Correlation Function Of Density Fluctuations Research Articles

Overview of new MAST physics in anticipation of first results from MAST Upgrade

The mega amp spherical tokamak (MAST) was a low aspect ratio device (R/a = 0.85/0.65 ~ 1.3) with similar poloidal cross-section to other medium-size tokamaks. The physics programme concentrates on addressing key physics issues for the operation of ITER, design of DEMO and future spherical tokamaks by utilising high resolution diagnostic measurements closely coupled with theory and modelling to significantly advance our understanding. An empirical scaling of the energy confinement time that favours higher power, lower collisionality devices is consistent with gyrokinetic modelling of electron scale turbulence. Measurements of ion scale turbulence with beam emission spectroscopy and gyrokinetic modelling in up-down symmetric plasmas find that the symmetry of the turbulence is broken by flow shear. Near the non-linear stability threshold, flow shear tilts the density fluctuation correlation function and skews the fluctuation amplitude distribution. Results from fast particle physics studies include the observation that sawteeth are found to redistribute passing and trapped fast particles injected from neutral beam injectors in equal measure, suggesting that resonances between the m = 1 perturbation and the fast ion orbits may be playing a dominant role in the fast ion transport. Measured D–D fusion products from a neutron camera and a charged fusion product detector are 40% lower than predictions from TRANSP/NUBEAM, highlighting possible deficiencies in the guiding centre approximation. Modelling of fast ion losses in the presence of resonant magnetic perturbations (RMPs) can reproduce trends observed in experiments when the plasma response and charge-exchange losses are accounted for. Measurements with a neutral particle analyser during merging-compression start-up indicate the acceleration of ions and electrons. Transport at the plasma edge has been improved through reciprocating probe measurements that have characterised a geodesic acoustic mode at the edge of an ohmic L-mode plasma and particle-in-cell modelling has improved the interpretation of plasma potential estimates from ball-pen probes. The application of RMPs leads to a reduction in particle confinement in L-mode and H-mode and an increase in the core ionization source. The ejection of secondary filaments following type-I ELMs correlates with interactions with surfaces near the X-point. Simulations of the interaction between pairs of filaments in the scrape-off layer suggest this results in modest changes to their velocity, and in most cases can be treated as moving independently. A stochastic model of scrape-off layer profile formation based on the superposition of non-interacting filaments is in good agreement with measured time-average profiles. Transport in the divertor has been improved through fast camera imaging, indicating the presence of a quiescent region devoid of filament near the X-point, extending from the separatrix to ψn ~ 1.02. Simulations of turbulent transport in the divertor show that the angle between the divertor leg on the curvature vector strongly influences transport into the private flux region via the interchange mechanism. Coherence imaging measurements show counter-streaming flows of impurities due to gas puffing increasing the pressure on field lines where the gas is ionised. MAST Upgrade is based on the original MAST device, with substantially improved capabilities to operate with a Super-X divertor to test extended divertor leg concepts. SOLPS-ITER modelling predicts the detachment threshold will be reduced by more than a factor of 2, in terms of upstream density, in the Super-X compared with a conventional configuration and that the radiation front movement is passively stabilised before it reaches the X-point. 1D fluid modelling reveals the key role of momentum and power loss mechanisms in governing detachment onset and evolution. Analytic modelling indicates that long legs placed at large major radius, or equivalently low at the target compared with the X-point are more amenable to external control. With MAST Upgrade experiments expected in 2019, a thorough characterisation of the sources of the intrinsic error field has been carried out and a mitigation strategy developed.

Assessing degrees of entanglement of phonon states in atomic Bose gases through the measurement of commuting observables

We show that measuring commuting observables can be sufficient to assess that a bipartite state is entangled according to either nonseparability or the stronger criterion of 'steerability'. Indeed, the measurement of a single observable might reveal the strength of the interferences between the two subsystems, as if an interferometer were used. For definiteness we focus on the two-point correlation function of density fluctuations obtained by in situ measurements in homogeneous one-dimensional cold atomic Bose gases. We then compare this situation to that found in transonic stationary flows mimicking a black hole geometry where correlated phonon pairs are emitted on either side of the sonic horizon by the analogue Hawking effect. We briefly apply our considerations to two recent experiments.

Theoretical study of stimulated and spontaneous Hawking effects from an acoustic black hole in a hydrodynamically flowing fluid of light

We propose an experiment to detect and characterize the analog Hawking radiation in an analog model of gravity consisting of a flowing exciton-polariton condensate. Under a suitably designed coherent pump configuration, the condensate features an acoustic event horizon for sound waves that at the semiclassical level is equivalent to an astrophysical black hole horizon. We show that a continuous-wave pump-and-probe spectroscopy experiment allows to measure the analog Hawking temperature from the dependence of the stimulated Hawking effect on the pump-probe detuning. We anticipate the appearance of an emergent resonant cavity for sound waves between the pump beam and the horizon, which results in marked oscillations on top of an overall exponential frequency dependence. We finally analyze the spatial correlation function of density fluctuations and identify the hallmark features of the correlated pairs of Bogoliubov excitations created by the spontaneous Hawking process, as well as novel signatures characterizing the emergent cavity.

Mean flows and blob velocities in scrape-off layer (SOLT) simulations of an L-mode discharge on Alcator C-Mod

Two-dimensional scrape-off layer turbulence (SOLT) code simulations are compared with an L-mode discharge on the Alcator C-Mod tokamak [Greenwald et al., Phys. Plasmas 21, 110501 (2014)]. Density and temperature profiles for the simulations were obtained by smoothly fitting Thomson scattering and mirror Langmuir probe (MLP) data from the shot. Simulations differing in turbulence intensity were obtained by varying a dissipation parameter. Mean flow profiles and density fluctuation amplitudes are consistent with those measured by MLP in the experiment and with a Fourier space diagnostic designed to measure poloidal phase velocity. Blob velocities in the simulations were determined from the correlation function for density fluctuations, as in the analysis of gas-puff-imaging (GPI) blobs in the experiment. In the simulations, it was found that larger blobs moved poloidally with the E × B flow velocity, vE, in the near-SOL, while smaller fluctuations moved with the group velocity of the dominant linear (interchange) mode, vE + 1/2 vdi, where vdi is the ion diamagnetic drift velocity. Comparisons are made with the measured GPI correlation velocity for the discharge. The saturation mechanisms operative in the simulation of the discharge are also discussed. It is found that neither sheared flow nor pressure gradient modification can be excluded as saturation mechanisms.

Density-density correlations on a gelatin films surface

The spatial correlations on the surface of films obtained from aqueous solutions of gelatin during cooling from 320 to 293 K have been studied using electron microscopy. It has been shown that the total density fluctuation correlation function on the scale R > 4 nm can be represented in the form h(R) ∼ R−nexp(−R/ζ), where the correlation radius ζ coincides with the hydrodynamic radius of a macromolecule. Unfolding of macromolecules in the coil → helix transformation leads to a decrease in the density and the fractal dimension of the physical network of pinnings of macromolecules and to variation in the index n of the power term in the function h(R) from n = 1 to 2, due to the transition from a continual type of disorder to a cellular type in a solid.

Slip boundary conditions in nanofluidics from the molecular theory of solvation

We propose the derivation and calculation of the hydrodynamic slip length from the first principles of statistical mechanics, based on a combination of linear response theory and equilibrium molecular theory of solvation. The slip length obtained is independent of the type of flow and is related to the fluid organisation near the solid surface, as governed by the solid–liquid and liquid–liquid interactions. In a wide range of shear rates and surface–liquid interactions, the slip length is expressed in terms of the Green–Kubo–Nakano relations as a function of the anisotropic inhomogeneous time correlation function of density fluctuations of the liquid in contact with the surface. The presented treatment generalises the phenomenological definition of the friction coefficient and the slip length to a tensor quantity, which reflects an anisotropic nature of an ordered crystalline surface. We derive generic analytical expressions for the liquid–surface friction coefficient and slip length for an arbitrary surface–liquid interaction potential. We further illustrate it by numerical calculations for the case of a laminar flow of several molecular liquids and water, at ambient conditions in contact with the (100) FCC surface of gold, copper and nickel modelled using all-atom or united-atom models for liquids and the Steele potential for crystalline surfaces. The values obtained for slip length range from few to hundreds of nanometres and are consistent with experimental measurements. We also calculate pressure and temperature dependence on the slip length for water in a wide range of these thermodynamic parameters. The information obtained is intended to be used, in particular, to control or manipulate the flow in electrokinetic processes.

Energy relaxation in dense, strongly coupled two-temperature plasmas

A quantum kinetic approach for the energy relaxation in strongly coupled plasmas with different electron and ion temperatures is presented. Based on the density operator formalism, we derive a balance equation for the energies of electrons and ions connecting kinetic, correlation, and exchange energies with a quite general expression for the electron-ion energy-transfer rate. The latter is given in terms of the correlation function of density fluctuations which allows for a derivation of increasingly realistic approximation schemes including a coupled-mode expression. The equilibration of the contributions of the total energy including the species temperatures in dense hydrogen and beryllium relevant for inertial confinement fusion is investigated as an example.

Monte Carlo simulation of light scattering spectra in atomic gases

The correlation function of density fluctuations in Argon is calculated by a direct simulation Monte Carlo method. The procedure is validated by comparison with analytical results for the continuum and the collisionless regimes. In the transitional regime deviations from the BGK equation results are found and discussed.

Diagrammatic Formulation of the Kinetic Theory of Fluctuations in Equilibrium Classical Fluids. V. The Short Time Approximation for the Memory Function

The correlation function for density fluctuations in a monatomic fluid obeys a formally exact kinetic equation containing a memory function. A previously derived short time approximation (STA) for this memory function is tested by comparing its predictions with the results of molecular dynamic simulations of a dense Lennard-Jones fluid at a variety of temperatures. This approximation takes into account the contribution to the correlation function of uncorrelated repulsive binary collisions. The qualitative changes of predicted correlation functions with temperature and wave vector are generally correct. The major exception to this is the transverse current correlation function for small wave vector. The quantitative accuracy is better at short times than long times and better at high temperatures than low temperatures. The major failing of the STA is its underestimation of the amplitudes of the negative dips in the current autocorrelation functions and of the temperature dependence of the amplitudes of the dips. Despite its deficiencies in predicting the time dependence of current correlation functions, the STA gives accurate results for the self-diffusion coefficient and the shear viscosity coefficient at the highest temperatures studied.

Diagrammatic kinetic theory for a lattice model of a liquid. I. Theory.

We present a diagrammatic formalism for the time correlation functions of density fluctuations for an excluded volume lattice gas on a simple d-dimensional hypercubic lattice. We consider a multicomponent system in which particles of different species can have different transition rates. Our theoretical approach uses a Hilbert space formalism for the time dependent dynamical variables of a stochastic process that satisfies the detailed balance condition. We construct a Liouville matrix consistent with the dynamics of the model to calculate both the equation of motion for multipoint densities in configuration space and the interactions in the diagrammatic theory. A Boley basis of fluctuation vectors for the Hilbert space is used to develop two formally exact diagrammatic series for the time correlation functions. These theoretical techniques are generalizations of methods previously used for spin systems and atomic liquids, and they are generalizable to more complex lattice models of liquids such as a lattice gas with attractive interactions or polymer models. We use our formalism to construct approximate kinetic theories for the van Hove correlation and self-correlation function. The most simple approximation is the mean field approximation, which is exact for the van Hove correlation function of a one component system but an approximation for the self-correlation function. We use our first diagrammatic series to derive a two site multiple scattering approximation that gives a simple analytic expression for the spatial Fourier transform of the self-correlation function. We employ our second diagrammatic series to derive a simple mode coupling type approximation that provides a system of equations that can be solved for the self-correlation function.

A Multicomponent Fluid System in the Critical Region in the Presence of Spatial Confinement

A spatially confined multicomponent fluid system is treated, which is located in the immediate neighborhood of the critical state. Pair correlation functions of density fluctuations are found for this system, as well as the shifts of the critical parameters due to the finiteness of the system dimensions. It is demonstrated that the behavior of a multicomponent mixture in the vicinity of the critical state is largely isomorphic to the behavior of pure fluid.

Dynamics in colloidal liquids near a crossing of glass- and gel-transition lines.

Within the mode-coupling theory for ideal glass transitions, the mean-squared displacement and the correlation function for density fluctuations are evaluated for a colloidal liquid of particles interacting with a square-well potential for states near the crossing of the line for transitions to a gel with the line for transitions to a glass. It is demonstrated how the dynamics is ruled by the interplay of the mechanisms of arrest due to hard-core repulsion and due to attraction-induced bond formation as well as by a nearby higher-order glass-transition singularity. Application of the universal relaxation laws for the slow dynamics near glass-transition singularities explains the qualitative features of the calculated time dependence of the mean-squared displacement, which are in accord with the findings obtained in molecular-dynamics simulation studies by Zaccarelli et al. [Phys. Rev. E 66, 041402 (2002)]. Correlation functions found by photon-correlation spectroscopy in a micellar system by Mallamace et al. [Phys. Rev. Lett. 84, 5431 (2000)] can be interpreted qualitatively as a crossover from gel to glass dynamics.

The Critical Phase Changes of a Forward-scattered Light in a Nonpolar Binary Liquid Mixture

In a previous paper 1 the author and his coworkers have obtained analytic results for the phase changes of the forward-scattered light in a nonpolar chiral fluid, when the incident light is completely linearly polarized above (or below) the horizontal at 45 o , using the formalism given by Barron 2 and the theory for the dielectric tensor developed by Lee and Kim. 3 The method is so effective that it enables us to calculate phase changes of the forward-scattered light in the critical region with the aid of Ornstein-Zernike theory of the correlation function of density fluctuations. Then, the method was extended to a binary liquid mixture composed of a chiral solute and an achiral solvent. 4 The mixture is more suitable for experimental verification than the pure fluid. The basic difference with the latter is that there is one more extra variable in the mixture, i.e., the concentration fluctuations. The results for the azimuth and ellipticity changes were extended in the critical region that the Fisher theory holds. 5 The purpose of the present paper is to discuss the phase changes in the critical region of a nonpolar binary liquid mixture, using the Fisher form for the correlation function of concentration fluctuations. 6,7 The explicit results are given without the detailed procedures, which were derived in the previous papers. 1,4

The critical phase changes of a forward-scattered light in an isotropic chiral fluid

The phase changes of the forward-scattered light is explicitly obtained in an isotropic critical chiral fluid by using the Fisher correlation function of density fluctuations, when the incident light is completely linearly polarized above (or below) the horizontal at 45°. A new experiment method is suggested to measure the critical exponent for correlation function by using the result of azimuth changes of the scattered light.

The three-point function in large-scale structure – I. The weakly non-linear regime in N-body simulations

ABSTRA C T This paper presents a comparison of the predictions for the two- and three-point correlation functions of density fluctuations, j and z, in gravitational perturbation theory (PT) against large cold dark matter (CDM) simulations. This comparison is made possible for the first time on large weakly non-linear scales (. 10 h 21 Mpc) thanks to the development of a new algorithm for estimating correlation functions for millions of points in only a few minutes. Previous studies in the literature comparing the PT predictions of the three-point statistics with simulations have focused mostly on Fourier space, angular space or smoothed fields. Results in configuration space, such as those presented here, were limited to small scales where leading-order PT gives a poor approximation. Here we also propose and apply a method for separating the first-order and subsequent contributions to PT by combining different output times from the evolved simulations. We find that in all cases there is a regime where simulations do reproduce the leading-order (tree-level) predictions of PT for the reduced three-point function Q3 , z/j 2 . For steeply decreasing correlations (such as the standard CDM model) deviations from the tree-level results are important even at relatively large scales, . 20 Mpc h 21 . On larger scales j goes to zero and the results are dominated by sampling errors. In more realistic models (such as the LCDM cosmology) deviations from the leading-order PT become important at smaller scales r . 10 Mpc h 21 , although this depends on the particular three-point configuration. We characterize the range of validity of this agreement and show the behaviour of the next-order (one-loop) corrections.

The origin of the Vogel–Fulcher law near the liquid–glass transition

Taking into account the scattering processes due to random eigenfrequencies and random hopping matrices, we calculate the correlation functions of density fluctuations associated with the particle--hole pairs in the intraband and interband. The correlation functions for the intraband yield sound velocity (proportional to the mean atomic velocity) and diffusion (equal to the relaxation time of atoms multiplied by square of the sound velocity). Those for the interband yield phonons and viscosity (equal to the Maxwell relaxation time multiplied by square of the phonon velocity). The relaxation times and the transport coefficients are governed by the Vogel–Fulcher law through the hopping matrices. Phonons exist in both phases, but sound disappears below the freezing point. The Stokes’ law between diffusion and viscosity holds at higher temperatures, but breaks at lower temperatures.

On the Vogel-Fulcher law near the liquid-glass transition

Abstract The correlation functions of density fluctuations associated with the particle-hole pairs in the intraband and interband are calculated. The correlation functions for the intraband yield the sound velocity (proportional to the mean atomic velocity) and diffusion (equal to the relaxation time of atoms multiplied by the square of the sound velocity). Those for the interband yield phonons and viscosity (equal to the Maxwell relaxation time multiplied by the square of the phonon velocity). Diffusion, viscosity, sound velocity, the mean atomic velocity and relaxation times obey the Vogel-Fulcher law. Phonons exist in both phases, but sound disappears below the freezing point.

A quantum field theory of diffusion near the liquid-glass transition

Randomly distributed atoms are considered in the harmonic potentials with random eigenfrequencies. The hopping of atoms makes energy dispersion curves of an atom. The correlation functions of density fluctuations in the intraband transition yield the freezing point, where the hopping is prohibited. The entropy due to the density fluctuations in the intraband transition yields the hopping matrix which shows the Vogel-Fulcher law. The diffusion is determined by the randomness of the eigenfrequencies and the hopping matrices. The diffusion coefficient is proportional to the first or third power of the configurationally averaged hopping matrix at high or low temperatures, respectively, and obeys the Vogel-Fulcher law. The randomness of eigenfrequencies and hopping matrices also yields the relaxation time of atoms, which obscures the liquid-glass transition.

Inverse problems in radar measurements of stochastic processes in the ionosphere

This paper offers a short introduction to certain radar studies of the ionosphere. Emphasis will be placed on applications of ‘‘incoherent scatter’’ radars relying upon stochastic Thomson scattering to make remarkably detailed measurements of the temperature, density, composition, and other physical parameters in the Earth’s ionosphere. The total backscattered power is proportional to the density, but all the other parameters are derived by detailed inspection of the correlation function of density fluctuations. Measuring the correlation function is quite difficult because of the very small scattering cross section. Furthermore, the ionosphere is ‘‘overspread,’’ which means that density fluctuation decorrelates in significantly less time than it takes a radio pulse to transit the region of interest. Simply stated, this means that one cannot measure the correlation function of the ionosphere with a periodic pulse train. Following a very quick tour of the plasma physics and theoretical correlation function (the forward problem) the intricate interplay of the transmitter waveform design and the inverse problem for parameter estimation—including some unique robustness issues—is shown. It will also be shown that the highest performance transmitter waveforms today are themselves stochastic (or nearly stochastic) processes. The results have a pleasing analogy to the quantum mechanics notion of ‘‘observables,’’ and shed light on the very nature of measurement.

On the degress of circularity for various kinds of polarized light in a nonpolar liquid mixture

The explicit results of the degrees of circularity for various kinds of completely polarized light in a nonpolar binary liquid mixture composed of an optically active solute and an optically nonactive solvent are obtained with the aid of the Ornstein-Zernike form for the correlation function of density fluctuations. The results are discussed in detail far from and near the critical point and compared with circular intensity differences.