbits and pseudoregular eigenfunctions.

In the model of electrostatic aggregation in two dimensions introduced by Shraiman and Bensimon, finite-time singularities leading to cusps in the interface occur for a special set of initial conditions. We show that this instability is essentially independent of initial conditions. Also, necessary and sufficient conditions for conservation of the number of critical points are stated for a generalized version of the problem.