Showing papers on "Correlation function (statistical mechanics) published in 1975"
TL;DR: In this article, a simple but relable parametrization of the direct correlation function d (r) for hard spheres is proposed, which is shown to be very poor over all but fairly reasonable in the most important region r σ.
Abstract: The direct correlation function c (r) and in particular the ’’tail’’ of the direct correlation function d (r) are calculated for hard spheres from reliable expressions for the radial distribution function of the hard sphere fluid. This permits a direct examination of the Percus–Yevick (PY) approximation d (r) =0. It is found that the PY approximation is very poor over‐all but fairly reasonable in the most important region r ? σ. Finally a simple but relable parametrization of d (r) for hard spheres is proposed.
229 citations
TL;DR: In this article, the scaling transformation is used in numerical calculations of the properties of the two-dimensional Ising model near its critical point, and the best approximation is seen to lead to 1 part in ${10}^{4}$ accuracy for the two largest scaling indices.
Abstract: Scaling transformations are used in numerical calculations of the properties of the two-dimensional Ising model near its critical point. When compared with the exact Onsager solution, the best approximation is seen to lead to 1 part in ${10}^{4}$ accuracy for the two largest scaling indices. This rather accurate calculation is obtained by utilizing a scaling transformation which depends upon a parameter. The parameter is set by demanding that two different evaluations of the magnetic eigenvalue agree with one another. One evaluation is found via the standard eigenvalue method; the other comes from a consistency condition for the spin-spin correlation function. This condition may also be used to distinguish scaling eigenvalues from other eigenvalues.
151 citations
93 citations
TL;DR: In this paper, the spectral decay rate at high frequency was used as a criterion for positive definability of the correlation function. And the spectral spectral spectral properties of the spectral transform of a correlation function were investigated.
Abstract: Objective analyses using the so-called method of optimum interpolation incorporates statistical information on the variable(s) by means of the covariance or correlation functions. The concern in this contribution is with some properties of the analytic forms of the correlation functions that are used to model the statistical structure. First, some attention is directed to the question of fitting the various analytic forms (containing adjustable constants) to samples of actual correlations. All but one of the candidate forms were indistinguishable on the basis of the residuals of the statistical fitting procedure. Second, the criterion of positive-definiteness of the correlation function is extended to stipulate that the transform (or spectrum) of the function should possess some features of the spectra of actual variables—the most important one being the spectral decay rate at high wavenumber. Again, all but one of the candidate forms (the same one as above) had transforms that were acceptable. T...
78 citations
TL;DR: In this paper, a high pressure study of the depolarized Rayleigh scattering in acetone, benzene, and methyl iodide is presented, where the density dependences of the angular velocity and angular position correlation functions are determined.
Abstract: Results of high pressure study of the depolarized Rayleigh scattering in acetone, benzene, and methyl iodide are presented The density dependences of the angular velocity and angular position correlation functions are determined The angular velocity correlation function of each liquid shows a negative overshoot In addition, benzene at all pressures and methyl iodide at high pressures exhibit additional oscillations in the angular velocity correlation function The details of the correlation functions are discussed in terms of ''collision interrupted'' models such as the well known J‐diffusion model It is concluded that a more general description than J diffusion capable of allowing incomplete loss of momentum upon collision is needed The results are compared to studies of reorientation by means of NMR Finally, the relation between shear viscosity and reorientation in these liquids is discussed
69 citations
TL;DR: In this article, the light-scattering spectrum of liquid triphenyl phosphite (TPP) has been studied over a temperature range in which the parameter (k 2η/Γoρ) varies from 0·17 to 50, where k is the scattering vector, η the shear viscosity, ρ the mass density and Γ 0 the rotational correlation frequency determined from the HH spectrum.
Abstract: The light-scattering spectrum of liquid triphenyl phosphite (TPP) has been studied over a temperature range in which the parameter (k 2η/Γoρ) varies from 0·17 to 50, where k is the scattering vector, η the shear viscosity, ρ the mass density and Γ0 the rotational correlation frequency determined from the HH spectrum. The VH spectrum changes markedly in character over this temperature range and can be explained in terms of molecular rotations coupled to shear modes. A single coupling parameter, dependent upon a time-dependent correlation function, is required to fit the spectrum; this parameter (R) is about equal to 0·45 and is independent of temperature and of k. The spectrum is a function of the parameter (k 2η/Γ0ρ) which includes variations with viscosity, temperature and scattering angle. A statistical theory is presented which yields a different explicit expression for the spectrum from that obtained previously; the difference is essential at large values of (k 2η/Γ0ρ). The coefficient of shear viscos...
59 citations
50 citations
TL;DR: In this article, a digital clipped auto correlation spectroscopy was used to investigate the kinetic features of thermodynamic fluctuation processes which relax on time scales from 10−6 to 102 s. The results indicate a nonexponential correlation function that can be fitted with the decay function exp - (t/τβ)β, where β and τβ are adjustable parameters.
Abstract: Digital clipped auto correlation spectroscopy was used to investigate the kinetic features of thermodynamic fluctuation processes which relax on time scales from 10−6 to 102 s. The autocorrelation function associated with structural relaxation in Na2O· K2/O·6SiO2, glass was determined in the annealing region. The results indicate a nonexponential correlation function that can be fitted with the decay function exp - (t/τβ)β, where β and τβ are adjustable parameters. The parameter β which characterizes the deviation from a simple exponential, was essentially constant (β= 0.57 ± 0.02) from 468 ° to 550 ° C. The characteristic time constant, τβ ranged from 1.35 × 10−2 to 6.30 s. The computed average relaxation times were compared with shear viscosity data reported for a similar glass; both the temperature dependence and the magnitude agreed well with the predictions of viscoelastic theory.
36 citations
TL;DR: In this paper, the authors applied the renormalization-group method to the analysis of phase transitions in systems where the order parameter s is coupled to a nonordering additional variable y. They found a variety of critical and tricritical behaviors or first-order transitions as a function of physical variables and possible macroscopic constraints imposed on the system.
Abstract: The renormalization-group method is applied to the analysis of phase transitions in systems where the order parameter s is coupled to a nonordering additional variable y. A variety of critical and tricritical behaviors or first-order transitions is found as a function of the physical variables and possible macroscopic constraints imposed on the system. For s2y coupling, the correlation function of y was found to be governed by a correlation length which is proportional to that of the order parameter, and by a critical index ηy∙ηy=2-2α; α here is the specific-heat exponent of the appropriate unconstrained system. The singular part of the susceptibility, χy, has a critical exponent equal to α, the true specific-heat exponent. When the coupling is s2y2, a weaker singularity of χy appears. The crossover between this behavior and the one typical to s2y coupling is calculated. (y) has a singular part with an exponent 1-α, in the unconstrained case. The breakdown of the scaling law related to the correlation function of y in the constrained case is discussed.
35 citations
TL;DR: In this article, the threshold conditions for a phase transition to a negative pion condensate in neutron matter were studied, where the Δ isobar was allowed to enter as an intermediate particle in the polarisation bubbles of the curent-current correlation function.
31 citations
TL;DR: In this article, the generalized second moment of a rotational spectral band is related to an angular momentum−orientation time−correlation function for the ensemble of molecules, and it is demonstrated that previous equations relating these spectral transforms to angular momentum correlation functions are recovered in the limit of short times.
Abstract: A general expression is derived that relates the generalized second moment of a rotational spectral band to an angular momentum−orientation time−correlation function for the ensemble of molecules. It is demonstrated that previous equations relating these spectral transforms to angular momentum correlation functions are recovered in the limit of short times. The extended diffusion models proposed by Gordon are used to calculate cross−correlation functions as a function of time, and the results are compared with several approximations, including the decoupling assumption widely used in spin−rotation relaxation. The cross correlations that appear in the spin−rotation problem are considered in detail. In contrast to the behavior of the spectral transform cross−correlation functions, it is explicitly demonstrated that the spin−rotation correlations for the extended diffusion models vanish after one collision.
TL;DR: In this article, the M− and J−diffusion models were extended to the particular case of the perpendicular vibration of a linear molecule and the spectral intensity was obtained in a simple analytical form.
Abstract: The M− and J−diffusion models, which are proposed by Gordon to describe the reorientation of molecules, and which were applied to parallel vibrations only, are extended to the particular case of the perpendicular vibration of a linear molecule. In the case of the M−diffusion model, the method presented here yields a simple analytical form for the correlation function of the perpendicular vibration. In the case of the J−diffusion model, the spectral intensity is also obtained in a simple analytical form. Previously published experimental data on compressed gases are compared with the theoretical results.
TL;DR: In this paper, the inelastic neutron scattering cross section for dense nitrogen gas at several pressures near 240 atmospheres and room temperature has been measured in the region of the first diffraction peak, and the Van Hove self-correlation function is calculated from models and subtracted from these cross sections to yield ∂S d(Q, ω)/∂p.
Abstract: The inelastic neutron scattering cross section for dense nitrogen gas at several pressures near 240 atmospheres and room temperature has been measured in the region of the first diffraction peak. The Van Hove self-correlation function is calculated from models and subtracted from these cross sections to yield ∂S d(Q, ω)/∂p. This derivative is related to an integral over the time-dependent triplet correlation function and models for the triplet function are tested. A simple model, based on pair functions, gives moderate agreement.
TL;DR: In this paper, the authors have calculated the time-dependent spin-spin correlation function for the classical Heisenberg chain by computer simulation at a series of temperatures and showed clearly the existence in the non-ordered system of spinwave excitations at sufficiently low temperatures.
TL;DR: In this paper, general relations are presented to relate the fluctuation thermodynamic properties of multicomponent electrolyte solutions (concentration derivatives of the chemical potential, partial molar volume, and compressibility) to integrals of the total correlation function (Kirkwood-Buff solution theory) and of the nondivergent portion of the direct correlation function.
Abstract: General relations are presented to relate the fluctuation thermodynamic properties of multicomponent electrolyte solutions (concentration derivatives of the chemical potential, partial molar volume, and compressibility) to integrals of the total correlation function (Kirkwood-Buff solution theory) and of the nondivergent portion of the direct correlation function. Detailed expressions are given for the single-salt, single-solvent system. It appears that the direct correlation function expressions may be of more practical use in developing correlations for solution properties because, unlike the total correlation functions, terms exist which distinguish between cation and anion interactions with water and with other ions.
TL;DR: In this article, a theory for the long time tails on the velocity correlation function for finite systems is developed, which is used to calculate the system size dependence of the self-diffusion coefficient in two dimensions and the form of the "tail" in small two dimensional systems.
Abstract: A theory is developed for the now well‐known ’’long time tails’’ on the velocity correlation function for finite systems. The theory is used to calculate the system size dependence of the self‐diffusion coefficient in two dimensions and the form of the ’’tail’’ in small two dimensional systems. The relationship between a three dimensional system with one small edge and a true two dimensional system is investigated.
TL;DR: In this paper, the authors compare the theoretical rotational velocity correlation functions with the Fourier transform of the optical absorption coefficient per unit length [α(ω)] in the time domain and find that the Wyllie-Larkin treatment results in an excessive apparent mean square torque, although having the correct short-time zero slope.
Abstract: The Brot–Larkin and Wyllie–Larkin models of permanent-dipolar librational and relaxational molecular motion in the liquid and plastic crystal (rotator) phases are tested in the time domain by comparing the theoretical rotational velocity correlation functions with the Fourier transform of the optical absorption coefficient per unit length [α(ω)]. The discord between model and observation is particularly acute at short times. The Brot–Larkin function does not have the necessary zero slope of the correlation function at t→ 0 because of its use of an effectively infinite intermolecular mean square torque. The Wyllie–Larkin treatment also results in an excessive apparent means square torque, although having the correct short-time zero slope. These features are reflected in the fact that the theoretical curves are too broad in the frequency domain, i.e., do not correctly predict the sharp high frequency fall off in α(ω). This has the further consequence that the model rotational velocity correlation functions do not show the pronounced short time oscillations of the experimental curves. Qualitative improvements are suggested.
TL;DR: In this paper, the Fourier transform of the product of the radial distribution function g(r) and the pair potential phi (r) of the reference liquid is used to calculate the form of Scc(q) for a liquid mixture of rare gas atoms and some liquid metal alloys.
Abstract: In the conformal solutions model for liquid mixtures the deviation of the concentration-concentration correlation function Scc(q) from its value c(1-c) in an ideal solution is given by the Fourier transform of the product of the radial distribution function g(r) and the pair potential phi (r) of the reference liquid. The form of Scc(q) is calculated for choices of the reference liquid appropriate to: (a) a liquid mixture of rare gas atoms and (b) some liquid metal alloys. As yet there is no suitable experimental data on an alloy for which the conformal solution model is known to be applicable, but it is shown that the data of Ruppersberg (1973) on Li-Pb alloys can be usefully interpreted in the light of the results presented. Some comments are also made on Cu6Sn5.
TL;DR: For the Heisenberg model, it was shown in this paper that the correlation functions are analytic inh andT if Re(h)≠0 andT is positive. But this is not the case for the non-Heisenberg models.
Abstract: It is shown for the Heisenberg model that the correlation functions are analytic inh andT if Re(h)≠0 andT is positive.
TL;DR: The triplet correlation function for liquid rubidium is determined by molecular dynamics as mentioned in this paper, based on a realistic pair potential for a density rho = 0.0107 AA-3 and a temperature T=319K.
Abstract: The triplet correlation function for liquid rubidium is determined by molecular dynamics. Calculations were based on a realistic pair potential for a density rho =0.0107 AA-3 and a temperature T=319K. The results are compared with various models.
01 Apr 1975
TL;DR: In this paper, the first and second order probability densities of the scattered field from chaff clouds are derived starting from first principles, and the utility of second order statistics in studying the effects of chaff clutter fluctuations on advanced radars such as moving target indicator is demonstrated.
Abstract: : Starting from first principles, the first and second order probability densities of the scattered field from chaff clouds are derived. Auto-correlation functions and power spectra of the received voltage, radar cross section and phase are obtained. All the mathematical derivations are explained in full detail. For the simple case of a spherically uniform distribution of relative speed of the dipoles, it is shown that an integral relation exists between the speed distribution function and the intensity auto- correlation function. The utility of second order statistics in studying the effects of chaff clutter fluctuations on advanced radars such as moving target indicator is demonstrated. Finally, numerical results are included both from an actual experiment and calculations based on assumed dipole velocities. Although it was not possible to compare the two unrelated events, there were definite trends of similarity in the data.
TL;DR: The broadening of a helix–coil transition due to base pair heterogeneity is calculated on the basis of a cumulant perturbation expansion in the quasi‐grand ensemble, and this “renormalization” seems to stabilize the perturbations expansion.
Abstract: The broadening of a helix–coil transition due to base pair heterogeneity is calculated on the basis of a cumulant perturbation expansion in the quasi-grand ensemble. In this ensemble the fictitious, homogeneous chain, to which the perturbation is referred, automatically decreases its correlation length as the heterogeneity increases. This “renormalization” seems to stabilize the perturbation expansion, in view of the good agreement between the present results and the exact theory of a heterogeneous polypeptide helix–coil transition. For the DNA model in which ring entropy is included, the transitions is found to be extremely narrow for an infinite random chain with conventional parameters. A tentative reconciliation of this result with contradictory calculations of some other workers is offered on the basis of end effects, coarse graining, or approximation to the ring entropy. An application of the new method to DNA with a non-random base pair distribution requires evaluation of the correlation function between molecular states (helix or coil), at different sites of the reference chain. The evaluation is reduced to quadrature, but numerical calculations have been made only for the random chain.
TL;DR: In this paper, the spin dependent pair distribution function and static spin susceptibility are numerically calculated from the calculated magnetic structure factor, and the results for the pair distribution functions for antiparallel spins are found to be positive for the entire range of real metallic density of electrons.
Abstract: The approximation method developed by Lobo et al. for classical distribution function is applied to the spin dependent Wigner distribution function in a quantum system. A quantum mechanical expression for generalized spin susceptibility of an interacting electron gas is derived. Self-consistent numerical calculations of magnetic structure factor and generalized spin susceptibility are carried out for an electron gas at metallic density of electrons. Spin dependent pair distribution function and static spin susceptibility are numerically calculated from the calculated magnetic structure factor. The results for the pair distribution function for antiparallel spins are found to be positive for the entire range of real metallic density of electrons.
TL;DR: Bethe's method of equilibrium statistical mechanics is generalized to treat Glauber's time-dependent Ising models in two and three dimensions by incorporating specific short-range correlations into the distribution function as mentioned in this paper.
Abstract: Bethe's method of equilibrium statistical mechanics is generalized to treat Glauber's time-dependent Ising models in two and three dimensions. This constitutes the first-order approximation which improves the mean-field theory (the zeroth-order approximation) by incorporating specific short-range correlations into the distribution function. We first calculate the single-spin expectation value. Although the dynamic equation may need to be solved by using a numerical method, its approximate solution in the pretransitional region can be studied in detail. We then calculate the two-spin correlation function. This requires yet another extension of Bethe's method. Again approximate solutions can be obtained in the pretransitional region. Our results for the static pair correlation functions are equivalent to that of Elliott and Marshall, but our derivation is simpler. We also calculate the time-dependent correlation function for the magnetization, and reproduce the fluctuation-dissipation relation.
01 Feb 1975
TL;DR: In this article, a new statistical model is proposed to predict the correlation function of unstructured black and white digital facsimiles, which is based on a first-order Markov process representation for pictures.
Abstract: A new statistical model is proposed to predict the correlation function of unstructured black and white digital facsimiles. The derivation of the model could be extended to structured printed materials and is based on a first-order Markov process representation for pictures.