Modified Distribution Function Research Articles

Second order hydrodynamics based on effective kinetic theory and electromagnetic signals from QGP

We study the thermal dilepton and photon production from relativistic heavy ion collisions in presence of viscosities by employing the recently developed second order dissipative hydrodynamic formulation estimated within a quasiparticle description of thermal quantum chromo-dynamics (QCD) medium. The sensitivity of shear and bulk viscous pressures to the temperature dependence of relaxation time is analyzed within one- dimensional boost invariant expansion of quark gluon plasma (QGP). The dissipative corrections to the phase-space distribution functions up to first order in gradients are obtained from the Chapman–Enskog like iterative solution of effective Boltzmann equation in the relaxation time approximation. Thermal dilepton and photon production rates for QGP are calculated by employing this viscous modified distribution function. Yields of these particles are quantified for the longitudinal expansion of QGP with different temperature dependent relaxation times. Our analysis employing this second order hydrodynamic model indicates that the spectra of dileptons and photons get enhanced by both bulk and shear viscosities and are well behaved. Also, these particle yields are found to be sensitive to relaxation time. Further, we do a comparison of these particle spectra with a standard hydrodynamic formulation.

Modeling and analysis of nonlinear axial friction forces of tripod joints considering roughness characteristics of contact pairs

The nonlinear axial friction force (NAFF) induced by contact and friction between components of tripod joints constitutes an important dynamic characteristic of drive-shaft systems, which will lead to deterioration of the NVH (noise, vibration and harshness) performance of the vehicle. Based on the fractal theory, this paper proposes a novel modeling and analysis method for accurate estimation of NAFF considering micro contact and friction characteristics of contact pairs. An analytical model is developed for estimating the NAFF of tripod joints considering micro contact and friction characteristics of contact pairs. In the process of modeling, a modified distribution function is proposed to describe distributions of contact asperities between the curved surfaces to describe the contact and friction states between the rollers and the tracks inside tripod joints more accurately. The developed analytical model suggests that NAFF is related to the fractal parameters, the material parameters, the operating conditions and the distribution of asperities. The effectiveness of the analytical model is demonstrated by comparing the model-predicted NAFF with the measured data. Based on the developed model of NAFF, the influences of important input parameters on the magnitude of NAFF are qualitatively analyzed. Subsequently, Sobol' global sensitivity analysis method is used to further investigate the influences of important input parameters on the magnitude of NAFF. The results show that the proposed modeling and analysis method for estimating NAFF are essential to characterize relationships among NAFF and fractal parameters, material properties and distribution of asperities from a micro level, which could facilitate design optimization of the tripod joints.

From nucleation and coarsening to coalescence in metastable liquids.

The transition of a metastable liquid (supersaturated solution or supercooled melt) occurring from the intermediate stage (where the crystals nucleate and grow) to the concluding stage (where the larger particles evolve at the expense of the dissolution of smaller particles) is theoretically described, with allowance for various mass transfer mechanisms (reaction on the interface surface, volume diffusion, grain-boundary diffusion, diffusion along the dislocations) arising at the stage of Ostwald ripening (coalescence). The initial distribution function (its 'tail') for the concluding stage (forming as a result of the evolution of a particulate assemblage during the intermediate stage) is taken into account to determine the particle-size distribution function at the stage of Ostwald ripening. This modified distribution function essentially differs from the universal Lifshitz-Slyozov (LS) solutions for several mass transfer mechanisms. Namely, its maximum lies below and is shifted to the left in comparison with the LS asymptotic distribution function. In addition, the right branch of the particle-size distribution lies above and is shifted to the right of the LS blocking point. It is shown that the initial 'tail' of the particle-size distribution function completely determines its behaviour at the concluding stage of Ostwald ripening. The present theory agrees well with experimental data. This article is part of the theme issue 'Patterns in soft and biological matters'.

Correction of Axial Force Distribution Function of Flexible Single Pile

By introducing the stiffness ratio between the pile end soil and the pile side soil, the existing axial force distribution function is modified, and the axial force of the model pile is calculated by the modified distribution function. The calculated results are consistent with the experimental results, indicating that the modified axial force distribution function can more accurately reflect the axial force variation of the single pile.

Modified theory of gravity and clustering of multi-component system of galaxies

In this paper, we analyze the clustering of galaxies using a modified theory of gravity, in which the field content of general relativity has been be increased. This increasing in the field content of general relativity changes the large distance behavior of the theory, and in weak field approximation, it will also modify the large distance behavior of Newtonian potential. So, we will analyzing the clustering of multi-component system of galaxies interacting through this modified Newtonian potential. We will obtain the partition function for this multi-component system, and study the thermodynamics of this system. So, we will analyze the effects of the large distance modification to the Newtonian potential on Helmholtz free energy, internal energy, entropy, pressure and chemical potential of this system. We obtain also the modified distribution function and the modified clustering parameter for this system, and hence observe the effect of large distance modification of Newtonian potential on clustering of galaxies.

Implementation and Identification of Preisach Parameters: Comparison Between Genetic Algorithm, Particle Swarm Optimization, and Levenberg–Marquardt Algorithm

Electromagnetic simulations of devices with soft materials and the study of the influence of cutting edge deformations/stresses on the degradation of magnetic performances require the identification of hysteresis parameters. Hence, computationally efficient implementation and identification of the hysteresis model is needed to obtain a solution with high accuracy within a reasonable time. Many studies on hysteresis identification have been reported, but few compare different methods to identify the optimal one. In the present study, we focus on the Preisach hysteresis model combined with the Lorentz modified distribution function to describe the magnetic behavior of a fully processed nonoriented Fe–3wt% Si steel sheet under static excitation. Three different identification techniques are implemented and evaluated: particle swarm optimization (PSO), genetic algorithms (GAs), and nonlinear least-squares approximation based on the Levenberg–Marquardt (LM) method. We evaluate each approach with regard to the accuracy, central processing unit computation time, and repeatability of the results. All the techniques perform well in this high-nonlinearity problem. The root-mean-square error is < 3%. However, the implementation of the GA is more complex than the PSO and LM methods. The optimized parameters are obtained in a few minutes in the case of the LM method, but a few hours are required for both the other techniques. Therefore, the LM method is the most suitable technique for the identification of Preisach hysteresis.

Correlated fluctuations near the QCD critical point

In this paper, we introduce a freeze-out scheme for the dynamical models near the QCD critical point through coupling the decoupled classical particles with the order parameter field. With a modified distribution function that satisfies specific static fluctuations, we calculate the correlated fluctuations of net protons on the hydrodynamic freeze-out surface. A comparison with recent STAR data shows that our model calculations could roughly reproduce energy dependent cumulant $C_4$ and $\kappa \sigma^2$ of net protons through tuning the related parameters. However, the calculated $C_2$ and $C_3$ with both Poisson and Binomial baselines are always above the experimental data due to the positive contributions from the static critical fluctuations. In order to qualitatively and quantitatively describe the experimental data, the dynamical critical fluctuations and more realistic non-critical fluctuation baselines should be investigated in the near future.

A Chemical Mechanical Planarization Model Including Global Pressure Distribution and Feature Size Effects

In this paper, we propose a chip-scale Chemical mechanical polishing (CMP) model based on the Greenwood–Williamson theory combining the effects of both the pad bulk deformation and the feature size, which are mainly responsible for global pressure distribution and feature-scale removal rate variation, respectively. A reference camber is first constructed to simulate the pad bulk deformation, which can be directly introduced into the global pressure distribution computation for incorporating the long-range characteristics of chip profiles. Then, both the effective curvature and the modified distribution function of pad asperity height are involved in the material removal rate. The model is first verified through some experimental data from CMP test structures and found to be in good agreement with the test data. For further validating the rationality and accuracy of the present model, some simulations are compared with our experimental data and discussed in detail.

An improved Moving Kriging-based meshfree method for static, dynamic and buckling analyses of functionally graded isotropic and sandwich plates

A meshfree method with a modified distribution function of Moving Kriging (MK) interpolation is investigated. This method is then combined with a high order shear deformation theory (HSDT) for static, dynamic and buckling analyses of functionally graded material (FGM) isotropic and sandwich plates. A meshfree method uses the normalized form of MK interpolation under a new quartic polynomial correlation to build the basis shape functions in high order approximations. The Galerkin weak form is used to separate the system equations which is numerically solved by meshfree method. A rotation-free technique extracted from isogeometric analysis is introduced to eliminate the degrees of freedom of slopes. Then, the method retains a highly computational effect with a lower number of degrees of freedom. In addition, the requirement of shear correction factors is ignored and the traction free is at the top and bottom surfaces of FGM plates. Various thickness ratios, boundary conditions and material properties are studied to validate the numerical analyses of the rectangular and circular plates. The numerical results show that the present theory is more stable and well accurate prediction as compared to three-dimensional (3D) elasticity solution and other meshfree methods in the literature.

Angular momentum transfer and the size���mass relation in early-type galaxies

Early-type galaxies (ETGs) define a narrow strip in the size–mass plane because of the observed correlation between the effective radius Reff and the total stellar mass M★. When expressed in logarithmic units, a linear relation, log Reff∝γ log M★, is indeed observationally found, but the slope γ deviates from the canonical γ= 1/2 value which can be naively predicted for a spherically symmetric isotropic system. We propose here that a transfer of angular momentum to the stellar component induces an anisotropy in the velocity space, thus leading to a modified distribution function (DF). Assuming an Osipkov–Merritt like anisotropic DF, we derive an analytic relation between the slope γ of the size–mass relation and the slope α of the angular momentum term in the DF. With this simple model, we are then able to recover the observed γ value provided α is suitably set. It turns out that an anisotropy profile which is tangential inside ∼0.6ra and radial outside, with ra the anisotropy radius, is able to reproduce the observed size–mass relation observed for massive (M★≥ 2 × 1010h−1 M⊙) elliptical galaxies.

Effect of shear viscosity on spectra, elliptic flow, and Hanbury Brown–Twiss radii

I calculate the first correction to the thermal distribution function of an expanding gas due to shear viscosity. With this modified distribution function I estimate viscous corrections to spectra, elliptic flow, and HBT radii in hydrodynamic simulations of heavy ion collisions using the blast wave model. For reasonable values of the shear viscosity, viscous corrections become of order one when the transverse momentum of the particle is larger than 1.7 GeV. This places a bound on the $p_{T}$ range accessible to hydrodynamics for this observable. Shear corrections to elliptic flow cause $v_{2}(p_{T})$ to veer below the ideal results for $p_{T} \approx 0.9$ GeV. Shear corrections to the longitudinal HBT radius $R^{2}_{L}$ are large and negative. The reduction of $R_{L}^2$ can be traced to the reduction of the longitudinal pressure. Viscous corrections cause the longitudinal radius to deviate from the $\frac{1}{\sqrt{m_T}}$ scaling which is observed in the data and which is predicted by ideal hydrodynamics. The correction to the sideward radius $R^{2}_{S}$ is small. The correction to the outward radius $R^{2}_{O}$ is also negative and tends to make $R_{O}/R_{S} \approx 1$.

Simulations of the instability of the m=1 self-shielding diocotron mode in finite-length non-neutral plasmas

The “self-shielding” m=1 diocotron mode in Malmberg–Penning traps has been known for over a decade to be unstable for finite length non-neutral plasmas with hollow density profiles. Early theoretical efforts were unsuccessful in accounting for the exponential growth and/or the magnitude of the growth rate. Recent theoretical work has sought to resolve the discrepancy either as a consequence of the shape of the plasma ends or as a kinetic effect resulting from a modified distribution function as a consequence of the protocol used to form the hollow profiles in experiments. Both of these finite length mechanisms have been investigated in selected test cases using a three-dimensional particle-in-cell code that allows realistic treatment of shape and kinetic effects. A persistent discrepancy of a factor of 2–3 remains between simulation and experimental values of the growth rate. Simulations reported here are more in agreement with theoretical predictions and fail to explain the discrepancy.

Application of Monte Carlo simulation to optimal maintenance scheduling in a parallel-redundant system

A modified distribution function is created based on the assumption that the component useful life phase is associated with an exponential distribution and the wearout period is subject to a Weibull distribution (/spl beta/>1). Using the modified distribution function, a Monte Carlo simulation approach is used to obtain the total failure frequency and optimum maintenance interval for a parallel-redundant system where each component can enter its wearout period. Two examples are used to illustrate the simulation procedure. The effect of the Weibull parameter /spl beta/ on the optimum maintenance interval for a parallel-redundant system is illustrated.

A numerical method for studying nonlinear bubble oscillations in acoustic cavitation

A numerical method is developed for the study of the behaviour of a gas bubble in ultrasonically induced cavitation. This method is based on the solution of the full Navier-Stokes equation for the two-fluid system consisting of the gas inside the bubble and the liquid surrounding it, following ideas originally introduced for the analysis of multi-component fluid flows. Analysis of acoustic cavitation must be done taking into account the compressibility of the gas bubble and for this purpose the Navier-Stokes equation is coupled with an equation of state for the gas; our model also considers the presence of viscosity and surface tension, thus allowing surface oscillations of the bubbles. To avoid numerical problems in the solution of the Navier-Stokes equation two different grids are introduced, an Eulerian one for the ‘background’, where the Navier-Stokes equation is solved, and another moving one for the interface; this second grid is explicitly tracked and properly modified during motion and is responsible for the behaviour of the bubble. The transfer of information between the Eulerian grid and the interface grid is obtained with the aid of a lattice modified distribution function. The method is tested analyzing forced oscillations of cavitation bubbles excited by ultrasonic standing waves at different frequencies and pressure amplitudes, showing characteristic behaviour of nonlinear dynamical systems; frequency spectra are obtained, stability analysis is performed and strong dependence from initial conditions is showed; comparisons with previous different approaches are also performed.

An estimation of the distribution functions of doped Tb 3+ and Nd 3+ in glasses by fluorescence measurements

The distribution functions of doped Tb 3+ and Nd 3+ in silicate, germanate, borate and phosphate glasses, in which cross-relaxation among active ions plays an important role, have been estimated by fluorescence measurements. In order to analyze both the steady state and the transient fluorescence characteristics, a non-linear model for the resonant energy transfer has recently been developed. The experimental fluorescence intensity and decay curves, measured as a function of acceptor concentration, suggested a modified distribution function, which included the effect of the segregation of donors and acceptors. The number of segregated Tb 3+ ions at the second nearest neighbor position around a Tb 3+ ion was estimated to be 0.1–3.4, depending on the host glass.

A non-linear analysis of energy transfer in highly Tb 3+-doped glasses

We present a comprehensive model for the resonant energy transfer of an electronic excitation between energy donors and energy acceptors. The model is based on a coupled non-linear rate equation for the excited state of donors and acceptors and on a modified distribution function of interacting ions. The non-exponential decay of the fluorescence intensity just after flash excitation has been consistently understood as a result of the non-linear term of the rate equation by a numerical analysis. The transient and steady state fluorescence of Tb 3+ in various glass matrices has been investigated as a function of the concentration of donors and acceptors, determining the physical parameters of the resonant energy transfer process. This model is especially valid for analyzing systems with high donor-acceptor concentration and those under high excitation such as up-conversion laser materials.

Multifractal nature of cluster size distributions on Bethe lattices

It is demonstrated that in the union space of all Bethe lattices with different coordination numbers ( σ + 1), the cluster number generating function takes the form of different moments of the perimeter ( t)-size ( s) distribution function g st with respect to the variable exp(− s). In that extended space, the moments exhibit a typical multifractal (MF) behavior in terms of the scaling variable exp(−( t + s)) with analytically determined MF exponents. It is also shown that the logarithm of g st is a homogeneous function of the variables s and ( t + s). This homogeneity property is used to show that the same MF behavior could be obtained in terms of exp(− s) as a scaling variable. However, in that case a modified distribution function has to be formed by raising g st to the power 1 σ .

The structure of dust tails of comets II. The tail and dust content of comet Arend-Roland

The modified distribution function of dust particles, f( γ), which can be determined from tail brightness profiles on the basis of mechanical theory, is discussed with special regard to its reliability and accuracy. Physical significance of f( γ) is also discussed in terms of dust model parameters, and it is shown that f( γ), if treated carefully, will serve as an effective tool in studying cometary dust. Four isophotes of Comet Arend-Roland, 1957 III, in the orange-red light (λλ 0.53–0.68 micron) obtained by Ceplecha (1958), are analysed by the numerical method described in Paper I (KIMURA and LIU 1975) with some improvements and higher approximations. The distribution f( γ) thus obtained shows a bimodal character with peaks at γ = 0.10 and 0.010 with a relative height ratio of 1 to 0.6. Dust emission rate, which is assumed to follow the inverse square law of heliocentric distance, is estimated to give P d C sca = (1.3±0.5) × 10 9 cm 2/sec, where P d is the rate of particle emission at 1 a.u. and the C sca is a mean effective cross section of particles for light scattering including the phase effect (the scattering angles of present interest range from about 80 to 100 degrees).

Amplification of whistler waves in the magnetosphere

The effect of a succession of narrow band morse pulses, propagating in the whistler mode, on a steady state distribution function of electrons trapped in the Earth's magnetic field is studied. The pulses have the effect of diffusing the particles in both pitch angle and energy, and the corresponding pitch angle diffusion coefficients and energy diffusion coefficients are computed. Because of the definite relation between the change in magnetic moment and energy of a particle which occur at resonance, particles are forced to move on given surfaces in velocity space, so that the diffusion is one-dimensional. The corresponding single diffusion coefficient is computed and a one-dimensional Fokker Planck equation derived. A Kennel and Petschek type initial distribution function is used and a time development of this distribution function is considered. It is found that a slot develops in the distribution function at a velocity corresponding to the frequency of the morse pulses. Growth rates for the modified distribution function are computed, and the behaviour of the growth rates lead us to postulate a mechanism for the explanation of the artificially stimulated emissions. The mechanism explains some of the important observational results for ASE such as the offset frequency and the fact that emissions are often triggered near the equator. An extension of this work may also explain the observed preferential triggering at half the minimum gyrofrequency.

The momentum distribution in liquid helium

A calculation is reported of the momentum distribution in liquid helium at absolute zero using a pair wave function and a modified superposition approximation. A value of 0 13 for the fraction of particles in the zero-momentum state is obtained. The approximations obtained for the momentum distribution function for small values of k is shown to be inaccurate. The use of an exact result for nk for small k leads to a modified distribution function which gives 0 25 for the fraction of particles in the zero-momentum state.