Showing papers on "Correlation function (statistical mechanics) published in 1987"

Origin of the scattering peak in microemulsions

Abstract: From a Landau theory the static scattering intensity distribution I(q) of microemulsions is obtained. As essential ingredient we have included a negative gradient term in the free energy expression. The form of I(q)∼(a2+c1q2+c2q4)−1 yields for a2>0, c1 0 a single broad scattering peak and a q−4 decay at large q, both properties experimentally observed for a variety of microemulsions containing comparable amounts of water and oil. The peak originates from the modulation in the corresponding space correlation function given by γ(r)=(d/2πr)⋅e−r/ξ⋅sin(2πr/d). It is shown that the scattering intensity relation describes experimental literature data remarkably well, using only three fit parameters.

Difference in X-ray scattering between metallic and non-metallic liquids due to conduction electrons

Abstract: The X-ray scattered intensity from a liquid metal as an electron-ion mixture is described using the structure factors, which are exactly expressed in terms of the static and dynamic direct correlation functions. This intensity for a metal is shown to differ from the usual scattered intensity from a non-metal in two points: the atomic form factor and the incoherent (Compton) scattering factor. It is shown that the valence electron form factor, which constitutes the atomic form factor in a liquid metal, leads to a determination of the electron-electron and electron-ion structure factors by combining the ionic structure factor. The authors clarify some confusion which appeared in the proposal by Egelstaff et al. (1974) for extracting the electron-electron correlation function in a metal from X-ray and neutron scattering experiments. A procedure to extract the electron-electron and electron-ion structure factors in a liquid metal is proposed on the basis of the formula for the scattered intensity derived.

Thermal conductivity of the Lennard‐Jones liquid by molecular dynamics calculations

Abstract: Precise results for the thermal conductivity of the Lennard‐Jones liquid obtained by equilibrium molecular dynamics (MD) are presented. These are compared in detail with previous nonequilibrium MD results. Effects due to the truncation of the potential and the particle number dependence are considered. The contributions of the partial correlation functions to the total one were separately calculated. Main results are: (i) in contrast to the viscosity, the thermal conductivity is rather insensitive to the MD conditions, even for the state corresponding virtually to the triple point of argon. The reason for this originates from the simple short ranged time decay of the correlation function. (ii) For the states considered, the partial correlation function involving the ‘‘potential–potential’’ term governs the transport coefficient. (iii) Nonequilibrium MD and MD give consistent values, except for the nonequilibrium method devised by Heyes which generates data far off the range permitted by the error bars. (iv) The computed thermal conductivities fall well in line with experimental data for argon.

Critical correlations in a Z-invariant inhomogeneous ising model

Abstract: We study the critical spin-spin correlation function in the Z -invariant inhomogeneous Ising model, which includes the rectangular, triangular, honeycomb, and checkerboard lattices as special cases. Using quadratic difference equations we derive various explicit representations for the correlation function, confirming and simplifying also a miraculous result of Baxter.

The direct correlation function in hard sphere fluids

Abstract: The pair correlation function in a homogeneous hard sphere fluid at various densities has been measured in a large system, using the Monte Carlo method. The corresponding direct correlation function C2(r) has been determined directly from these measurements, and will be given here in closed form. We conclude that density functional models that neglect the effect of C3 and higher order direct correlation functions, that are defined in bulk fluids, are not able to describe an inhomogeneous hard sphere fluid accurately.

Neutron Spin Echo Study of Dynamic Correlations Near Liquid-Glass Transition

Abstract: Neutron spin echo measurements have been performed on the mixed ionic system Ca0.4K0.6(NO3)1.4 around the glass transition temperature Tg in order to determine the time dependence of the density-density correlation function q(t). This quantity plays a central role in recent mode-coupling theories of the glass transition in schematic models. On the liquid side of the transition our results reveal that there are two distinct slow relaxation processes. The faster one is situated in the 10-13-10-12s time domain. The second one (a) shows a rapid slowing down on T approaching the transition, which scales with the Stokes-Einstein diffusion constant (b) has a stretched exponential form exp (-(t/τ)β] with β 0.6 and (c) tends to become the non-ergodic fraction of the structure factor at the transition. The amplitude ratio of the second and the first relaxation steps varies between 3 and 6 in a smooth oscillatory fashion over the studied wave-number range of q = 0.1-2.1 A-1. These findings are in qualitative agreement with theoretical predictions, but only if the theoretically relevant glass transition temperature T0 is assumed to lie considerably above the phenomenological glass temperature Tg. This implies that the theory breaks down at relaxation times longer than some 10-5s. In contrast, the q dependence of the characteristic time τ of the slower relaxation step revealed a fully unexpected feature, a strong maximum at q0/2, where q0 is the position of the first peak in the structure factor. We propose that this slowing down of the mass density fluctuations reflects the maximum of the charge density structure factor. This new phenomenon is expected to allow the study of the static charge density correlations directly in the q space via the induced dynamic effects.

Correlation in atomic scattering.

Abstract: Correlation due to the Coulomb interactions between electrons in many-electron targets colliding with charged particles is formulated, and various approximate probability amplitudes are evaluated. In the limit that the electron-electron, 1/r/sub i//sub j/, correlation interactions are ignored or approximated by central potentials, the independent-electron approximation is obtained. Two types of correlations, or corrections to the independent-electron approximation due to 1/r/sub i//sub j/ terms, are identified: namely, static and scattering correlation. Static correlation is that contained in the asymptotic, e.g., bound-state, wave functions. Scattering correlation, arising from correlation in the scattering operator, is new and is considered in some detail. Expressions for a scattering correlation amplitude, static correlation or rearrangement amplitude, and independent-electron or direct amplitude are derived at high collision velocity and compared. At high velocities the direct and rearrangement amplitudes dominate. At very high velocities, ..nu.., the rearrangement amplitude falls off less rapidly with ..nu.. than the direct amplitude which, however, is dominant as electron-electron correlation tends to zero. Comparisons with experimental observations are discussed.

Correlation functions in a one-dimensional Bose gas

Abstract: The problem of calculating correlation functions is considered for the one-dimensional Bose gas with a delta-function repulsive interaction between particles. The method of calculation is based on the algebraic Bethe ansatz.

One-dimensional Gibbs states and Axiom A diffeomorphisms

Abstract: On etudie la mecanique statistique en equilibre de systemes reticules classiques a une dimension a interactions a decroissance exponentielle. Pour de tels systemes, les transformees de Fourier des fonctions de correlation de paires sont meromorphes dans une bande, et les residus des poles peuvent s'exprimer en termes de distributions de Gibbs. On applique ces resultats aux diffeomorphismes de l'axiome A de Smale

The collapse transition of linear polymers on fractal lattices

Abstract: The authors show that for linear polymers a collapse transition exists on several fractal lattices and obtain exact results for the critical exponents at this transition. A 'rod-like' phase is found in some cases at intermediate temperatures, between the swollen phase and the collapsed phase. They introduce infinitesimal recursion relations with correlation function rescaling as the formal limit of a class of fractals, which give a better approximation to Euclidean 2D lattices. The gyration radius exponent at the transition temperature lies in the range nu 1=0.546+or-0.010, in good agreement with a recent transfer matrix calculation. The possible relevance of anisotropy at the collapse transition is discussed.

A scattering model for perfectly conducting random surfaces II. Range of validity

Abstract: A recent st udy of the range of validity of the surface scattering model by Eftimiu using a two-dimensional surface is re-examined with a more realistic surface correlation function that is everywhere-differentiable and vanishes gradually at infinity. Ii is shown that a sufficient condition on the range of validity is that cos2 θ [(kσ)2/(kc)1/2]exp{−[2kc(1−sinθ)]1/2} ≪ 1 where k is the electrical wave-number, 0 is the angle of incidence and a and c are the surface standard deviation and the surface correlation parameter respectively.

Ground-state properties of Heisenberg spin chains.

Abstract: The ground-state properties of some finite antiferromagnetic Heisenberg rings with N spins at the isotropic point have been investigated using a modified Lanczos method. Rings with spin 1 and (3/2 have been studied and new results are presented for the correlation functions and the structure factor up to N=16 (12) for S=1 ((3/2). Evidence in support of Haldane's conjecture is found. The correlation functions for S=(3/2 behave as predicted by Affleck.

Example of the description of dissipative processes in terms of reversible dynamic equations and some comments on the fluctuation-dissipation theorem

Abstract: The dynamics of a macroscopic oscillator which is interacting with a heat reservoir, which also consists of oscillators, is analyzed. This problem, which can be solved exactly in its general form in both the classical and quantum-mechanic cases, is used as an example for a study of the transition from a purely dynamic description to a statistical description. Since the system of linear oscillators is not ergodic, an averaging procedure must be regarded as taking an average over the time or over repeated measurements on a unique dynamic trajectory. Depending on the nature of the quadratic form of the potential energy, the oscillations of a macroscopic oscillator can decay in various ways, including exponentially, in the initial stage of the evolution. After a Poincare cycle, the system returns to its initial state, and the damping of the oscillations gives way to a growth. The reversibility of the motion means that the Green's function of the system of oscillators is of odd parity in the time. Equilibrium fluctuations of a macroscopic oscillator are examined. In the classical case the Callen-Welton fluctuation-dissipation theorem can be formulated as follows: The derivative of the coordinate correlation function is proportional to the Green's function of the macroscopic oscillator. In a description in terms of frequencies, the odd parity of the Green's function gives rise to an imaginary part of the Fourier transform of this function in the fluctuation-dissipation theorem. This result is a consequence of the reversibility of the motion in time. The fluctuation-dissipation theorem is proved for Hamiltonian systems without dissipation, but it also applies to systems with dissipation. The exact microscopic Green's function is replaced in this case by the Green's function of a simplified phenomenological description, which explicitly contains dissipative parameters. In the quantum-mechanical case, the results are analogous. The classical and quantum-mechanical versions of the Nyquist relation which follow from the fluctuation-dissipation theorem when the Green's function is approximated by an exponentially damped sinusoidal oscillation are discussed.

Correlation functions of hard body fluids from thermodynamic properties of their mixtures

Abstract: An exact relation between the structural properties of a fluid and the thermodynamic properties of an infinitely dilute mixture is derived. A recurrence formula for the virial coefficients of the infinitely dilute mixtures is proposed and used to calculate the background correlation function of the hard sphere and dumbbell fluids. The results are superior to those obtained by other methods currently available.

On the dielectric susceptibility of classical Coulomb systems. II

Abstract: This paper deals with the shape dependence of the dielectric susceptibility (equivalently defined, in a canonical ensemble, by the mean square fluctuation of the electric polarization or by the second moment of the charge-charge correlation function) of classical Coulomb systems. The concept of partial second moment is introduced with the aim of analyzing the contributions to the total susceptibility of pairs of particles of increasing separation. For a diskshaped one-component plasma with coupling parameter γ=2 it is shown, numerically and algebraically for small and large systems, that (1) the correlation function of two particles close to the edge of the disk decays as the inverse of the square of their distance, and (2) the susceptibility is made up of a bulk contribution, which saturates rapidly toward the Stillinger-Lovett value, and of a surface contribution, which varies on the scale of the disk diameter and is described by a new law called the “arc sine” law. It is also shown that electrostatics and statistical mechanics with shape-dependent thermodynamic limits are consistent for the same model in a strip geometry, whereas the Stillinger-Lovett sum rule is verified for a boundary-free geometry such as the surface of a sphere. Some results of extensive computer simulations of one- and two-component plasmas in circular and elliptic geometries are shown. Anisotropy effects on the susceptibilities are clearly demonstrated and the “arc sine” law for a circular plasma is well confirmed.

Comments On The Correlation Length

Abstract: Surface finish has vertical and transverse attributes. In optics the first is characterized by the root-mean-square roughness, and it is natural to seek an analogous length parameter, or "correlation length", to characterize its transverse properties. This paper evaluates various candidates for this purpose by examining their robustness to the finite-bandwidth effects inherent in all real measurement processes. The critical consideration is the magnitude of the correlation length with respect to the range of surface wavelengths in-cluded in the measurement: When the length parameters of the surface fall outside the measurement bandwidth it is possible to get totally spurious values for the measured parameters, and when they fall inside the band-width, the measured and true values can still differ by significant factors. This sensitivity of the correlation function to bandwidth effects arises from the fact that the measurement process modifies it by convolution; in contrast to the finish power spectrum, where the modifications involve simple multiplication. The present results argue against the use of length parameters derived from the measured correlation function as precise measures of the transverse character of surface finish, although they may still be useful for compar-ative and diagnostic purposes. The most precise characterization of the combined vertical and transverse properties of surface finish are given in terms of its power spectral density.

Computations on the dimensional crossover in collective pinning

Abstract: The displacement correlation function introduced by Larkin and Ovchinnikov for pinned flux lines in a superconducting layer is evaluated explicitly in order to study the thickness dependence. Expressions for the transverse and longitudinal correlation lengths R/sub c/ and L/sub c/ are derived and a criterion for the crossover from two- to three-dimensional collective pinning is obtained. Using these expressions, computations are carried out for amorphous superconductors. L/sub c/ turns out to drop exponentially fast in three dimensions, while R/sub c/ changes gradually through the crossover remaining close to its 2D value. The explicit thickness dependence is logarithmic, so that the pure 3D case is only accessible in a very limited experimental regime.

The direct correlation function of a one-dimensional Ising model

Abstract: The strictly finite range of the direct correlation function for a homogeneous nearest neughbor Ising chain is shown to persist in the presence of arbitrary site-dependent coupling constants and an arbitrary external field. A method is developed to examine the range of the direct correlation function for many-neighbor interactions. It is found from numerical examples that, in general, third-neighbor and higher interactions induce long-range direct correlations, as does the presence of a field in the second-neighbor case.

Distribution function of two cavities and Percus–Yevick direct correlation functions for a hard sphere fluid in D dimensions: Overlap volume function representation

Abstract: A one‐parameter (‘‘smeared diameter’’) overlap volume function representation of the distribution function of two cavities and of the Percus–Yevick direct correlation functions, for a hard‐sphere fluid in D dimensions, is analyzed and found to be very effective.

Equilibrium properties of a multi-component ionic mixture: I. Sum rules for correlation functions

Abstract: Equilibrium statistical methods are used to derive sum rules for two- and three-particle correlation functions of a multi-component ionic mixture. Some of these rules are general consequences of the electrostatic character of the interaction, whereas others depend on specific thermodynamic properties of the system. The first group of rules follows from the BBGKY hierarchy and a clustering hypothesis for Ursell functions. The sum rules of the second group are obtained by describing the system with the help of a restricted grand-canonical ensemble in which the particle numbers of the various components in the mixture fluctuate under the condition that the total charge in the system remains constant.