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Showing papers on "Correlation function (statistical mechanics) published in 1972"


Journal ArticleDOI
TL;DR: The first order electric field correlation function of laser light scattered by polydisperse solutions of macromolecules can be written as a sum or distribution of exponentials, with decay rates proportional to the diffusion coefficients of the solute molecules as discussed by the authors.
Abstract: The first order electric field correlation function of laser light scattered by polydisperse solutions of macromolecules can be written as a sum or distribution of exponentials, with decay rates proportional to the diffusion coefficients of the solute molecules. It is shown that the logarithm of this correlation function is formally equivalent to a cumulant generating function. A method is described by which the distribution function of the decay rates (and thus the extent of polydispersity) can be characterized, in a light scattering experiment, by calculation of the moments or cumulants. The systematic and random statistical errors in the calculated cumulants are discussed.

2,613 citations


Journal ArticleDOI
TL;DR: In this paper, a modification of an earlier theory of Singwi et al of electron correlations at metallic densities is presented, which allows for the change of the pair correlation function in an external weak field via the density derivative of the equilibrium pair correlation functions.
Abstract: In this paper we present a modification of an earlier theory of Singwi et al of electron correlations at metallic densities The modification consists in allowing for the change of the pair correlation function in an external weak field via the density derivative of the equilibrium pair correlation function This results in a new expression for the local-field correction The present theory has the merit of satisfying almost exactly the compressibility sum rule and of giving a satisfactory pair correlation function Results of self-consistent numerical calculations for the static pair correlation function, correlation energy, compressibility, and plasmon dispersion relation for the electron liquid in the metallic-density range are presented For those interested in the application of the results of the present paper, numerical values of the local-field correction as a function of wave number have been tabulated in the density range ${r}_{s}=1\ensuremath{-}6$

732 citations


Journal ArticleDOI
TL;DR: In this article, a statistical expression for the surface tension of a liquid-vapor interface is derived, which contains the density profile and the direct correlation function in the vicinity of the interface.
Abstract: We derive a statistical mechanical expression for the surface tension of a liquid-vapor interface. It contains the density profile and the direct correlation function in the vicinity of the interface. When a local Ornstein-Zernike approximation is made on the direct correlation function, the surface-tension theory of Fisk and Widom is obtained.

231 citations



Journal ArticleDOI
TL;DR: In this article, the authors used the Monte Carlo method to obtain the triplet correlation function for low density fluid argon with three different temperatures, assuming a Lennard Jones type potential.
Abstract: The triplet correlation function for low density fluid argon was obtained by the Monte Carlo method for three different temperatures, assuming a Lennard‐Jones‐type potential. At low densities the agreement with the prediction of the Kirkwood superposition assumption (SA) was found to be better than 10% for most values. Significant differences between the two were found at small distances, where the SA gives unreliable estimates, and at the peaks of the triplet function, where the SA predicts too large a value.

68 citations


Journal ArticleDOI
TL;DR: In this paper, a simple closed form for the Fourier transform of the correlation function of an order-parameter correlation function was obtained and its dependence on wave vector and the reciprocal correlation length was examined.
Abstract: A central problem in the theory of phase transitions is the calculation of the order-parameter correlation function. Here we study the correlation function of an $n$-component order parameter whose configuration energy is determined by the usual Landau functional. Using a screening approximation, we obtain a simple closed form for the Fourier transform of the correlation function and examine its dependence on wave vector $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ and the reciprocal correlation length $k$.

52 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the thermodynamic behavior of a classical system with short-range interactions reduces, when the system becomes infinite in at least one dimension, to the calculation of the largest eigenvalue of the transfer matrix.
Abstract: In this work we briefly review the Ornstein-Zernike prediction for the decay of correlation functions, extend it to treat the decay of correlation near surfaces, and then contrast this prediction with the exactly known results for the two-dimensional Ising model. We develop the transfer-matrix approach to classical statistical mechanics in sufficient generality for its use in later papers in this series, where it is employed to derive general forms for the decay of correlation functions in Ising models away from the critical point, which provide a clear explanation of the failure of the Ornstein-Zernike theory for the two-dimensional Ising model. In particular, we show that the thermodynamic behavior of a classical system with short-range interactions reduces, when the system becomes infinite in at least one dimension, to the calculation of the largest eigenvalue of the transfer matrix. Using the Perron-Fr\"obenius theorem, we show that for a system infinite in no more than one dimension, an arbitrary correlation function defined on the system decays at least exponentially fast. One is able to predict whether the decay of correlation is monotone or oscillatory on the basis of the largest few eigenvalues of the transfer matrix.

51 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that in the case of the three non-polar solvents, broadening due to vibrational fluctuations is small and that the band shape comes essentially from vibration-rotation coupling.
Abstract: Extensive measurements of vibrational and, in a few cases, pure rotational Raman spectra of HCl, DCl, HBr and DBr dissolved in SF6, C2F6, CCl4 and SO2 have been carried out at room temperature. Band profiles of both isotropic and anisotropic scattering have been studied and discussed in the light of existing theories on molecular motions in liquids. From isotropic scattering band-fitting it is concluded that, in the case of the three non-polar solvents, broadening due to vibrational fluctuations is small and that the band shape comes essentially from vibration-rotation coupling. Thus information about reorientational motions can be obtained directly from anisotropic scattering. For SO2 solutions this statement is no longer valid because of the important contribution of vibrational fluctuations to the profile of the isotropic scattering. In this case the rotational correlation function can be determined, as for pure liquid hydrochloric and hydrobromic acids, by elimination of the vibrational correlation fu...

40 citations


Journal ArticleDOI
TL;DR: In this paper, high-temperature series for the inverse spin-spin correlation function of the Ising ferromagnet are presented for the three cubic lattices and the critical correlations are analyzed for scaling behavior in both position and momentum space.
Abstract: High-temperature series for the inverse spin-spin correlation function of the $s=\frac{1}{2}$ Ising ferromagnet are presented for the three cubic lattices. Series are tabulated to twelfth order in $K=\ensuremath{\beta}J$ in zero magnetic field and to eighth order in finite field. Direct series expansions for the true correlation length are derived. The critical correlations are analyzed for scaling behavior in both position and momentum space. The critical amplitudes which characterize the correlations both at ${T}_{c}$ and in the Ornstein\char22{}Zernike region are determined. The Fisher\char22{}Burford scattering approximant is found to be consistent with the data. Finally, the consequences of the universality hypothesis are formulated for the correlations. Correlation-amplitude data for the three cubic lattices are shown to be in remarkable agreement with a formulation of universality which requires that the free energy in a region of diameter given by the coherence length be scale independent.

36 citations


Journal ArticleDOI
TL;DR: In this article, the equilibrium properties of a one-dimensional plasma model for numerical simulation using finite-size particles (clouds) with spatial grids were investigated experimentally and compared with the theoretical predictions.
Abstract: Some of the equilibrium properties of a one‐dimensional plasma model for numerical simulation using finite‐size particles (clouds) with spatial grids were investigated experimentally and compared with the theoretical predictions. Fluctuation spectrum of the electric field, correlation function, and Landau damping of the Fourier modes, drag on a particle, diffusion in velocity space, Debye shielding, wakes due to fast particles, and the frequency spectrum of the Fourier modes were measured. The experimental results give good agreement with the theoretical predictions of a kinetic theory for finite‐size particles, showing the reduction of interactions at short wavelengths and, hence, reduction of collisional effects.

29 citations



Journal ArticleDOI
TL;DR: In this paper, it is argued that the collision distribution which determines the rotational diffusion process need not necessarily be a Poisson distribution in liquids with strong interactions and the distribution is governed by the frequency distribution of the medium.
Abstract: Starting with an m‐diffusion model a matrix description is given of the rotational motion of a dipole molecule undergoing frequent collisions. This treatment gives rise to an analytical expression for the dipole correlation function and for the angular momentum correlation function in which a limited number of parameters from the model appear. It is argued that the collision distribution which determines the rotational diffusion process need not necessarily be a Poisson distribution. In liquids with strong interactions the distribution is governed by the frequency distribution of the medium. This leads to the inclusion of a librational motion in the rotational diffusion model. A comparison of simulations with different collision distributions and experimental data is given.

Journal ArticleDOI
TL;DR: In this paper, the free transverse vibrations of elastic rectangular plates with random material properties are considered and the perturbation technique is applied and statistical characteristics of the random eigenvalues are determined.


Journal ArticleDOI
TL;DR: In this paper, the authors derived time correlation functions for gases of linear molecules in a static homogeneous magnetic field from the set of transport relaxation equations obtained from the linearized Waldmann-Snider equation.
Abstract: Abstract Time correlation functions for gases of linear molecules in a static homogeneous magnetic field are derived from the set of transport relaxation equations obtained from the linearized Waldmann-Snider equation. In particular, the complex auto-and cross correlation functions of the friction pressure tensor and the tensor polarization are considered. They are of importance for the viscosity, the depolarized Rayleigh spectrum, and flow birefringence. For the gases HD and CO2 their real and imaginary parts are plotted as functions of a reduced time and discussed. Lastly, the influence of a constant magnetic field on the flow birefringence property of a gas of linear molecules in a Couette flow arrangement is treated. The principal axes of the dielectric tensor are rotated about the field direction. For two field orientations the respective angles of rotation are determined. The eigenvalues of the dielectric tensor reveal the tendency of the optical and transport properties of the gas to decouple for increasing field strengths

Journal ArticleDOI
TL;DR: In this paper, the Gibbs distribution for a polymer molecule contains implicitly the correlation functions for velocity and position of the constituent monomers, and the velocity correlation function is calculated, exploiting the markovian structure of the problem in phase space.
Abstract: The Gibbs distribution for a polymer molecule contains implicitly the correlation functions for velocity and position of the constituent monomers. The purely spatial part, ignoring potentials, gives the random flight distribution; the velocity correlation function is calculated, exploiting the markovian structure of the problem in phase space. It is shown that the problem can be reduced to an eigenfunction problem and hence solved. The form of the correlation function is quite accurately given by ( nu n1- nu n2)2=(2kT/m)(1-exp(-c mod n1-n2 mod )) where nu n1, nu n2 are the velocities of the n1th and n2th monomers and c equivalent to 1.

Journal ArticleDOI
TL;DR: In this article, the ground state of an electron gas is studied by means of the method of correlated basis functions developed by Wu and Feenberg, and the optimum form of the Bijl-Dingle-Jastrow-type wave function of a charged-boson gas obtained with the application of the paired-phonon analysis is used as the correlation factor in a trial wave function describing the electron gas.
Abstract: The ground state of an electron gas is studied by means of the method of correlated basis functions developed by Wu and Feenberg. The optimum form of the Bijl-Dingle-Jastrow-type wave function of a charged-boson gas obtained with the application of the paired-phonon analysis is used as the correlation factor in a trial wave function describing the electron gas. Numerical results are given for the correlation energy per particle and the radial distribution function at metallic densities. The high-density limit of the system is briefly discussed.

Journal ArticleDOI
TL;DR: In this article, a new method of correlation-function display and peak detection based on the polarity correlator is described, where the overload states of a parallel array of integrating counters, when displayed at regular intervals of elapsed integration time, define the correlation function.
Abstract: A new method of correlation-function display and peak detection, based on the polarity correlator, is described. It is shown that the overload states of a parallel array of integrating counters, when displayed at regular intervals of elapsed integration time, define the correlation function. The system has been especially designed for use in correlation flowmeters.

Journal ArticleDOI
TL;DR: In this article, expectation values of various atomic operators are given for a 1s1s′ correlation split wave function, and are used in a simple variational calculation allowing for angular correlation in the 1S state.
Abstract: Expectation values of various atomic operators are given for a 1s1s′ correlation split wavefunction, and are used in a simple variational calculation allowing for angular correlation in the 1S state. The radial and angular correlation energy contributions obtained are roughly additive.

Journal ArticleDOI
TL;DR: It is shown that in the latter case in the limit of large N the statistics of clusters is fully described by the cluster densities pl and pm, and a correlation function g&R) for which a susceptibility theorem is derived.

Journal ArticleDOI
TL;DR: In this paper, the transverse correlation and transverse modes in the dynamical region in a simple liquid are studied on the basis of the generalized Langevin equation under the assumption of Gaussian decay of the time correlation of the random forces.
Abstract: Properties of the transverse correlation and transverse modes in the dynamical region in a simple liquid are studied on the basis of the generalized Langevin equation. Under the assumption of Gaussian decay of the time correlation of the random forces, the molecular expressions for the transverse current correlation function and the transverse response function are obtained. Numerical calculations are made for the liquid argon at T =84.25 K. The analysis of spatial correlations shows that the local correlations in a liquid may conveniently be described by two parts, the radial and the angular parts. The damping of the correlation Δ ( k ) shows a very weak dependence on the wave number k . The dispersion relation of the transverse mode is obtained from the maximum of the correlation function. The analytical properties of the pole of the response function shows the possibility of the propagation of the transverse mode in the case where G / Δ 2 >0.294, G being related with the modulus of rigidity.

Journal ArticleDOI
TL;DR: In this article, the form and decay time of the intensity correlation function of the light from a single-mode He:Ne laser have been measured in the neighborhood of threshold, and the experimental results are found to be substantially in agreement with the nonlinear oscillator theory of the laser, without adjustment of any arbitrary scaling parameters.
Abstract: The form and the decay time of the intensity correlation function of the light from a single-mode He:Ne laser have been measured in the neighborhood of threshold. The experiments were performed at several predetermined values of the pump parameter a , ranging from a = -1 to a = 4 . By combination of the experimental results with the theoretically predicted eigenvalues appearing in the expansion of the correlation function, it was possible also to determine the amplitudes of the first two terms in the expansion over the same range. The experimental results are found to be substantially in agreement with the nonlinear oscillator theory of the laser, without adjustment of any arbitrary scaling parameters.

Journal ArticleDOI
TL;DR: In this paper, various spectral relationships are developed for energy-bounded and powerbounded functions in terms of a running-parameter (time) Fourier Transform, and the resultant spectral densities and auto-(or cross-) correlation functions are initially generalized for N dimensions, and then specialized in more familiar one-dimensional relationships.

Journal ArticleDOI
TL;DR: In this paper, the dipole correlation function governing electronic absorption is e−iωOt cosγt with spectrum ω 0±γ, associated with vertical excitation, and a procedure is outlined for characterizing the state reached first after interaction with light is initiated.
Abstract: For a particular model with two electronic states, each with two vibrations, the dipole correlation function governing electronic absorption is e−iωOt cosγt with spectrum ω0±γ. The function starts as e−iωOt (γ≪ω0), with Fourier transform peaking around ω0 instead of ω0±γ, and this is associated with vertical excitation. After a time t∼2/γ the spectrum goes over into the normal one. As a generalization, a procedure is outlined for characterizing the state reached first after interaction with light is initiated. Finally it is suggested that one can understand aspects of internal conversion by analogy with the case of vertical excitation.

Journal ArticleDOI
TL;DR: In this paper, the susceptibility of a model polymer having rotating dipolar sidechains with nearest neighbor interactions and diagonal visocosity was studied. And the authors were able to produce a Cole-Cole plot which approaches the origin with finite slope at high frequencies, as does much experimental data.
Abstract: We study the susceptibility of a model polymer having rotating dipolar sidechains with nearest neighbor interactions and diagonal visocosity. The susceptibility can be found exactly (1) at zero frequency for arbitrary interaction strength, and (2) at zero interaction strength for arbitrary frequency and viscosity. In addition, a high‐temperature expansion exists for arbitrary frequency both in the viscous limit and for finite viscosity. We also study the time‐dependent correlation function 〈 cosφi(t) cosφi+1(0)〉 for arbitrary viscosity and weak interaction, and the possibility of a more complex diagonal friction law. In no case are we able to produce a Cole‐Cole plot which approaches the origin with finite slope at high frequencies, as does much experimental data.


Journal ArticleDOI
M. Nambu1
TL;DR: In this article, new collision terms are obtained within the framework that the ternary correlation function is expressed in terms of the products of the single particle and pair correlation functions, and the Balescu-Lenard collision term is modified.

Journal ArticleDOI
TL;DR: In this paper, a statistical study of the correlations of the complex poles of the unitary collision matrix is carried out, and it is shown that both for the elastic and the inelastic scattering the correlation coefficient of the two total widths is always very small.
Abstract: A statistical study of the correlations of the complex poles of the unitary collision matrix is carried out. It is shown that both for the elastic and the inelastic scattering the correlation coefficient of the two total widths is always very small. A simple relation satisfied by the correlation coefficient of the real parts of the complex poles is given. The distribution of the single width is calculated and compared with the Porter‐Thomas distribution and the one obtained by a numerical calculation. Some other interesting results, like the energy correlation function for the purely elastic scattering cross section and a relation satisfied by the resonance parameters for the fluctuation calculation, are also given.

Journal ArticleDOI
J. J. van Aartsen1
01 Jun 1972
TL;DR: In this article, a new theory is developed which takes the correlation function approach, but in such a way that the resulting expressions can be applied to obtain meaningful parameters from experimental data, which is accomplished by expansion of the general correlation function G (r) into a series of spherical harmonics and introducing a number of rather general assumptions.
Abstract: Theories for calculating the scattering of light from solid polymeric materials can be divided into two classes. The particle scattering theories using models of the scattering entities and treating the total scattering in a “dilute solution” approximation form one of these. The statistical theories utilizing correlation functions constitute the other class. They are better in principle but cannot be easily connected with the known physical structure of the material. Moreover, the existing theories are either too simple or too complicated to be of practical use. In the present article a new theory is developed which takes the correlation function approach, but in such a way that the resulting expressions can be applied to obtain meaningful parameters from experimental data. The derivation is accomplished by expansion of the general correlation function G (r) into a series of spherical harmonics and introducing a number of rather general assumptions. It is also pointed out that cross correlations are intractable. This new theory is particularly suited to compare the two types of light scattering theories as applied to macroscopically isotropic materials. It also shows that the determination of the usual scattering components is not enough to obtain the maximum amount of information possible.

Journal ArticleDOI
Abstract: The Mayer series for the pair correlation functions of the Gaussian mixture has been determined to ninth power of the density (11‐point graphs). The model is a mixture of A and B in which the AA and BB interaction potentials are zero. The AB interaction is such that the Mayer f function, exp (−vAB/kBT) −1, is a negative Gaussian. The direct correlation functions and radial distribution functions are evaluated using Pade approximants. Comparison is made with solutions of the Percus‐Yevick and hypernetted chain equations.