By combining the hypothesis that the thermodynamic curvature is the correlation volume with the hypothesis that the free energy is the inverse of the correlation volume, I propose that the thermodynamic curvature is proportional to the inverse of the free energy near the critical point. This hypothesis leads to a partial differential geometric equation for the free energy which a generalized homogeneous function reduces to a third-order nonlinear ordinary differential equation whose solution is consistent with two-scale factor universality. The resulting scaled equation of state is, overall, in very good agreement with mean-field theory, the three-dimensional Ising model, and experiment for the pure fluid. Universal ratios among the critical amplitudes are also in good agreement with known values. For the non-mean-field theory exponents, the solution considered here is not analytic in the whole one-phase region; the second derivative of the free energy suffers a discontinuity.

The viscosity coefficient of iodine in the temperature range 500≤T≤3000 K is calculated. Because of the low dissociation energy of the ${\mathrm{I}}_2$ molecules, the dissociation degree of the gas increases quickly with temperature, and I+${\mathrm{I}}_2$ and I+I collisions must be taken into account in calculations of viscosity at temperatures greater than 1000°. Several possible channels for atom-atom interaction are considered, and the resulting collision integrals are averaged over all the important channels. It is also shown that the rigid-sphere model is inaccurate in predictions of the viscosity. The approach of the present work is general and can be used for other diatomic gases with arbitrary dissociation degree.

We report experimental results of Rayleigh-Brillouin spectra of xenon along the isotherm ${\mathit{T}}_1$=293.26 K at high pressures, up to 130 atm. For this purpose we use the available measurements of thermodynamic and transport properties for the numerical simulation of the model. We clearly identify a pressure domain centered at the pressure ${\mathit{P}}_1$=64 atm corresponding to the critical density at ${\mathit{T}}_1$. In the pressure domain outside the pressure range defined by ${\mathit{P}}_1$∓25 atm, the full Rayleigh-Brillouin spectra are described by hydrodynamic theory. Inside this pressure range, experimental spectra are only partially described by the model used, but nevertheless we confirm the existence of a diffusive mode partially detected by Cannell and Benedek [Phys. Rev. Lett. 25, 1157 (1970)]. This mode reaches a maximum of intensity and width at half height for the pressure ${\mathit{P}}_1$. We discuss an explanation for the critical perturbation of the Brillouin line with this mode.

In this paper some insight into the behavior of the one-particle distribution function describing the Knudsen gas in a convex-plane domain is given. The influence of different kinds of boundary conditions is studied, namely of those following ergodic or chaotic reflection laws and of those with a stable periodic orbit. By applying numerical and theoretical methods it is shown that, under appropriate boundary conditions and after a sufficiently long period of time, there appear enclaves free of particles in the configuration space, and moreover the collection of limiting velocities reached by every particle is finite. On the basis of this result we prove that the one-particle distribution function becomes nonanalytic after a sufficiently long period of time, independent of the initial conditions. On the other hand, it turns out that in our model ergodicity or chaotic properties of the reflection law imply ergodicity or chaotic behavior of the Knudsen gas, respectively. From this result some related properties of the one-particle distribution function of the corresponding Knudsen gas for long periods of time are deduced.

This paper reports the behavior of the probability density functions (PDF’s) of temperature differences, between different times, but measured at the same point, which is at the center of a helium-gas cell. One objective of this work is to study how these PDF’s evolve as the separation time increases. Data from seven Rayleigh numbers (Ra), which range from ${10}^9$ to ${10}^{15}$ and are above what is called the soft-to-hard turbulence transition, are studied. These PDF’s are symmetric and non-Gaussian, and are fitted approximately by a stretched-exponential form ${\mathit{e}}^{\mathrm{-}\mathit{c}\mathrm{\Vert }\mathit{x}\mathrm{\Vert }\mathrm{\beta }}$. As the separation time τ increases, the parameter β starts from 0.51±0.05 (for the smallest separation), remains approximately constant for τ≤${\mathrm{\tau }}_1$, then increases, and finally at τ=${\mathrm{\tau }}_2$, it saturates to 1.7±0.1 (1.6±0.1 for the two largest Ra studied). For Ra<7.3×${10}^{10}$, β increases as ${\mathrm{\tau }}^{0.27\pm 0.03}$, while for Ra≥7.3×${10}^{10}$, the increase has to be described by two powers: for lower τ, β first increases slower as ${\mathrm{\tau }}^{0.15\pm 0.03}$, then for τ≥${\mathrm{\tau }}_{\mathit{b}}$, it increases as ${\mathrm{\tau }}^{0.27\pm 0.02}$. Attempts are made to understand the origin of these time scales and to relate them to time scales previously identified in the problem.

By extending a recent calculation of self-diffusion in shearing systems [Cummings et al., J. Chem. Phys. 94, 2149 (1991)] we derive a Green-Kubo formula for the thermal-conductivity tensor in fluids subject to planar Couette flow. These expressions are valid for weak heat flows in strongly shearing fluids.

The fractal dimension of propagating interfaces in turbulence is analyzed using a variant of a method recently used to obtain intermittency corrections to the fractal dimension of diffusive-scalar interfaces. The fractal dimension is found to be the same as for diffusive-scalar interfaces.

Arguments are presented for understanding the selection of the speed and the nature of the fronts that join stable and unstable states on the supercritical side of first-order phase transitions. It is suggested that from compact support, nonpositive-definite initial conditions, observable front behavior occurs only when the asymptotic spatial structure of a trajectory in the Galilean ordinary differential equation (ODE) corresponds to the most unstable temporal mode in the governing partial differential equation (PDE). This selection criterion distinguishes between a ‘‘nonlinear’’ front, which has its origin in the first-order nature of the bifurcation, and a ‘‘linear’’ front. The nonlinear front has special properties as a strongly heteroclinic trajectory in the ODE and as an integrable trajectory in the PDE. Many of the characteristics of the linear front are obtained from a steepest-descent linear analysis originally due to Kolmogorov, Petrovsky, and Piscounov [Bull. Univ. Moscow, Ser. Int., Sec. A 1, 1 (1937)]. Its connection with global stability arguments, and in particular with arguments based on a Lyapunov functional where it exists, is pursued. Finally, the point of view and results are compared and contrasted with those of van Saarloos [Phys. Rev. A 37, 211 (1988); 39, 6367 (1989)].

We calculate both the base flow and its stability to order ${\mathrm{\Omega }}^2$ for a Taylor-Couette system subjected to a Coriolis force, where Ω is a dimensionless measure of the system’s rotation rate. A generalized linear eigenvalue problem is posed for the stability calculation. The first-order correction to the eigenvalue is found to be zero in accordance with the system’s symmetry. The second-order correction is calculated using the zeroth- and first-order eigenfunctions and a suitably defined inner product. The calculation yields an eigenvalue in quantitative agreement with experimental results that show that the Coriolis force stabilizes the fluid against the onset of the primary instability. The analysis indicates that the primary instability leads to time-independent, nonaxisymmetric Taylor vortices, also in agreement with experiments. The range in Ω for the validity of the perturbation analysis decreases as the radius ratio increases.

In the present paper, we propose another translational-dilation method (TDM) to construct Sierpinski carpets. Based on the proposed TDM, we present a quasianalytic method, the method of self-similar solutions, to solve heat-transfer problems of two-dimensional self-similar structure Sierpinski carpets. The numerical computations are performed for the self-similar solution method. The numerical results show good agreement with those of the finite-difference method.

Detailed measurements of tilt angle, polarization, and susceptibility have been carried out in the vicinity of the electric-field-induced smectic-A–ferroelectric-smectic-C critical point in order to determine the critical exponents β, γ, and δ describing the temperature dependences of order parameter and susceptibility, and the course of the critical isotherm. For β, the mean-field value of 0.5 is found. The results for γ and δ are less conclusive since values slightly smaller than the mean-field values are indicated.

The selection of self-similar fingers growing in a wedge is examined numerically both in the convergent and divergent flow regime. For divergent flows, a bifurcation occurs in the spectrum of the relative finger width λ versus σ, the effective surface tension parameter, when it is very small (σ<0.005). By comparison with experimental data, we show that it can explain the tip-splitting instability usual to any radial growth process. At finite surface tension (σ>0.005), a universal selection law is revealed numerically for any sector value, in both growth regimes.

An additional two-particle interaction is introduced in the Widom model of microemulsions in order to produce three-phase coexistence between the oil-rich, water-rich, and disordered fluid phases in accordance with experimental results. We test the resulting model numerically through Monte Carlo simulations. We observe a hysteresis in the order parameter and a jump in the surface tension at the transition between the disordered fluid and water-rich and oil-rich phases, thus signifying a first-order transition with three-phase coexistence.

A model of x-ray diffraction for thin smectic-A liquid-crystal films is presented. The effect of the smectic-layer displacement fluctuations and correlations and the molecular form factor on the interlayer structure and the x-ray-diffraction pattern is discussed. In thin films the influence of the displacement-displacement correlation function on the x-ray-diffraction pattern is very small and can be neglected in the analysis of experimental data. On the other hand, both the displacement-fluctuation term (Debye-Waller factor) and molecular form factor produce strong measurable effects and so can be determined. We discuss the dependence of displacement fluctuations, calculated in the framework of the Landau–de Gennes model, on the smectic elastic constants and the smectic-vapor surface tension and show that these constants can be determined from the x-ray-diffraction pattern. The analysis of the hydrodynamic (collective) layer fluctuations and the individual molecular-motion fluctuations shows that the latter can be neglected in comparison to the former. The fluctuation amplitudes predicted by the model agree within 5% with the recent experimental measurements performed on smectic-I on -C films.

In thin smectic films the fluctuation amplitudes σ(r)=[〈${\mathit{u}}^2$(r)〉${]}^{1/2}$ are only ∼4 Å compared with ∼8 Å in a macroscopic sample. The fluctuations are suppressed at the two free surfaces by the large surface tension, grow away from each surface, and have a parabolic profile near the center of the film. We argue that one of the reasons for surface freeezing in smectic liquid crystals is the quench of the layer fluctuations by the large surface tension. However, we also show that in systems with small surface tension the fluctuations at the surface are in fact larger than the ones in the interior of the system. The growth of the diffuse scattering, due to the displacement-displacement correlations, with the thickness of the smectic film is discussed and shown to evolve towards the structure predicted for large samples by Gunther, Imry, and Lajzerowicz [Phys. Rev. A 20, 1733 (1980)]. The model for the displacement layer fluctuations including the director as an independent variable is presented. Furthermore, the coupling between the layer fluctuations and the nematic order parameter in smectic liquid crystals is qualitatively discussed. It is argued that the compressional modes induce the nematic order-parameter fluctuations and that a large fluctuation profile may induce the smectic-A–smectic-C phase transition in thin films. Eventually in tilted smectic liquid crystals the layer fluctuation profile may induce a tilt profile. Finally, it is shown that the presented model can be applied to smectic systems other than smectic-A; we give explicit formulas for the x-ray-scattering intensity from the smectic-${\mathit{A}}_{\mathit{d}}$ films and also calculate the fluctuations amplitudes for the stratified smectic-I on -C system.

In this paper, a model for water-oil-surfactant mixtures, which we have previously studied on two- and three-dimensional lattices, is now studied on a one-dimensional lattice. In this case we are able to obtain exact results, whereas on the higher-dimensional lattices it was necessary to use approximations. This one-dimensional model produces correlation and structure functions that are similar to those obtained for the disordered phase on the two- and three-dimensional lattices. The disorder line is obtained from the water-water correlation function and the Lifshitz line is derived from the water-water structure function. One or the other of these lines is typically used to divide the disordered phase into a region of ordinary disordered fluid and a microemulsion region. Both these lines calculated exactly for the one-dimensional lattice behave similarly to their counterparts on the two- and three-dimensional lattices calculated by various approximations.

A second-order density-functional theory is used to study the isotropic-nematic transition in a system of hard ellipsoids of revolution. The direct pair-correlation functions of the coexisting isotropic liquid that enter in the theory as input information are obtained from solving the Ornstein-Zernike equation using the Percus-Yevick closure relation. The spherical harmonic expansion coefficients of the correlation functions obtained from this solution are in good agreement with those found from computer simulations. We find that a system spontaneously transforms to a nematic phase when the structural parameter denoted by c${\mathrm{^}}_{22}^{\mathrm{}\left(0\right)\mathrm{}}$ attains a value close to 4.40. This value of c${\mathrm{^}}_{22}^{\mathrm{}\left(0\right)\mathrm{}}$ depends, although very weakly, on the value of the length-to-width ratio of the molecules. The transition parameters we found are in very good agreement with the results generated by computer simulations. By using the harmonic coefficients of the direct pair-correlation functions, we have calculated the Frank elastic constants of the nematic phase.

The Hagen-Poiseuille flow of liquidlike drops formed during capillary condensation of Lennard-Jones molecules through narrow cylindrical pores is studied by molecular-dynamics simulations. The observed time evolution of the system depends upon the strength of the wall-fluid potential and the strength of the gravitational-type driving force. In the case of weak adsorbing walls, the whole drop slips along the pore, and during the flow changes in its local structure are observed. For stronger fluid-wall interaction and for larger driving forces, the phenomenon of vanishing of the drop has been found.

The liquid-crystalline phase behavior of a fluid of biaxial hard particles (spheroplatelets) is studied using a scaled particle calculation of the fluid configurational entropy combined with a cell description of translational order. If translational ordering is ignored, the density versus particle biaxiality phase diagram displays a cusp-shaped biaxial-nematic phase intervening between two uniaxial nematic phases. The location of the crossover from rodlike to platelike uniaxial-nematic behavior is in agreement with previous bifurcation-analysis results. The density discontinuity at the isotropic-nematic transition decreases as this crossover is approached from both the rodlike and platelike sides, becoming vanishingly small at the crossover point itself. When the possibility of translational order is considered, the phase diagram displays three distinct smectic-A phases, in addition to the two uniaxial nematic phases, and only a small remnant of the biaxial-nematic phase. One of the three smectic phases has in-layer orientational order, while the other two have in-layer isotropic order.

Molecular-dynamics (MD) simulations have been carried out on a two-component Lennard-Jones system, quenched into supercooled and amorphous states. Two different regimes of viscous behavior are found in the time window accessible in MD simulation studies (of the order of nanoseconds if units appropriate for argon are used). The results for the time dependence of the self-intermediate scattering function ${\mathit{F}}^{\mathit{s}}$(q,t) show two different slow relaxation processes, where the slowest (α relaxation) can be represented by a stretched exponential, exp[-(t/${\mathrm{\tau }}_{\mathrm{rel}}$${)}^{\mathrm{\beta }}$]. In the frequency domain this gives rise to a quasielastic peak, and it is found that its area, the nonergodicity parameter ${\mathit{f}}^{\mathit{s}}$(q)==a, shows an anomalous decrease when increasing the temperature towards a critical value ${\mathit{T}}_{\mathit{c}}$. This happens in the supercooled-liquid regime, and it is one of the basic predictions of the recent mode-coupling theory for the liquid-glass transition problem. In the strongly supercooled-liquid regime the diffusion is of the hopping type, and it is found to be strongly cooperative in nature.

Experimental studies of free convection in binary mixtures have revealed that the use of a lateral stress-free boundary induces a stabilization of localized traveling waves.