For the n-vector model of the general type a renormalization-group (RG) scheme in momentum space is developed. Using this scheme we obtain a generalized exact equation of the renormalization group. The main idea of this approach is not to allow the zero-order critical Hamiltonian to move away from the critical surface under the influence of the RG transformation. The equation found differs from the known ones in that all redundant operators are excluded. On the basis of the developed approach we argue that critical exponents are functionals of some function introducing the cutoff into the phase-transition theory.

The leading quantum corrections for the scattering of a particle in one dimension from a pulsed potential are discussed. Depending on the model parameters, the scattering may be either regular or chaotic. The quantum corrections are obtained by applying a semiclassical formalism that gives the distribution of scattered momenta down to a resolution of order ${\mathrm{ħ}}^{4/5}$. At this resolution, the main quantum effect is the softening of the classical rainbow singularities into peaks with widths of order ${\mathrm{ħ}}^{2/3}$. Quantum corrections for examples of both regular and chaotic scattering are compared with exact quantum results, and a criterion is given for estimating the resolution at which the corrections first become significant.

We study numerically the scaling behavior of disordered sandpile automata with preferred direction on a two-dimensional square lattice. We consider two types of bulk defects that modify locally the dynamic rule: (i) a random distribution of holes, through which sand grains may leave the system, and (ii) several models with a random distribution of critical heights. We find that at large time and length scales the self-organized critical behavior, proved exactly in the pure model, is lost for any finite concentration of defects both in the model of random holes and in those models of random critical heights in which the dynamic rule violates the height conservation law. In the case of the random critical height model with the height-conserving dynamics, we find that self-organized criticality holds for the entire range of concentrations of defects, and it belongs to the same universality class as the pure model. In the case of random holes we analyze the scaling properties of the probability distributions P(T,p,L) and D(s,p,L) of avalanches of duration larger than T and size larger than s, respectively, at lattices with linear size L and concentration of defect sites p. We find that in general the following scaling forms apply: P(T)=${\mathit{T}}^{\mathrm{-}\mathrm{\alpha }}$scrP(T/x,T/L) and D(s)=${\mathit{s}}^{\mathrm{-}\mathrm{\tau }}$scrD(s/m,s/${\mathit{L}}^{\nu }$), where x≡x(p) and m≡m(p) are the characteristic duration (length) and the characteristic size (mass) of avalanches for a given concentration of defects.

The power-law behavior of the distributions still persists for length scales T≪x(p) and mass scales s≪m(p). The characteristic length x(p) and mass m(p) are finite for small concentrations of defects and diverge at p→0 according to the power law x(p)∼${\mathit{p}}_{\mathit{x}}^{\mathrm{-}\mathrm{\mu }}$ and m(p)∼${\mathit{p}}_{\mathit{m}}^{\mathrm{-}\mathrm{\mu }}$, with the numerically determined values of the exponents close to ${\mathrm{\mu }}_{\mathit{x}}$=1 and ${\mathrm{\mu }}_{\mathit{m}}$=1.5. The finite size of the lattice may affect the measured probability distributions if for a given concentration of defects the characteristic length x(p) exceeds the lattice size L. A finite-size scaling analysis for the mass distribution yields the exponent ν=1.5, while the duration of the avalanches scales linearly with the size. We also determine the exponent D=1.5 that connects the mass and the duration of avalanches.

The nature of the roughness of the growing surface modeled by the Kardar-Parisi-Zhang (KPZ) equation has been further studied by defining a ‘‘domain’’ structure for it. This was done by mapping the height h(x,t) of the surface onto a ‘‘spin’’ S(x,t)=sgn[h(x,t)-〈h(t)〉], where 〈h(t)〉 is the mean height of the growing surface at time (t). It was then found that the growth of the surface has useful analogies with the domain-coarsening process in nonequilibrium systems quenched into an ordered phase. Thus, in d=1+1, the average size 〈l〉 of spin domains grows as ${\mathit{t}}^{1/3}$ and the domain-size distribution P(l,t) is consistent with ${\mathit{l}}^{\mathrm{-}3/2}$${\mathit{f}}_1$(l/${\mathit{t}}^{2/3}$). We find that the autocorrelation function A(t)=〈S(x,t)S(x,0)〉 decays as ${\mathit{t}}^{\mathrm{-}\mathrm{\alpha }}$, with α(d=1+1)=1.0±0.08 and α(d=2+1)=1.5±0.1. The form of P(l,t) in higher dimensions and the connection between the spin autocorrelation function and the height autocorrelation function are discussed. For computational convenience, the KPZ equation was studied by transforming it to the problem of directed polymers in a random potential.

Different microscopic models exhibiting self-organized criticality are studied numerically and analytically. Numerical simulations are performed to compute critical exponents, mainly the dynamical exponent, and to check universality classes. We find that various models lead to the same exponent, but this universality class is sensitive to disorder. From the dynamic microscopic rules we obtain continuum equations with different sources of noise, which we call internal and external. Different correlations of the noise give rise to different critical behavior. A model for external noise is proposed that makes the upper critical dimensionality equal to 4 and leads to the possible existence of a phase transition above d=4. Limitations of the approach of these models by a simple nonlinear equation are discussed.

A detailed molecular-dynamics study is presented concerning the dynamic one-particle properties of binary and ternary mixtures of hard-sphere-like particles. We have studied the self-diffusion coefficient for each component of the mixture as a function of the density, and as a function of the difference in particle diameter. The results are compared with the predictions of the Enskog theory. At moderate densities positive deviations from the Enskog theory are found; at high densities we find negative deviations. It is observed that the amplitudes of these deviations differ for each component of the mixture. For moderate densities it is shown that these amplitudes have a particle-diameter dependence that is similar to that of the amplitudes of the long-time tails of the velocity-autocorrelation functions. This finding indicates that the vortex processes that give rise to the long-time tails are also responsible for the nonexponential decay of the velocity-autocorrelation function at shorter times.

The Boltzmann entropy of a dilute gas under uniform shear flow is analyzed. The entropy variation associated with viscous heating is evaluated and compared with the local equilibrium expression. For interaction potentials other than the Maxwell potential, significant discrepancies are found that can be related to the difference between thermodynamic quantities, defined from the entropy, and kinetic quantities, defined by means of local equilibrium. The discrepancies change, but remain relevant, when artificial external forces are introduced in order to create an ideal stationary state.

The ballistic limit of isotropic elastic scattering in an infinite slab of thickness L is studied using rigorous lower bounds for the transmission-time distribution. In this model particles with velocity v enter the slab normal to its face. They then scatter isotropically with mean free path λ. In the ballistic limit, where λ/L→∞, the transmission-time distribution moments 〈${\mathit{t}}^{\mathit{n}}$〉 might be expected to converge to those of the input distribution, properly translated in time. It is shown, however, that all moments with n>2 diverge as ${\mathit{c}}_{\mathit{n}}$${\mathit{L}}^2$${\mathit{v}}^{\mathrm{-}\mathit{n}}$${\lambda }^{\mathit{n}\mathrm{-}2}$. Comparison between Monte Carlo simulations and rigorous, analytic lower bounds for the moments show excellent agreement in the ballistic limit. It is conjectured that the anomalous asymptotic behavior of the moments is completely due to trajectories involving at most two scattering events.

We derive approximate relations between the transport coefficients of a dilute gas starting from the linearized equations of hydrodynamics.

We present experimental results for pattern formation in a thin fluid layer heated time periodically from below. They were obtained with computer-enhanced shadowgraph flow visualization and with heat-flux measurements. The experimental cell was cylindrical, with a radius-to-height ratio of 11.0. The temperature of the top plate was held constant while that of the bottom plate was modulated sinusoidally so that the reduced Rayleigh number ε≡ΔT/Δ${\mathit{T}}_{\mathit{c}}$-1 had the form ε(t)=${\mathrm{\epsilon }}_0$+δ sin(ωt). Here the time t and frequency ω are scaled by the vertical thermal diffusion time. Experiments were performed within the ranges 8.0≤ω≤18.0, 0.4≤δ≤3.3, and -0.2≤${\mathrm{\epsilon }}_0$≤0.6. Measurements of the convective threshold shift ${\mathrm{\epsilon }}_{\mathit{c}}$(δ,ω) were in good agreement with theoretical predictions. Comparisons were made with theoretical predictions of a range ${\mathrm{\epsilon }}_{\mathit{A}}$(δ,ω)≤${\mathrm{\epsilon }}_0$<${\mathrm{\epsilon }}_{\mathit{R}}$(δ,ω) (${\mathrm{\epsilon }}_{\mathit{A}}$<${\mathrm{\epsilon }}_{\mathit{c}}$,${\mathrm{\epsilon }}_{\mathit{R}}$≳${\mathrm{\epsilon }}_{\mathit{c}}$) where only a hexagonal pattern with downflow at the cell centers is predicted to be stable, a range ${\mathrm{\epsilon }}_{\mathit{R}}$≤${\mathrm{\epsilon }}_0$<${\mathrm{\epsilon }}_{\mathit{B}}$(δ,ω) where both hexagonal and roll patterns are expected to be stable, and a range ${\mathrm{\epsilon }}_0$≥${\mathrm{\epsilon }}_{\mathit{B}}$ where only a roll pattern should be stable. At low modulation amplitudes (δ≲1.2 for ω=15) only rolls were observed over the range of ${\mathrm{\epsilon }}_0$ studied, although the rolls appeared perturbed for ${\mathrm{\epsilon }}_{\mathit{A}}$≤${\mathrm{\epsilon }}_0$<${\mathrm{\epsilon }}_{\mathit{R}}$. At moderately high amplitudes (1.2≲δ≲2.3 for ω=15), a cellular pattern with local sixfold symmetry and downflow at the cell centers, which was reproducible from one cycle to the next, was observed over the range ${\mathrm{\epsilon }}_{\mathit{A}}$≲${\mathrm{\epsilon }}_0$≲${\mathrm{\epsilon }}_{\mathit{R}}$. Over the range ${\mathrm{\epsilon }}_0$≳${\mathrm{\epsilon }}_{\mathit{B}}$ roll-like patterns were observed. Over the range ${\mathrm{\epsilon }}_{\mathit{R}}$≲${\mathrm{\epsilon }}_0$≲${\mathrm{\epsilon }}_{\mathit{B}}$, where theory predicts bistability of rolls and hexagons, a coexistence between the two patterns was found. At high values of δ (δ≳2.3 for ω=15), a pattern consisting of randomly placed cells and short roll segments that was reproducible from one cycle to the next was observed in all three regions. At still higher values of δ (δ>3.0 for ω=13), this pattern was observed to be irreproducible from one cycle to the next. The transition from patterns resembling those predicted by the deterministic theory to irreproducible random patterns as δ is increased is presumed to be due to stochastic perturbations. These perturbations appear to play an important role in those parameter ranges where the amplitude of the pattern decays to a microscopic value during part of the modulation cycle.

In a linear stability analysis of axisymmetic Taylor vortex flow between counterrotating cylinders we find a parity-breaking bifurcation and a Hopf bifurcation. They are due to an internal resonance of modes with wave number q and 2q and do not break the azimuthal symmetry of the flow. They may lead to drift waves and standing waves, respectively. The standing waves resemble an optical mode, in which adjacent vortex pairs have opposite phase. A temporally periodic driving of the system is expected to excite and stabilize such standing waves. An analysis of the amplitude-phase equations for parity-breaking bifurcations shows that the drift waves can be localized although the parity-breaking bifurcation itself is supercritical. This result applies also to the other systems in which localized drift waves are observed as, for instance, in directional solidification.

A phenomenon of spontaneous separation of components in an initially uniform fluid mixture is found experimentally. A qualitative explanation of the effect is proposed in terms of nonparallel streamlines in the medium.

We calculate the field due to lines and planes of dipoles, and use this to calculate the bulk and surface energies of various dipolar-lattice structures. The calculation technique is simpler than traditional Ewald methods, and is well suited to surface-energy calculations. We find strongly anisotropic surface energies for the lattices; the more closely packed lattices have negative surface energies for directions approximately parallel to the dipolar axis. These results should be applicable to ferro- and electrorheological fluids.

Dielectric studies of a thermotropic-liquid-crystalline substance [pentylcyanobiphenyl (5CB)] under high pressure are reported. Measurements of a complex dielectric permittivity were carried out in the pressure range 0.1–160 MPa, the frequency range 1 kHz–13 MHz, and the temperature range 287–343 K. Static permittivities measured at constant pressures are governed by the [2S(T)+1]/T factor. An increase in permittivity with pressure at a constant temperature is interpreted as a result of an increasing molecular ordering and of a breaking of the dipole-dipole correlations between neighboring molecules. The relaxation time ${\mathrm{\tau }}_{\mathrm{\parallel }}$ characterizing molecular rotations around the short axes in the nematic phase is analyzed as a function of temperature and pressure, yielding an activation enthalpy of approximately 60 kJ ${\mathrm{mol}}^{\mathrm{-}1}$, which decreases with increasing pressure, and an activation volume of about 60 ${\mathrm{cm}}^3$ ${\mathrm{mol}}^{\mathrm{-}1}$, decreasing with increasing temperature. An activation energy in isochoric conditions was evaluated to be 37 kJ ${\mathrm{mol}}^{\mathrm{-}1}$ at ${\mathit{V}}_{\mathit{m}}$=247 ${\mathrm{cm}}^3$ ${\mathrm{mol}}^{\mathrm{-}1}$, 25 kJ ${\mathrm{mol}}^{\mathrm{-}1}$ at 243 ${\mathrm{cm}}^3$ ${\mathrm{mol}}^{\mathrm{-}1}$. A relation between the different activation parameters is discussed.

A sui generis second-order transition in water solutions of Rochelle-salt crystal has been seen during measurements of both the solubility and solution densities versus temperature. Both the volume-expansion coefficients and the temperature differential of solution densities for several compositions along the solubility curve present a jump discontinuity around a saturation temperature of 24 °C. Electrical measurements revealed a small peak in the dielectric constant, showing a minute ferroelectric component of solute near 24 °C in solution. The loss tangents of solution and the dissolution enthalpy (drawn from the solubility curve) show other characteristics. The excess volume of solute vanishes around 31 °C, showing an ‘‘ideal’’ behavior of solution in this region. All data seem to indicate the tartrate molecule ${\mathrm{NaKC}}_4$${\mathrm{H}}_4$${\mathrm{O}}_6$ associates with the four water molecules in solution, just like the water molecules undergoing crystallization. This quasicrystalline structure of the solute is stable in solutions up to about 40 °C. A tentative interpretation as a critical phenomenon is made for the volume-expansion coefficient. This response function with a cusplike singularity at 24 °C has a critical point exponent of about 1/2. A one-to-one correspondence within four temperature ranges (between 16–18 °C and 58 °C), of the properties of the crystal and solute Rochelle salt, has been experimentally proved in five types of measurements.

The behavior of low-dimensional manifolds such as interfaces, membranes, or polymers in an external potential is studied in d=${\mathit{d}}_{\mathrm{\parallel }}$+${\mathit{d}}_{\mathrm{\perp }}$ dimensions. If the potential has several minima, the manifolds can undergo nontrivial transitions from a localized to a delocalized state at a characteristic temperature T=${\mathit{T}}_{\mathrm{*}}$. For interfaces in d=1+1, several universality classes must be distinguished which can be determined exactly by transfer-matrix methods. The same classification also applies to (self-intersecting) polymers in d=1+${\mathit{d}}_{\mathrm{\perp }}$. Within a functional renormalization-group approach, the critical behavior is governed by a line of fixed points in close analogy with unbinding transitions. However, in contrast to unbinding, this fixed point line now contains the trivial fixed point at which one can determine the spectrum of eigenperturbations exactly. The number of relevant eigenperturbations increases monotonically with the decay exponent τ=${\mathit{d}}_{\mathrm{\parallel }}$/ζ where the roughness exponent ζ characterizes the delocalized manifold.

The structure and dynamics of helium at 5 eV over the density range 1.5≤ρ≤8 g/${\mathrm{cm}}^3$ were studied using self-consistent-field molecular dynamics. Representative results for the pair-correlation function, velocity autocorrelation function, and the average charge density surrounding an atom are presented. Many-atom screening is found to have an important effect on all of these quantities above 0.3 g/${\mathrm{cm}}^3$. Above 2 g/${\mathrm{cm}}^3$, screening effects appear to saturate, and the coefficient of diffusion tends to the limit of a one-component plasma.

When considering critical phenomena in liquid phases we show that the critical exponents can be calculated using the classical components of the liquid-state theory (Ornstein-Zernike equation, n-body direct correlation function, etc.) to which we must add a renormalization procedure. In this approach, we consider the Callan-Symanzik equation as the natural closure of the Ornstein-Zernike equation. Our work allows one to elucidate some points that are generally not investigated in the standard theory. For example, at the liquid-gas critical point, we show that two coupling constants must be introduced; they are not independent. Their definition is given at a microscopic level and their dimensions are calculated. Moreover, we point out the role of the n-body direct correlation functions. Specially, the influence of the three-body correlation functions on the critical exponents is emphasized. In the case of a binary mixture we show that the critical exponents remain those of the pure liquid along the line of critical points. However, some differences between the pure liquid and the mixture appear in the calculation. For instance, the structure of the coupling constants is much more complicated in the mixture than in the case of the pure liquid. This paper represents an extension of a recent work published by the authors [Phys. Rev. Lett. 67, 1598 (1991)].

The nematic to neat-soap (N-NS) and nematic to isotropic (N-I) phase transitions in binary solutions of cesium per-fluoro-octanoate (CsPFO) and water have been studied by high-precision calorimetry at 39.3, 43.6, 48.3, 49.0, 50.7, 52.0, and 55.3 wt. % CsPFO concentrations. The ratio of the heat-capacity amplitudes, above and below the N-NS transition, and the critical exponent α remain near 1 and 0, respectively, suggesting three-dimensional XY-type behavior. The coexistence range and the latent heat at the first-order N-I transition decrease with decreasing concentration. Extrapolation of the N-I coexistence width suggests existence of a Landau point at ∼28 wt. % concentration. The coefficients of expansion in the Landau–de Gennes free energy at the N-I transition and their concentration dependence have been determined. Coefficients of the cubic and the biquadratic terms become vanishingly small at low concentrations underscoring the need to use higher-order terms in the free energy.

The measurement of the saddle-splay surface elastic constant ${\mathit{K}}_{24}$ in a nematic liquid crystal is reported based on two independent deuterium nuclear-magnetic-resonance ${(}^2$H-NMR) experiments. Fifty years after the pioneering work of Oseen and Zocher, these measurements were made from observations of nematic director-field configurations and a configuration transition discovered in submicrometer-sized cylindrical cavities of Nuclepore membranes under selected surface preparations and wall curvatures. The experimental difficulties in separating the effect of anchoring energy from surface elastic energy (inherent in small confining volumes) were overcome by a unique use of NMR and the ability to predict stable nematic structures with elastic theory. Direct comparison of calculated ${}_{}{}^2{}_{}{}^{}\mathrm{NMR}$ spectral patterns to experiment is very sensitive to the details of the stable nematic director-field configuration in cylindrical cavities. Small differences in the director configuration imposed by the curvature or elastic properties of the nematic liquid crystal are strongly reflected in the shape of the spectral pattern. Different nematic structures with preferred perpendicular anchoring conditions such as the escaped radial and planar polar show markedly different patterns. Theoretical analysis reveals that a planar-polar structure is preferred in cavities with a high degree of curvature or sufficiently weak anchoring conditions at the cavity boundary.

The features of the planar-polar structure are described in terms of the dimensionless parameter ${\mathit{RW}}_0$/K, where R is the radius, ${\mathit{W}}_0$ is the molecular anchoring strength, and K is the bulk elastic constant in the one-constant approximation. At a sufficiently large radius or substantially strong anchoring conditions, the escaped-radial structure is favored and sensitive to the dimensionless surface parameter σ=${\mathit{RW}}_0$/K+${\mathit{K}}_{24}$/K-1. The ${}_{}{}^2{}_{}{}^{}\mathrm{NMR}$ technique unambiguously distinguishes between these two stable nematic director-field configurations and is sensitive to the surface parameters σ and ${\mathit{RW}}_0$/K. Theoretical analysis also reveals that the relative magnitude of the bend (${\mathit{K}}_{33}$) and splay (${\mathit{K}}_{11}$) bulk elastic constants plays a vital role in determining the molecular anchoring angle at the cavity wall of the escaped-radial configuration. As ${\mathit{K}}_{33}$/${\mathit{K}}_{11}$ increases, the molecular anchoring angle is shown to deviate further from its preferred perpendicular orientation in the presence of finite molecular anchoring to alleviate the expensive bend deformation. Experimentally, point defects occur in the escaped-radial configuration where the direction of bend is changed, resulting is a series of alternating hyperbolic and radial defects. An analytical trial function is constructed that describes these defect structures in the region of interest.