Calculation of static correlations in a fluid of linear molecules

Recently, both X-ray and neutron diffraction data have been obtained for liquid crystalline materials. While these results contain information about the instantaneous correlations of molecular positions and orientations in such fluids, no relevant calculations are available to aid the interpretation of the data. In this paper a formalism which may be used to describe static, molecular correlations in isotropic or anisotropic fluids of linear molecules is developed. To obtain the theory, the direct and total correlation functions of two molecules are expanded as series of spherical harmonics the arguments of which describe the molecular orientations. A relation between the expansion coefficients of these series is obtained by making use of the Ornstein-Zernike equation. It is shown that this equation may be solved in the Percus-Yevick or Optimized-Random-Phase approximations by a variational technique. This technique is used to find the total correlation function in the Percus-Yevick approximation of hard, linear molecules of various length-breadth ratios at several fluid densities.     

Pair correlation functions in nematics: Free-energy functional and isotropic-nematic transition

We develop a free-energy functional for an inhomogeneous system that contains both symmetry conserved and symmetry broken parts of the direct pair correlation function. These correlation functions are found by solving the Ornstein-Zernike equation with the Percus-Yevick closure relation. The method developed here gives the pair correlation functions in the ordered phase with features that agree well with the results found by computer simulations. The theory predicts accurately the isotropic-nematic transition in a system of anisotropic molecules and can be extended to study other ordered phases such as smectics and crystalline solids.     

Comments on the analytical solutions of the sticky hard rod model

The result that follows by taking the Percus-Yevick (PY) approximation from the exact analysis of the sticky hard rod model is shown to differ from that previously obtained by Tago, Y., and Katsura, S., 1975, Can. J. Phys. 53, 2587, and the correct PY compressibility and virial equations of state are given. It is also shown that Born-Green theory yields the exact solution for the sticky hard rod fluid.     

Analytical expressions for the correlation function of a hard sphere dimer fluid

A closed form expression is given for the correlation function of a hard sphere dimer fluid. A set of integral equations is obtained from Wertheim's multidensity Ornstein-Zernike integral equation theory with Percus-Yevick approximation. Applying the Laplace transformation method to the integral equations and then solving the resulting equations algebraically, the Laplace transforms of the individual correlation functions are obtained. By the inverse Laplace transformation, the radial distribution function (RDF) is obtained in closed form out to 3D (D is the segment diameter). The analytical expression for the RDF of the hard dimer should be useful in developing the perturbation theory of dimer fluids.     

Statistical mechanics of linear molecules II: Density expansions,intermolecular potentials and virial coefficients

In the Percus-Yevick and convolution-hypernetted-chain equations obtained in the previous paper, a density expansion for the correlation functions is introduced. To first order in the density, the so-obtained equations are identical and exact. By solving these, the pair correlation functions for linear molecules are obtained explicitly to first order in the density and for arbitrary order in the potential perturbation expansion. From these, the second and third virial coefficients can be extracted for all orders. A generalized charge is defined and used to give generalized multipole expansions for the intermolecular potential. Explicit expressions for this potential model are given up to fourth order. It is shown how the correlation functions, and second and third virial coefficients can be obtained to fourth order for any intermolecular potential with the same perturbation structure.     

A hard sphere fluid model for superionic conductors

We develop a theory for the mobile constituent of a superionic conductor using the Ornstein-Zernike integral equation for the pair correlation function of an inhomogeneous fluid. We solve this equation in the Percus-Yevick approximation using a simple decoupling procedure and hard core potentials. Comparison is made with molecular dynamics calculations on α-AgI.     

Integral equation theory for fluids ordered by an external field: separable interactions

The structural and thermodynamical properties of classical fluids orientationally ordered by an external field are investigated by means of integral equation theories. A general theoretical framework for handling these theories is developed and detailed for the particular case of separable interactions between fluid particles. This approach is then illustrated for the case of two (off lattice) models: the ferromagnetic Heisenberg model and a simple liquid crystal model, for which the numerical solution of integral equations such as the Percus-Yevick, the hypernetted chain, and the reference hypernetted chain closure equations are presented and compared with Monte Carlo simulation results and the analytical solution of the mean spherical approximation. The zero-field case is also examined, and the spontaneous ordering is analyzed in detail, mainly in what concerns the appearance of infinite wavelength singularity in the Ornstein-Zernike equation and the relation with the one-body closure equations and the long range orientational ordering that occurs. In particular, it is shown that the Wertheim one-body closure equation appears as a sum rule compatible with the Ornstein-Zernike equation. The relation between the elastic constant and the long range tail of the pair correlation function is made explicit. In particular, the long range behavior of the various terms in the expansion of the pair correlation function is depicted. The numerical investigation of the two models shows that it is not possible to discriminate between the four integral equations, as to which one would be the most accurate in all cases. The general trends in the thermodynamical and structural properties seem to indicate that the Percus-Yevick approximation is generally better in the strong ordering case, whereas the reference hypernetted chain approximation might be better suited to the study of the isotropic phase and the low ordering regimes.     

Analytic solution of the Percus-Yevick equation for sticky hard sphere potential

It is shown that within the Percus-Yevick approximation the radial distribution function for sticky (i.e. with a surface adhesion) hard spheres satisfies a linear differential equation with retarded right-hand side. Using the theory of distributions and the Green's function technique the analytic solution of this equation is found and explicit formulas are given enabling one to evaluate the radial distribution function both for sticky and non-attractive hard spheres for any distance and any density.     

Dislocation-point defect dynamics in two-dimensional anisotropic liquids: New aspects of melting

The interaction of dislocations and vacancies in a locally correlated liquid is studied within a second quantized linear response theory. One- and two-particle dislocation Green's functions are calculated in RPA, using Zubarev's method. The elementary excitation spectrum and elastic response coefficient are discussed. Non-conservative and mass-conserving processes are differentiated, and the anisotropy of the melting statistics is found to influence the solidification temperature of the liquid. Quantum effects on the melting of Bose systems are discussed. An explicit relation between the elastic response coefficient and Hooke's tensor, based on Kröner's theory, valid in three dimensions, is given in an appendix, as well as its reduction to the two-dimensional case. A comparison with a recent microscopic derivation of this linear response law taking also dynamic properties of the incompatibility tensor into account is presented on a qualitative level.     

Micromorphic Electromagnetic Theory and Waves

This paper introduces a continuum microelectromagnetic theory (also called micromorphic electromagnetic theory), to discuss electromagnetic phenomena in bodies with microstructures. Balance laws of microelectromagnetic media of the first-grade are given. Constitutive equations are developed. The field equations are obtained . It has been shown that, this theory gives rise to several new vector and tensor waves. A theorem of conservation of energy (Poynting type) is proved. Dispersion relations are obtained for both vector and tensor waves. Relations of tensor waves to microscopic phenomena (such as spin waves) are discussed.     

Random-walk theory and the decay of pair correlations in Ornstein-Zernike lattice systems

A random-walk formalism is applied to some general Ornstein-Zernike lattice systems to obtain information as to the asymptotic form of the total correlation function. Calculations in terms of the Percus-Yevick approximation are then presented for certain lattice gases with interactions extending over a few orders of neighbours, illustrating the circumstances in which the decay of the total correlation function may be monotonic or oscillatory.     

A self-consistent approach to linear and nonlinear transport in an electric field

The Boltzmann equation for charge carriers (obeying any statistics) is solved analytically in terms of electric-field-dependent relaxation times for a model collision operator. This operator is a linear, rotationally invariant one, which reduces to a multiplicative operator in each irreducible tensor subspace. The relaxation times are given by a self-consistency requirement. This gives (i) an exact solution of the Boltzmann equation for energy-independent multiplicative factors and (ii) exact asymptotic solutions for low- and high-field situations. Ohm's law is shown to hold exactly in case (i). Furthermore, a number of simple models is shown to be exactly soluble. For more complicated cases, perturbative and iterative methods of solution are discussed.     

The compressibility theorem for quantum simple fluids at equilibrium

An extension of the compressibility theorem for quantum simple fluids within the pathintegral approach is presented. First, it is demonstrated that in the absence of quantum exchange, the isothermal compressibility can be formulated in an exact manner with the use of the pair radial correlation function of the path-integral centroids corresponding to the particles of the fluid. This adds up to the two known formulations based on the pair correlations between true quantum particles, namely the instantaneous and the pair linear response correlations. To complement this extension, an exact Ornstein-Zernike equation for pair centroid correlations is derived, which permits accurate estimates for the isothermal compressibility to be obtained. Several fluids are studied, new numerical results for the latter quantity are reported to support the theoretical points, and some difficulties present in this sort of calculation are discussed. The systems studied are the following: the quantum hard sphere fluid with and without attractive Yukawa interaction, liquid helium-4 and liquid para-hydrogen. Finally, the possibilities of extending the theorem to deal with quantum exchange are considered, and it is shown that the extension and its computational Ornstein-Zernike scheme also hold for a Bose fluid.     

Permittivity of suspensions

The permittivity of a suspension treated as a system of hard spheres is calculated in the Born approximation. The structure of the suspension is described by the Percus-Yevick correlation function. The permittivity of the system under consideration is expressed through a nonlocal susceptibility whose spatial extension is determined by the form factor of suspension particles and the characteristic value of the structure factor. It is shown that the permittivity of the suspension mixture is characterized by a spatial dispersion that manifests itself already in the first order of the perturbation theory. It is demonstrated that the concentration dependences of the extinction length and the transport length calculated from the obtained data on the permittivity tensor exhibit substantially nonlinear behavior. Within the range of applicability of the theory, the results obtained are in agreement with available experimental data.     

An exact operator solution of the quantum Liouville field theory

Exact, closed form results are given expressing the quantum Liouville field theory in terms of a canonical free pseudoscalar field. The classical conformal transformation properties and a Bäcklund transformation of the Liouville model are briefly reviewed and then developed into explicit operator statements for the quantum theory. This development leads to exact expressions for the basic operator functions of the Liouville field: ?_μΦ, and e^nΦ. An operator product analysis is then used to construct the Liouville energy-momentum tensor operator, which is shown to be equal to that of a free pseudoscalar field. Dynamical consequences of this equivalence are discussed, including the relation between the Liouville and free field energy eigenstates. Liouville correlation functions are partially analyzed, and remaining open questions are discussed.     

Polarization effects in non-linear scattering

The problem of non-linear scattering of unpolarized light is discussed. New relationships between the scattering intensities for unpolarized and polarized incident light are given. It is shown that the usual equation connecting the depolarization ratios for plane and unpolarized light does not hold for non-linear scattering processes such as hyper-Rayleigh and hyper-Raman scattering. A new relation between the two is given in terms of two additional intensity ratios. An expression for the depolarization ratio for unpolarized light is also given in terms of the hyperpolarizability tensor.     

Nonlinear massive spin-2 field generated by higher derivative gravity

We present a systematic exposition of the Lagrangian field theory for the massive spin-2 field generated in higher-derivative gravity upon reduction to a second-order theory by means of the appropriate Legendre transformation. It has been noticed by various authors that this nonlinear field overcomes the well-known inconsistency of the theory for a linear massive spin-2 field interacting with Einstein’s gravity. Starting from a Lagrangian quadratically depending on the Ricci tensor of the metric, we explore the two possible second-order pictures usually called “(Helmholtz-)Jordan frame” and “Einstein frame.” In spite of their mathematical equivalence, the two frames have different structural properties: in Einstein frame, the spin-2 field is minimally coupled to gravity, while in the other frame it is necessarily coupled to the curvature, without a separate kinetic term. We prove that the theory admits a unique and linearly stable ground state solution, and that the equations of motion are consistent, showing that these results can be obtained independently in either frame (each frame therefore provides a self-contained theory). The full equations of motion and the (variational) energy-momentum tensor for the spin-2 field in Einstein frame are given, and a simple but non-trivial exact solution to these equations is found. The comparison of the energy-momentum tensors for the spin-2 field in the two frames suggests that the Einstein frame is physically more acceptable. We point out that the energy-momentum tensor generated by the Lagrangian of the linearized theory is unrelated to the corresponding tensor of the full theory. It is then argued that the ghost-like nature of the nonlinear spin-2 field, found long ago in the linear approximation, may not be so harmful to classical stability issues, as has been expected.     

Density fluctuations and the structure of a nonuniform hard sphere fluid

We derive an exact equation for density changes induced by a general external field that corrects the hydrostatic approximation where the local value of the field is adsorbed into a modified chemical potential. Using linear response theory to relate density changes self-consistently in different regions of space, we arrive at an integral equation for a hard sphere fluid that is exact in the limit of a slowly varying field or at low density and reduces to the accurate Percus-Yevick equation for a hard core field. This and related equations give accurate results for a wide variety of fields.