Showing papers on "Correlation function (statistical mechanics) published in 1978"
TL;DR: In this paper, the authors present a phenomenological model of intermittency called the P-model and related to the Novikov-Stewart (1964) model, which is dynamical in the sense that they work entirely with inertial-range quantities such as velocity amplitudes, eddy turnover times and energy transfer.
Abstract: We present a phenomenological model of intermittency called the P-model and related to the Novikov-Stewart (1964) model. The key assumption is that in scales N &2-” only a fraction /3n of the total space has an appreciable excitation. The model, the idea of which owes much to Kraichnan (1972, 1974)’ is dynamical in the sense that we work entirely with inertial-range quantities such as velocity amplitudes, eddy turnover times and energy transfer. This gives more physical insight than the traditional approach based on probabilistic models of the dissipation. The P-model leads in an elementary way to the concept of the self-similarity dimension D, a special case of Mandelbrot’s (1974, 1976) ‘fractal dimension’. For threedimensional turbulence, the correction B to the Q exponent of the energy spectrum is equal to +( 3 - D) and is related to the exponent p of the dissipation correlation function by B = Qp (0.17 for the currently accepted value). This is a borderline case of the Mandelbrot inequality B < Qp. It is shown in the appendix that this inequality may be derived from the Navier-Stokes equation under the strong, but plausible, assumption that the inertial-range scaling laws for second- and fourth-order moments have the same viscous cut-off. The predictions of the P-model for the spectrum and for higher-order statistics are in agreement with recent conjectures based on analogies with critical phenomena (Nelkin 1975) but generally diasgree with the 1962 Kolmogorov lognormal model. However, the sixth-order structure function (8v6(Z)) and the dissipation correlation function (e(r) e(r + 1)) are related by
911 citations
TL;DR: In this article, the authors derived closed equations for the velocity correlation tensor and for the mean-squared displacement of a particle suspended in a stationary homogeneous turbulent flow, with an arbitrary linear law of fluid-particle interaction.
Abstract: The closed equations for the velocity correlation tensor and for the mean-squared displacement of a particle suspended in a stationary homogeneous turbulent flow, with an arbitrary linear law of fluid-particle interaction, are obtained using two assumptions suggested previously for the problem of turbulent self-diffusion: the ‘independence approximation’ and the Gaussian property of the functional distribution of particle velocities. The numerical solution of the derived equations is given for an isotropic system with a model turbulence spectrum. The following characteristics of the particle motion are obtained: ( a ) the mean kinetic energy, ( b ) diffusivity, ( c ) rate of energy dissipation, ( d ) velocity correlation function, and ( e ) the correlation function of the relative fluid-particle velocity. The impact of various spectral modes on the characteristics of the particle motion is discussed.
128 citations
TL;DR: In this paper, the authors derived the two-point correlation function for a distinguishable-particle hopping on a one-dimensional linear chain with all sites equivalent, and the solution is obtained from a multiple-scattering equation derived from first principles.
Abstract: We derive the two-point correlation function for a distinguishable-particle hopping on a one-dimensional linear chain with all sites equivalent. The solution is obtained from a multiple-scattering equation derived from first principles. The results are similar to results obtained from computer experiments and phenomenological arguments.
116 citations
82 citations
TL;DR: The velocity measuring correlation sonar as discussed by the authors employs a planar array of receiving transducers spaced in the directions along which velocity components parallel to the plane of the array are to be measured, and includes means for transmitting a series of two or more identical pulses which are separated by a time interval selected in accordance with transducer separation and the estimated velocity components.
Abstract: The velocity measuring correlation sonar disclosed employs a planar array of receiving transducers spaced in the directions along which velocity components parallel to the plane of the array are to be measured, and includes means for transmitting a series of two or more identical pulses which are separated by a time interval selected in accordance with transducer separation and the estimated velocity components so as to place the expected point of maximum correlation of the echo return from one pulse with that from a following pulse within the boundaries of a set of spatial sample points representing relative spacings between pairs of receiving transducers. Correlation measurements are made corresponding to these relative spacings of the receiving transducers, with each such measurement being treated as a sample of a space-time correlation function of predetermined shape. The location of the peak of this function in each of the directions of interest is estimated by curve fitting techniques, and yields the velocity vector in that direction scaled by the inter-pulse time interval. The velocity component normal to the plane of the array may be derived by estimating the location of the correlation peak as a function of time and/or phase, using similar curve fitting techniques.
76 citations
TL;DR: In this article, the authors derived an expression for the velocity autocorrelation function in a simple liquid and applied it to a model of liquid rubidium in order to see how the collective modes in the system influence the individual particle motion.
Abstract: By constructing a microscopic form for the velocity field the author's have derived an expression for the velocity autocorrelation function in a simple liquid. It represents an analysis of the correlation function in terms of the longitudinal and transverse momentum current densities and correctly describes both the long and short time behaviour. The authors have applied the result to a model of liquid rubidium in order to see how the collective modes in the system influence the individual particle motion. It is found that the longitudinal current component is responsible for the oscillatory behaviour of the velocity autocorrelation function; the principle peak in the associated frequency spectrum is generated by the coupling of the particle velocity to the transverse current; the coupling to the longitudinal current produces the peak or shoulder at higher frequencies which is observed in computer experiments; and the diffusion coefficient is determined almost entirely by the transverse current component.
67 citations
TL;DR: In this paper, the influence of the nonperturbative structure of the quantum-chromodynamic vacuum on the short distance behavior of hadronic currents is discussed and the dilute-gas approximation is systematically used.
Abstract: The influence of the nonperturbative structure of the quantum-chromodynamic vacuum on the short-distance behavior of hadronic currents is discussed. The dilute-gas approximation is systematically used. We show how to calculate in this approximation arbitrary Green's functions and that the effects of tunneling (instantons) are summarized by an appropriate effective Lagrangian. These methods are applied to the two-point current correlation function, which is explicitly calculated in the dilute-gas approximation. We estimate the numerical size of the instanton effects for e/sup +/e/sup -/ annihilation and find them to be strongly momentum dependent and large. The qualitative features of these corrections suggest an explanation of precocious scaling.
60 citations
TL;DR: In this article, a review of the results is presented with special emphasis on the differentation between short range order, as e.g. revealed from the pair distribution function, and long range correlations as proposed for the so-called bundle models.
Abstract: In order to describe the structure of polymers in the molten or vitreous state four different kinds od static correlation functions should be studied experimentally: (i) the pair distribution function of the monomer units, (ii) the distance distribution function of the repeating units of a single chain, (iii) the orientation correlation function of the bond vectors, and (iv) the density fluctuation correlation function of the bulk sample. The appropriate methods include: X-ray wide-angle scattering and electron diffraction, small-angle neutron diffraction of deuterium tagged molecules, depolarized light scattering and magnetic birefringence, small-angle X-ray and polarized light scattering. A review of the results is presented with special emphasis on the differentation between short range order, as e.g. revealed from the pair distribution function, and long range correlations as proposed for the so-called “bundle” models. It is shown that there exists no experimental evidence for such a structure besides the “grains” in electron micrographs.
54 citations
TL;DR: In this paper, the effect of segmental motion on the density-density correlation function of a viscous polymer liquid has been analyzed using a generalized relaxation equation developed by Zwanzig and Mori.
Abstract: The effect of segmental motion on the density–density correlation function of a viscous polymer liquid has been analyzed using a generalized relaxation equation developed by Zwanzig and Mori. It is shown that for polymer liquids of high viscosity, Brillouin scattering is closely associated with the structural relaxation associated with the motion chain segments. A single relaxation time theory is shown to yield good agreement with the experimental results on polypropylene glycol. The torsional motion involving a small number of monomer units is shown to be responsible for the dispersion and attenuation of the hypersonic wave. The fact that the Brillouin scattering spectrum of a polymer liquid is insensitive to the change of molecular weight is discussed. We have shown that temporal modulation of the spatial second moment of the intermolecular or intersegmental interaction energy is responsible for the relaxation process involved in Brillouin scattering.
50 citations
TL;DR: In this article, an analysis of the results of molecular dynamics investigations of thermodynamic and transport properties of simple liquids and dense gases is presented. Butler et al. make use of a simple particle in a dense medium of disordered heavy scatterers to study the relationship between the behavior of the temporal velocity correlation function of a particle and its spatial velocity correlation functions.
Abstract: An analysis is made of the results obtained in investigations of dense media by the molecular dynamics method. This method is based on mathematical simulation of the motion of a sufficiently large number of particles with a given interparticle interaction law. The attention is concentrated on new physical ideas about the nature of simple liquids and dense gases which have made their first appearance, have been derived, or confirmed in studies carried out by the molecular dynamics method. The principal laws of particle motion and their influence on the form of the temporal velocity correlation function are considered. Spatial and temporal correlations appearing in dense systems are studied and their role in the propagation of longitudinal and shear waves is discussed. An analysis is made of the results of molecular dynamics investigations of thermodynamic and transport properties of simple liquids and dense gases. The dynamics of a light classical particle in a dense medium of disordered heavy scatterers is discussed. Consideration is given to the close relationship between the behavior of the temporal velocity correlation function of a particle, its spatial velocity correlation function, and "percolation" in a random field of heavy scatterers.
48 citations
TL;DR: In this article, the exact analytical forms of the low-temperature thermodynamic quantities and correlation functions are obtained for classical one-dimensional Heisenberg ferromagnets with single-site anisotropy including both the systems with an easy axis and with a easy plane.
Abstract: The exact analytical forms of the low-temperature thermodynamic quantities and correlation functions are obtained for classical one-dimensional Heisenberg ferromagnets with single-site anisotropy including both the systems with an easy axis and with an easy plane. The problem has been managed on the basis of the functional integral method. The calculated results have a few leading terms in series-expansion with respect to the reduced temperature, exhibiting the characteristic points borne out in the recent numerical studies. In particular, the longitudinal correlation function and susceptibility of the system with an easy axis show Ising-like behaviours and the transverse counterparts of the system with an easy plane show isotropic XY- or Heisenberg-like behaviours. Physical implications of the calculated results are also discussed.
TL;DR: In this paper, the two-point correlation function for complex spectra described by the Gaussian Orthogonal Ensemble (GOE) is calculated, and its essential simplicity displayed, by an elementary procedure which derives from orthogonal invariance and the dominance of intrinsic binary correlations.
TL;DR: In this article, the effects of deviations from an ideal lamellar structure (infinite-size clusters of parallel layers of alternating electron densities) on the small-angle scattering curve are treated with the aid of the correlation function.
Abstract: The effects of deviations from an ideal lamellar structure (infinite-size clusters of parallel layers of alternating electron densities) on the small-angle scattering curve are treated with the aid of the correlation function. If surrounded by a matrix of the average electron density, reduction of the size of the clusters in the direction of the layer normals leads to a simple modification of the one-dimensional correlation function. Distortions giving rise to structures containing concentric layers have little effect on this function, whereas corrugation of the surfaces causes minor modifications. Second-order defects are shown to reduce the three-dimensional correlation function of the ideal structure γ°(r) according to γ(r) = γ°(r) exp (−2r/d), where d is the `distortion length'. This is the average length of the vectors for which the number of intersections with lamellar interfaces has changed by ±1 as a consequence of the distortions. Calculated diffraction curves show that the effects of reducing the cluster size and of increasing the width β of the lamellar thickness distribution function are very similar. However, changes in d and β affect the scattering curves in a different way, which, other conditions being favourable, may enable these parameters to be determined from observed scattering curves.
TL;DR: In this article, an exact relation between the two-point correlation function of the classical XY model and the free energy of a step associated with two screw dislocations on a crystal surface described by a solid-on-solid model was proven.
TL;DR: In this article, the diffusive motion of particles in a one-dimensional periodic potential is treated using the Fokker-Planck equation method, and a general method for calculating the correlation functions is presented for the velocity correlation functions.
Abstract: The diffusive motion of particles in a one-dimensional periodic potential is treated using the Fokker-Planck equation method First, a concise form of the Fokker-Planck equation and of the correlation function for this problem is set up By expanding the distribution function into suitable eigenfunctions, a general method for calculating the correlation functions is then given Finally, explicit calculations are presented for the velocity correlation functions, some of these are compared with those which were obtained by continued fraction methods
TL;DR: A simple analytical model is proposed to account for the contribution of the twiddle motion to the correlation function, and a satisfactory agreement between the theory and the measured angular dependence of the line shape is obtained.
TL;DR: In this article, a system of exact algebraic relations for spin correlation functions on any so-called "boundary set" of an Ising model on an arbitrary planar graph is derived.
Abstract: A system of exact algebraic relations is derived for the spin correlation functions on any so-called “boundary set” of an Ising model on an arbitrary planar graph. One way of expressing these relations is to say that every higher-order correlation function is equal to a Pfaffian of the pair correlation function on the same set of boundary spins.
TL;DR: In this paper, the fourth-order equation is used to provide expressions for the higher-order site-independent correlation functions M2n(n=3,4,5,...) in terms of the two and four-spin correlation function M2 and M4, at the critical point.
Abstract: In a new theory for the spin one-half Ising ferromagnet, the fourth-order equation is used to provide expressions for the higher-order site-independent correlation functions M2n(n=3,4,5,...) in terms of the two- and four-spin correlation functions M2 and M4, at the critical point. As an example, the expression for M6 is M42/M2-9M2M4-6M23. These expressions (for up to M60) are applied to the calculation of Tc for the SC, BCC, and FCC lattices. The results converge to, at most, 0.755, 0.799, and 0.819 respectively, which are within 2/3% of the accepted results 0.752, 0.794, and 0.816 of Sykes et al. (1972).
TL;DR: It is demonstrated that it is essential to consider the shape as well as the internal structure of bacteria in order to account fully for both the scattered intensity and the field self-correlation function as reduced from the intensity correlation function.
Abstract: An extensive study has been made of the angular dependence of the intensity as well as the quasi-elastic intensity correlation function of light scattered from a dilute suspension of E. Coli K12 bacteria in solution using the photon correlation technique. We have demonstrated that it is essential to consider the shape as well as the internal structure of bacteria in order to account fully for both the scattered intensity and the field self-correlation function as reduced from the intensity correlation function. The simple structure model previously proposed accounts well for both the intensity and diffusion data with a reasonable set of parameters. For the case of motile bacteria the observed field correlation function at small scattering angles is well represented by assuming a mixture of diffusive and straight line swimming motions of the bacteria. At larger scattering angles contributions of rotational motions become significant.
TL;DR: In this article, the broken symmetry concept of Goldstone is applied to an inhomogeneous fluid and specific calculations are performed for the liquid-vapor interface of a simple fluid, where the collective variable associated with the breaking of spatial symmetry is the normal displacement of the interface from its equilibrium position.
Abstract: The broken symmetry concept of Goldstone is applied to an inhomogeneous fluid. Specific calculations are performed for the liquid–vapor interface of a simple fluid. The collective variable associated with the breaking of spatial symmetry is the normal displacement of the interface from its equilibrium position. The static correlation function of this variable exhibits long‐range order. Associated with this symmetry breaking variable are propagating modes. The dispersion relation for these surface modes is found to be identical with the hydrodynamic result for capillary waves.
TL;DR: In this paper, the Buckingham-Darey flux law, an empirical equation that relates the volumetric flux density of water in an unsaturated soil to the gradient of the total potential, is derived from first principles by using the methods of statistical mechanics.
Abstract: The Buckingham-Darey flux law, an empirical equation that relates the volumetric flux density of water in an unsaturated soil to the gradient of the total potential, is derived from first principles by using the methods of statistical mechanics. The derivation given, a direct application of well-known techniques in nonequilibrium statistical mechanics, proceeds through a detailed molecular description of two laboratory experiments for measuring the hydraulic conductivity tensor of a homogeneous unsaturated soil. In the first experiment the steady flow of water is induced by a gradient in the matric potential, while in the second, flow is induced by a gradient in the gravitational potential. In both cases the appropriate form of the Buckingham-Darcy law is derived on the basis of a linear response approximation, and an expression for the hydraulic conductivity is given in terms of a time integral of the correlation function for the velocities of the water molecules in the soil. The problem of calculating the hydraulic conductivity of a soil thereby is reduced to quadrature and to the task of developing a molecular model of the velocity correlations among the water molecules. A recent successful model of this type is discussed briefly.
TL;DR: In this paper, the correlation function of temperature fluctuations around homogeneous stationary states of the ballast resistor is evaluated by extending the theory of thermal fluctuations around equilibrium states to non-equilibrium situations.
Abstract: The correlation function of temperature fluctuations around homogeneous stationary states of the ballast resistor is evaluated by extending the theory of thermal fluctuations around equilibrium states to non-equilibrium situations. It is found that equal-time temperature fluctuations become correlated over large distances if one approaches the critical point. At the “critical point” of the ballast resistor the correlation length and the amplitude of the temperature fluctuations diverge. Furthermore the theory predicts a critical slowing down for the time-dependent temperature fluctuation correlation function. The electric noise of the system is also analyzed.
TL;DR: In this paper, a hydrodynamic theory for the collective excitations in superionic conductors at low frequencies and long wavelengths is presented, and explicit expressions for dispersion and damping of the modes and for the density correlation function are worked out for high-symmetry directions and numerically evaluated for α-AgI.
Abstract: Using the Mori method we present a rigorous hydrodynamic theory for the collective excitations in superionic conductors at low frequencies and long wavelengths. By means of static and dynamic linear response theory all microscopic quantities of the theory are expressed as derivatives of the free energy or as transport coefficients. Explicit expressions for the dispersion and the damping of the modes and for the density correlation function are worked out for high-symmetry directions and numerically evaluated forα-AgI. In particular we show that the two non-propagating modes can give rise to a small central component (width~k2, constant intensity) and a broad central component (constant width fork→0, intensity ~k2) in scattering spectra. Extrapolating our theory to larger wave vectors we offer a new explanation for the broad central component found recently in neutron scattering spectra ofα-AgI.
TL;DR: In this article, one-dimensional discrete chaotic processes are studied from a statistical-dynamical point of view, and a set of equations which describe the behavior of the time correlation functions is derived with the aid of Mori's projector formalism.
Abstract: Abstract One-dimensional discrete chaotic processes are studied from a statistical-dynamical point of view. A set of equations which describe the behavior of the time correlation functions is derived with the aid of Mori’s projector formalism. A condition under which a process is Markoffian is obtained, and an approximate method is developed for a non-Markoffian process. As an illustration, a time correlation function for a simple system is calculated and the comparison with results of computer simulations is made. The relation between the instability of a trajectory and the characteristic time of chaotic motions is also discussed.
TL;DR: In this paper, the exact solution of the linear Boltzmann equation for one-dimensional hard rods is presented, in particular, the velocity correlation function and the Van Hove self-correlation function.
Abstract: We present the exact solution of the linear Boltzmann equation for one-dimensional hard rods. In particular, we compute the velocity correlation function and the Van Hove self-correlation function.
TL;DR: In this article, the time-displaced conditional distribution functions for an infinite, one-dimensional mixture of equal-mass hard rods of different diameters were calculated and the kinetic equation that describes the time dependence of the one-particle total distribution function was found to be non-Markovian.
Abstract: Time-displaced conditional distribution functions are calculated for an infinite, one-dimensional mixture of equal-mass hard rods of different diameters. The kinetic equation that describes the time dependence of the one-particle total distribution function is found to be non-Markovian, in contrast with the situation in systems of identical rods. The correlation function does not contain any isolated damped oscillation, except for systems of equal-diameter rods with discrete velocities. Thus, we generalize the one-component results of Lebowitz, Percus, and Sykes, removing some nontypical features of that system.
TL;DR: In this article, a projection operator method is presented, which provides the most efficient way for calculating the stationary behavior of nonlinear Brownian motion, and a continued fraction expansion of the Fourier-Laplace transform of the displacement correlation function or the spectral density is used.
Abstract: A projection operator method is presented, which provides the most efficient way for calculating the stationary behavior of nonlinear Brownian motion. A continued-fraction expansion of the Fourier-Laplace transform of the displacement correlation function or the spectral density is used. This method utilizes a successive optimization procedure on the nonlinear terms and includes the method of “statistical linearization” as the lowest order approximation. A systematic way to calculate the continued fraction numerically up to sufficient order for convergence is developed, which enables us to obtain the spectral density of a system previously uncomputable.
TL;DR: In this paper, a representation of the pair correlation function for the rectangular Ising model in zero magnetic field is derived using a new spinor technique; this enables the scaling limit to be established, as well as several analytical properties of the scaling functions.
Abstract: A representation of the pair correlation function for the rectangular Ising model in zero magnetic field is derived using a new spinor technique; this enables the scaling limit to be established, as well as several analytical properties of the scaling functions.