Average Distribution Function Research Articles

Equation of State for Complex Liquid Mixtures

Abstract The present work shows a successful extension of previous studies to molecular liquids for which the second virial coefficients are not known. Recent advances in statistical-mechanical theory of equilibrium fluids can be used to obtain an equation of state for compressed liquids. The average contact pair distribution function (G), which is at the heart of the equation of state, has the very simple form of a single curve in λb*ρ* (any substance). The effect of many-body forces may still be included in G. The temperature-dependent parameters in the equation can be calculated with reasonable accuracy for most practical purposes just from the experimental heat of vaporization and triple-point density. Thus, thermodynamic consistency is achieved by the use of two scaling parameters (ΔHν, ρt). In terms of these parameters, the reduced second virial coefficient (B2*(T)) obeys a single law of corresponding states in which α*(T) and b*(T) are only slightly modified. The correlations embrace the temperature range Tt ≤ T < Tc and can be used in a predictive mode. The remaining constant parameters are best found empirically from ρt data for pure dense fluids. We tested the equation of state on several liquid mixtures. The results indicate that the density of liquid mixtures can be predicted within about 5% at any pressure and temperature.

Theory and simulation of chain-molecule fluid structure

Results of NVT molecular-dynamics simulations are reported for modelled n-butane and n-pentane over a range of densities at temperatures of about 430 K. The model potential is the repulsive part of the Lennard-Jones potential with bond angles constrained at 109·47° and bond lengths of 0·153 nm, where σ = 0·3923 nm and ε/k = 72 K. Simulation results for average site-site distribution functions are compared with results of an approximate RISM theory. Similarly to the case of diatomic fluids, the accuracy of the approximate theory is only qualitative, but trends of key features of the fluid structure are well represented. The softness of the Lennard-Jones potential appears to play a significant role in smoothing away several potential sources of error in the approximate theory.

Analytical equation of state for molecular fluids: Kihara model for rodlike molecules.

We present an analytical equation of state for convex-molecule fluids in any number of dimensions, based on statistical-mechanical perturbation theory for hard convex bodies. The second virial coefficient is calculated exactly. Two temperature-dependent parameters in addition to the second virial coefficient arise, an effective molecular volume (or van der Waals covolume) and a scaling factor for the average contact pair distribution function of hard convex bodies. The equation is tested against computer simulations and perturbation calculations for the model of rodlike molecules interacting via a Kihara (12,6) potential. This model allows the two temperature-dependent parameters to be calculated from the intermolecular potential by simple quadrature, without further approximations for averaging over molecular orientations. The agreement is quite good.

Structure of a stage-3 Cs-graphite intercalate

The influence of interlayer coupling on the in-plane structural properties of the stage-3 caesium-graphite system is investigated using a multilayer molecular-dynamics simulation. We show that the in-plane structure, represented by the circularly averaged pair distribution function, is insensitive to the interlayer coupling over a wide range of temperatures. The structure of the layer is investigated as a function of the parameters of the Cs-Cs potential and of the barrier height for free diffusion created by the graphite periodic potential.

A covariant gravitational Landau equation for discreteness fluctuations

Starting from a relativistic Vlasov description for a system of stars that is not 'too relativistic', this paper examines the effects of discreteness fluctuations and shows that, in the limit that these may be associated with relatively proximate stars, the average distribution function f0 will satisfy the relativistic Landau equation first derived by Israel and Kandrup (1984). This alternative derivation of the Landau equation, which is both simpler and more intuitive, views discreteness effects as being only one particular type of fluctuation which one might envision; and, as such, suggests the possibility of an effective relaxation induced by collective fluctuations, which has already been explored in the Newtonian limit.

Dynamical evolution of three-dimensional kinetic electron-beam instabilities

We present a method of solution of the quasi-linear diffusion equation for the averaged distribution function. The criterion for its validity is discussed. Using the solution obtained, we study the dynamics of the relaxation of a weak kinetic electron-beam instability and show that it saturates asymptotically at zero energy level. In a self-consistent manner, the distribution function for the beam particles will be shown to deform to a flat shape.

Theory of trapped-ion-temperature-gradient-driven turbulence and transport in low-collisionality plasmas

A novel theory for the nonlinear evolution of the trapped-ion-temperature-gradient-driven mode, based on the turbulent trapping of resonant ions in the electrostatic potential of the waves, is proposed. A statistical description is adopted whereby the self-consistent evolution of the two-point correlation function of the trapped-particle distribution function is followed in phase space. Threshold-dependent, non-steady-state turbulence (nonlinear instability) is shown to develop when the decay of the correlation function is overcome by a source term that derives its free energy from the relaxation of the average distribution function. This nonlinear instability leads to anomalous thermal and particle transport that in turn reconfigure the equilibrium temperature and density profiles in such a way as to return the system toward its marginal point. Expressions for the nonlinear dispersion relation and threshold, as well as estimates of the thermal and particle transport level, are derived. The estimated flux levels are sufficiently high as to make any significant departure away from marginality unlikely. The scenario outlined serves to underscore the desirability for pellet injection in experimental devices such as the Compact Ignition Tokamak [Bull. Am. Phys. Soc. 32, 1921 (1987)] that operate in the very low ion collisionality regime where this mode would be expected to become relevant. As with a number of recent theories, the present work further reinforces the notion that unfavorably drifting trapped particles pose a serious menace to confinement and suggests inboard, off-axis radio-frequency heating as one means of reducing the size of this population, at least for the case of those energetic trapped particles created during auxiliary heating.

A model for ultrafast vibrational cooling in molecular crystals

A model is presented to describe vibrational cooling (VC) in crystals of large molecules. Vibrational cooling is the process by which a vibrationally excited crystal returns to the ground state. This process may consist of many sequential and parallel vibrational relaxation (VR) steps. The model describes a highly excited, vibrationally dense molecular crystal at zero and finite temperatures. An initially excited vibration relaxes via anharmonic coupling by sequential emission of many lattice phonons until all vibrational energy is destroyed. The time evolution of vibrational excitation probability is described with a Master equation. Various models for the phonon density of states, which exerts primary control over the VR process, are considered. It is found that VC occurs on a much slower time scale than VR, and that the rate of VC is only weakly dependent on temperature, even in systems where VR is highly temperature dependent. An important conclusion of this work is that vibrational cooling is described by an ensemble averaged vibrational population distribution function which moves to lower energy states and broadens as time increases. The motion to lower energy is described by a ‘‘vibrational velocity’’ (emitted energy per unit time) which is independent of temperature, while the width of the distribution increases with increasing temperature. The model is then used to calculate experimental observables including time resolved absorbance, emission, and Raman scattering following excitation of a high frequency vibration.

A semianalytical analysis of ion velocity distributions in the presence of combined neutral-beam and ICRF-heating

The energy clamping scheme with combined neutral-beam and ICRF-heating is considered. Assuming the neutral-beam particles to be injected almost isotropically, an approximate equation is derived for the pitch angle averaged distribution function. Comparison between results obtained from this equation and those of a full 2D numerical calculation, shows good agreement for physically significant quantities, like total absorbed power, power transfer to background plasma particles and fusion reactivities. This indicates that the equation for the averaged distribution function should be useful in situations where time consuming 2D numerical calculations of the distribution function are impractical, e.g., in plasma transport codes.

Study of γ′-precipitation kinetics in alloy 800 at 575° C by small angle neutron scattering

Small-angle neutron scattering has been used to study the characteristics of γ′-precipitation during ageing at 575°C in several alloy 800 heats, differing in the initial chemical composition; the effect of a pre-ageing treatment at 700°C and of creep tests have been studied by the same technique. For each investigated sample the volume fraction average size and size distribution function of the γ′-precipitates were determined and quantitatively compared with the prediction of the Lifshitz-Slyozov and Wagner model.

Linear transport theory in a random medium

The time-independent linear transport problem in a purely absorbing (no scattering) random medium is considered. A formally exact equation for the ensemble averaged distribution function 〈Ψ〉 is derived. Under the assumption of a two-fluid statistical mixture, with the transition from one fluid to the other assumed to be determined by a Markov process, an exact solution to this equation for 〈Ψ〉 is obtained. In the source-free case, this solution is shown to agree with the result obtained by ensemble averaging simple exponential attenuation. Several approximations to the exact equation for 〈Ψ〉 are considered, and numerical results given to assess the accuracy of these approximations.

Transport and diffusion in a statistical medium

Abstract The problem of formulating and solving linear kinetic (transport) equations in a random maedium is considered. In such a medium, the cross sections and external source are known only in a statistical sense. A projection operator technique is used to derive the appropriate equation for the ensemble averaged distribution function. This equation contains and infinite series, with the nth term involving an n-fold applicaytion of the inverse operator for a non-statistical transport equation. This multiple inverse operator (a multiple integral over Green's functions) acts on various correlation functions describing the statistical nature of the medium. Fokker-Planck approximation can be employed to localize these integral operators. For a static statistical mixture of two fluids, a stationary Markov model is assumed to compute the required correlations. Various explicit solutions to the ensemble averaged transport equation, both exact and approximate, are given for the purely absorbing (no scattering) ...

Radial distribution function analyses of polythiohydroquinone

In recent years, the RDF analysis [1-4] of X-ray diffraction patterns has become a powerful tool for studying the structures of non-crystalline disordered materials. This technique may be applied to systems for which the average radial distribution function is spherically symmetrical, such as pure liquids, solutions, glasses and finely divided powders. It yields information not only about structural parameters but also many physical parameters such as Debye temperature, coupling coefficients, coordination numbers and interatomic potentials. The polymer, polythiohydroquinone, reported here does not crystallize and therefore single crystal techniques cannot be used to obtain structural information. Hydroquinone and excess sulphur were mixed thoroughly and poured into a glass tube. The tube was flushed several times with nitrogen and placed in a furnace in a nearly horizontal position. The issuing hydrogen sulphide gas was trapped in a lead acetate solution. After several hours the tube was removed and allowed to cool. The material was washed with concentrated sulphuric acid, followed by concentrated alkali solution to remove low molecular weight products and finally with hot carbon disulphide solutions to remove excess sulphur. The yield was about 70%. X-ray diffraction patterns for the present sample were recorded with the help of a Phillips diffractometer on a strip chart, the scanning rate being 2 ° min -1 , using MoK~ radiation in the range s = 0.77 to s = 8.96 (where s = 4re sin 0/4). The background radiation was suppressed with the aid of a discriminator by running the diffractometer once without sample. Then the sample was mounted and the intensities recorded. Other corrections for polarization and absorption were carried out using procedures described by Kaelble [2]. Tubulated values of atomic scattering factors [5] and incoherent scattering factors [6] were used for obtaining independent scattering curves. The corrected intensities were scaled to electron units by the method described by Kroughmoe [7]. The contribution of the flat-faced diffractometer sample in the small-angle region and that due to multiple scattering were subtracted following the method described by Warren [8]. Following the procedure described by Kaelble, the radial distribution function for polyatomic samples, 47t?2~0(r) is given by

Long-time quasilinear evolution of the free-electron laser instability for a relativistic electron beam propagating through a helical magnetic wiggler

The long-time quasilinear development of the free-electron laser instability is investigated for a tenuous electron beam propagating in the z direction through a helical wiggler field B0=−B̂ cos k0zêx−B̂ sin k0zêy. The analysis neglects longitudinal perturbations (δφ≂0) and is based on the nonlinear Vlasov–Maxwell equations for the class of beam distributions of the form fb(z,p,t) =n0δ(Px)δ(Py)G(z,pz,t), assuming ∂/∂x=0=∂/∂y. The long-time quasilinear evolution of the system is investigated within the context of a simple ‘‘water-bag’’ model in which the average distribution function G0( pz,t) =(2L)−1∫L−L dz G(z,pz,t) is assumed to have the rectangular form G0( pz,t) =[2Δ(t)]−1 for ‖pz−p0(t)‖ ≤Δ(t), and G0( pz,t) =0 for ‖pz−p0(t)‖ >Δ(t). Making use of the quasilinear kinetic equations, a coupled system of nonlinear equations is derived which describes the self-consistent evolution of the mean electron momentum p0(t), the momentum spread Δ(t), the amplifying wave spectrum ‖Hk(t)‖2, and the complex oscillation frequency ωk(t) +iγk(t). These coupled equations are solved numerically for a wide range of system parameters, assuming that the input power spectrum Pk(t=0) is flat and nonzero for a finite range of wavenumber k that overlaps with the region of k space where the initial growth rate satisfies γk(t=0) >0. To summarize the qualitative features of the quasilinear evolution, as the wave spectrum amplifies it is found that there is a concomitant decrease in the mean electron energy γ0(t)mc2=[m2c4+e2B̂2/k20 +p20(t)c2]1/2, an increase in the momentum spread Δ(t), and a downshift of the growth rate γk(t) to lower k values. After sufficient time has elapsed, the growth rate γk has downshifted sufficiently far in k space so that the region where γk >0 no longer overlaps the region where the initial power spectrum Pk(t=0) is nonzero. Therefore, the wave spectrum saturates, and γ0(t) and Δ(t) approach their asymptotic values.

Effects of dextran polydispersity on red blood cell aggregation

B 512 F dextran fractions of different polydispersities were prepared by fractionnal precipitation and preparative elution chromatography. By combination of low angle laser light scattering (LALLS) and gel permeation chromatography (GPC), absolute average molecular weights (MWs) and molecular weight distribution functions (MWDs) were determined. Static and thermodynamical properties in terms of polymer dimensions and second virial coefficient of dilute solutions of dextran in water have been investigated. The results indicate that dextran macromolecules in water are rather compact and impenetrable coils. Measuring the disaggregation shear stress of dextran-induced red blood cell aggregates by laser light reflectometry, the macromolecular bridging energy was shown to depend upon dextran sample polydispersity. This reflects the weak and reversible character of red blood cell aggregation by dextran chains in physiological saline solution.

Quasilinear stabilization of the free electron laser instability for a relativistic electron beam propagating through a transverse helical wiggler magnetic field

A quasilinear model is developed that describes the nonlinear evolution and stabilization of the free electron laser instability in circumstances where a broad spectrum of waves is excited. The relativistic electron beam propagates perpendicular to a helical wiggler magnetic field B0=−B̂ cos k0 z êx−B̂ sin k0 z êy, and the analysis is based on the Vlasov–Maxwell equations assuming ∂/∂x=0=∂/∂ y and a sufficiently tenuous beam that the Compton-regime approximation is valid (δφ≂0). Coupled kinetic equations are derived that describe the evolution of the average distribution function G0( pz,t) and spectral energy density ℰk(t) in the amplifying electromagnetic field perturbations. A thorough exposition of the theoretical model and general quasilinear formalism is presented, and the stabilization process is examined in detail for weak resonant instability with small temporal growth rate γk satisfying ‖γk/ωk‖≪1 and ‖γk/k Δvz‖≪1. Assuming that the beam electrons have small fractional momentum spread (Δ pz/p0≪1), the process of quasilinear stabilization by plateau formation in the resonant region of velocity space (ωk−kvz=0) is investigated, including estimates of the saturated field energy, efficiency of radiation generation, etc.

Nonlinear perturbation approach to bunch lengthening and blow-up of energy spread

An analytical method is formulated in the nonlinear regime for longitudinal instabilities of a bunched beam in particle accelerators. In the present perturbation approach to the Vlasov equation, the average distribution function of particles which is stationary in the linear theory, is adiabatically changed by the nonlinear term of the second-order in perturbed quantities. A set of two differential equations for the bunch length (or the energy spread) and the perturbation amplitude is given in the absence or presence of the synchrotron frequency spread. Bunch lengthening in electron beams and the overshoot phenomenon in the bunch blow-up in proton beams are investigated systematically.

RENORMALIZED QUASI-LINEAR THEORY OF TURBULENCE IN NON-UNIFORM PLASMA——GENERALIZATION OF MISGUICH-BALESCU THEORY

The re-normalized quasi-linear theory of turbulence in uniform plasma, given by Mis-guich and Balescu, is generalized to the case of non-uniform plasma. In the weak-coupling and the weak non-uniformity approximation, the explicit expressions for the propagator and the average turbulent collision operator are obtained. Non-uniformity not only modifies the diffusion contribution, resonance broadening or frequency shift, Dupree damping and velocity-space slope effect of the average distribution function in these operators, but also producesnew differentio-exponential operators——the velocity differential operator in the propagatordue to non-uniformity of the time-space correlation function, and the space differential operator in the average turbulent collision operator due to non-uniformity of the average distribution function. Non-uniformity of the average distribution function prevents two free-stream propagators that are inverse to each other in the expression for the average turbulent collision operator in terms of the propagators from compensating for each other, and as a result the non-Markovian contribution in the propagator acted immediately on the average distribution function appears.

Exchange and transport time scales in the Seto Inland Sea

Five typical water volumes in the Seto Inland Sea are defined, and their average residence times, remnant functions, and residence time distribution functions are obtained, mainly from results of hydraulic model experiments; the average residence times and the functions well describe characteristics of exchange and transport of materials in the sea. A representative residence time, which is the average residence time of the total water in the inland sea, is about 15 months.

Growth rate of the clump instability

The instability of phase space density granulations or clumps has recently been reported. A theory for the growth rate of the clump instability in a one-dimensional, ion–electron plasma is presented. Growth is predicted to occur in regions of velocity space where the average particle distribution functions have opposing velocity gradients. The free energy required for instability is significantly below the threshold necessary for the linear instability of ion acoustic waves. The growth rate increases with fluctuation amplitude and is proportional to the inverse of the nonlinear trapping time. The effect of particle discreteness and clump self-energy on the growth rate is discussed. The growth rate is in reasonable agreement with recent particle simulations.