Evolution Of Distribution Function Research Articles

Chemical Kinetics Roots and Methods to Obtain the Probability Distribution Function Evolution of Reactants and Products in Chemical Networks Governed by a Master Equation.

In this paper first, we review the physical root bases of chemical reaction networks as a Markov process in multidimensional vector space. Then we study the chemical reactions from a microscopic point of view, to obtain the expression for the propensities for the different reactions that can happen in the network. These chemical propensities, at a given time, depend on the system state at that time, and do not depend on the state at an earlier time indicating that we are dealing with Markov processes. Then the Chemical Master Equation (CME) is deduced for an arbitrary chemical network from a probability balance and it is expressed in terms of the reaction propensities. This CME governs the dynamics of the chemical system. Due to the difficulty to solve this equation two methods are studied, the first one is the probability generating function method or z-transform, which permits to obtain the evolution of the factorial moment of the system with time in an easiest way or after some manipulation the evolution of the polynomial moments. The second method studied is the expansion of the CME in terms of an order parameter (system volume). In this case we study first the expansion of the CME using the propensities obtained previously and splitting the molecular concentration into a deterministic part and a random part. An expression in terms of multinomial coefficients is obtained for the evolution of the probability of the random part. Then we study how to reconstruct the probability distribution from the moments using the maximum entropy principle. Finally, the previous methods are applied to simple chemical networks and the consistency of these methods is studied.

Bimodal ion energy distribution functions in a hydrogen magnetized plasma

Direct measurements of temporal evolution of the ion energy distribution functions (IEDFs) in a pulsed magnetized hydrogen plasma with a complex spatial structure are presented. The results show a remarkable split of the curves into two peaks of energy that draw away as the spatial structure expands until they reach a complete separation. The behavior is explained in terms of × drift velocity plasma dynamics and its influence on the IEDFs, with consequences for ion temperature measurements. The development of a criterion to calculate the ion temperatures allows us to obtain the temporal evolution of this parameter for the three ion species, and .

Concerning pressure and entropy of shock-accelerated heliosheath electrons

We study the behaviour of shocked wind-electrons leaving wind-driving stars after undergoing the outer wind termination shock. As an example, we describe the evolution of the keV-energetic electron distribution function downstream of the heliospheric termination shock. We start from a kinetic transport equation in the bulk frame of the heliosheath plasma flow taking into account shock-induced electron injection, convective changes, cooling processes, and whistler wave-induced energy diffusion. From this equation we proceed to an associated pressure moment of the electron distribution function arriving at a corresponding pressure transport equation which describes the evolution of the electron pressure in the bulk frame of the plasma along the plasma flow lines. We assume that the local distribution function, in view of the prevailing non-LTE conditions, is represented by a local kappa function with local kappa parameters that vary with the streamline coordinate $s$ downstream of the solar wind termination shock. We obtain the solution for the electron pressure as a function of the streamline coordinate $s$ from the pressure transport equation and demonstrate that, connected with this pressure, one obtains an expression for the entropy of the electron fluid which can also be derived as a streamline function. We show that the heliosheath electron fluid can essentially be characterized as an isobaric and isentropic flow. These results allow to generally conclude that astrotail plasma flows are characterized as such flows.

Evolution of Retail Distribution Functions in Omnichannel Environment (Evidence from Bulgarian Retail Sector)

The evolution of the distribution, as a result of the development of new communication technologies and transformations in consumer behavior, is the subject of extensive academic studies. The purpose of this article is to investigate and summarize the changes in the retail distribution functions in an omnichannel environment in the context of economic and marketing theory. The role of distribution services, wholesale and retail trade is the starting point in the analysis of the added value of the trade sector. The author argues that in order to make sense of changes in the retail distribution functions, it is necessary to look at the buying, selling and exchange functions, marketing and supply chain logistics in an interconnected way. The article is based on the research of authors in the field of distribution and retail economics and marketing. Empirical material from Bulgarian retail sector based on data from National Statistical Institution and Euromonitor illustrates that retailing has started to operate in a omnichannel environment.

Coupled discrete unified gas kinetic scheme for the thermal compressible flows in all Knudsen number regimes

In this paper, a coupled discrete unified gas kinetic scheme (CDUGKS) with a flexible Prandtl number is developed for the thermal compressible flows in all Knudsen number regimes. Different from the existing thermal discrete unified gas kinetic scheme based on the Shakhov model, the proposed CDUGKS based on the total energy double-distribution-function model can well preserve the nonnegative property of the distribution function, especially for the strong shock in the continuum regime. In the CDUGKS, the velocity distribution function (VDF) is used to recover the compressible continuity and momentum equations, while the energy distribution function (EDF) is used to recover the energy equation. The VDF and EDF are evaluated in a similar way and then coupled via the thermal equation of state. With the un-splitting treatment of the particle transport and collision in the distribution function evolution and the flux evaluation, the time step in CDUGKS is not limited by the particle collision time. Furthermore, the CDUGKS is an asymptotic preserving scheme, in which the Navier-Stokes solution in the hydrodynamic regime and the free transport mechanism in the kinetic regime can be precisely recovered with the second-order accuracy in both space and time. Finally, several numerical experiments, including the weak shock tube and the strong one in the whole Knudsen number flows, as well as the two-dimensional Riemann problem and the Rayleigh-Taylor instability in both hydrodynamic regime and kinetic regimes, are performed to validate the method. Numerical results agree fairly well with other benchmark results in different flow regimes, which demonstrates the current CDUGKS is a reliable and efficient method for multiscale flow problems.

An improved discrete velocity method (DVM) for efficient simulation of flows in all flow regimes

In this paper, an improved discrete velocity method (DVM) is developed for efficient simulation of fluid problems in all flow regimes. Compared with the conventional explicit DVM, the present scheme could effectively remove its drawbacks of low accuracy and efficiency in continuum flow regime with no deterioration of its performance in rarefied flow regime. One of the novel strategies adopted in the new method is to introduce a prediction step for solving the macroscopic governing equation. By using the prediction step, the equilibrium state is first estimated before solving the discrete velocity Boltzmann equation (DVBE). As a result, the collision term in the DVBE can be discretized implicitly to improve the stability and efficiency of the conventional explicit DVM. Another contribution of the new method is to physically reconstruct numerical flux at the cell interface by incorporating the collision effect into the process. To maintain simplicity and efficiency of the conventional DVM, in the present scheme, the collision effect in the flux reconstruction at the cell interface is considered through the solution of the macroscopic governing equation. This can effectively control the effect of numerical dissipation in the process of updating the macroscopic flow variables in the continuum flow regime. Analyses indicate that the prediction step does not contribute to the evolution of distribution functions in the highly rarefied flow regime while dominating the solutions in the continuum flow regime. Accordingly, the improved scheme automatically converges toward the conventional explicit DVM in the free molecular flow regime and approaches the Navier-Stokes solver in the continuum flow regime.

Ion velocity analysis of rotating structures in a magnetic linear plasma device

The MISTRAL device is designed to produce a linear magnetized plasma column. It has been used a few years ago to study a nonlinear low frequency instability exhibiting an azimuthal number m = 2. By changing the experimental configuration of MISTRAL, this work shows experimental results on an m = 1 rotating instability with strongly different behavior. The spatio-temporal evolution of the ion velocity distribution function given by a laser-induced fluorescence diagnostic is measured to infer the radial and azimuthal velocities, ion fluxes, and electric fields. The naive image of a plasma exhibiting a global rotation is again invalidated in this m = 1 mode but in a different way. Contrary to the m = 2 mode, the rotation frequency of the instability is lower than the ion cyclotron frequency and ions exhibit a complex behavior with a radial outward flux inside the unstable arm and azimuthal ion fluxes always directed toward the unstable arm. The azimuthal ion velocity is close to zero inside the ionization region, whereas the radial ion velocity grows linearly with radius. The radial electric field is oriented inward inside the unstable arm and outward outside. An axial velocity perturbation is also present, indicating that contrary to the m = 2 mode, the m = 1 mode is not a flute mode. These results cannot be easily interpreted with existing theories.

Long-term evolution of electron distribution function due to nonlinear resonant interaction with whistler mode waves

Accurately modelling and forecasting of the dynamics of the Earth’s radiation belts with the available computer resources represents an important challenge that still requires significant advances in the theoretical plasma physics field of wave–particle resonant interaction. Energetic electron acceleration or scattering into the Earth’s atmosphere are essentially controlled by their resonances with electromagnetic whistler mode waves. The quasi-linear diffusion equation describes well this resonant interaction for low intensity waves. During the last decade, however, spacecraft observations in the radiation belts have revealed a large number of whistler mode waves with sufficiently high intensity to interact with electrons in the nonlinear regime. A kinetic equation including such nonlinear wave–particle interactions and describing the long-term evolution of the electron distribution is the focus of the present paper. Using the Hamiltonian theory of resonant phenomena, we describe individual electron resonance with an intense coherent whistler mode wave. The derived characteristics of such a resonance are incorporated into a generalized kinetic equation which includes non-local transport in energy space. This transport is produced by resonant electron trapping and nonlinear acceleration. We describe the methods allowing the construction of nonlinear resonant terms in the kinetic equation and discuss possible applications of this equation.

An efficient high-order finite difference gas-kinetic scheme for the Euler and Navier–Stokes equations

Based on the temporal evolution of the Navier-Stokes gas distribution function and Weighted Essential Non-Oscillatory (WENO) interpolation, a high-order finite difference gas-kinetic scheme (FDGKS) is constructed. Different from the previous high-order finite volume gas-kinetic method [Li, Xu, and Fu, J. Comput. Phys. vol. 229, pp. 6715 (2010)], which uses a discontinuous initial reconstruction at the cell interface, the present scheme is a finite-difference one with a continuous flow distribution at the grid point.The time-accurate solution of the gas distribution function permits the FDGKS to be a one-step high-order scheme without multi-step Runge-Kutta temporal matching, which significantly reduces the computational time. Many numerical tests in solving one and two-dimensional Euler and Navier-Stokes equations demonstrate that FDGKS is a highly stable, accurate, and efficient scheme, which captures discontinuities without oscillations.

Turbulence and cloud droplets in cumulus clouds

In this paper, we report on the successful and seamless simulation of turbulence and the evolution of cloud droplets to raindrops over 10 minutes from microscopic viewpoints by using direct numerical simulation. Included processes are condensation–evaporation, collision–coalescence of droplets with hydrodynamic interaction, Reynolds number dependent drag, and turbulent flow within a parcel that is ascending within a self-consistently determined updraft inside a cumulus cloud. We found that the altitude and the updraft velocity of the parcel, the mean supersaturation, and the liquid water content are insensitive to the turbulence intensity, and that when the turbulence intensity increases, the droplet number density swiftly decreases while the spectral width of droplets rapidly increases. This study marks the first time the evolution of the mass density distribution function has been successfully calculated from microscopic computations. The turbulence accelerated to form a second peak in the mass density distribution function, leading to the raindrop formation, and the radius of the largest drop was over 300 μm at the end of the simulation. We also found that cloud droplets modify the turbulence in a way that is unlike the Kolmogorov–Obukhov–Corrsin theory. For example, the temperature and water vapor spectra at low wavenumbers become shallower than in the inertial-convective range, and decrease slower than exponentially in the diffusive range. This spectra modification is explained by nonlinear interactions between turbulent mixing and the evaporation–condensation process associated with large numbers of droplets.

Elastic scattering of a quark from a color field: Longitudinal momentum exchange

Perturbative QCD in the small Bjorken $x$ limit can be formulated as an effective theory known as the Color Glass Condensate (CGC) formalism. The CGC formalism takes into account the dynamics of large gluon densities at small $x$ and has been successfully applied to Deep Inelastic Scattering (DIS) and particle production in high energy hadronic and nuclear collisions in the small $x$ kinematic region. The effective degrees of of freedom in CGC are Wilson lines which enter in the effective quark (and gluon) propagators and re-sum multiple soft scatterings from the small $x$ gluon field of the target. It is however known that the CGC effective theory breaks down when one probes the moderately large $x$ (high $p_t$) kinematics where collinear factorization and DGLAP evolution of parton distribution functions should be the right framework. Here we propose a general framework which may allow one to eventually unify the two approaches and to calculate pQCD cross sections in both small and large Bjorken $x$ regions. We take the first step towards this goal by deriving an expression for the quark propagator in a background field which includes scatterings from both small and large $x$ modes of the gluon field of the target. We describe how this quark propagator can be used to calculate QCD structure functions $F_2$ and $F_L$ at all $x$ and thus generalize the dipole model of DIS. We outline this approach can also be used to extend the so-called hybrid approach to particle production in the forward rapidity region of high energy hadronic and nuclear collisions to all $x$ and $p_t$ regions and speculate on how one may apply the same techniques to extend the McLerran-Venugopalan effective action used in high energy heavy ion collisions to include high $p_t$ physics.

NNLO solution of nonlinear GLR–MQ evolution equation to determine gluon distribution function using Regge like ansatz

In this work we have suggested a solution of the Gribov–Levin–Ryskin, Mueller–Qiu (GLR–MQ) nonlinear evolution equation at next-to-next-to-leading order (NNLO). The range of Q2 in which we have solved the GLR–MQ equation is Regge region of the range 6.5 GeV2≤Q2≤25GeV2 and so we have incorporated the Regge like behavior to obtain Q2 evolution of gluon distribution function G(x,Q2). We have also checked the sensitivity of our results for different values of correlation radius (R) between two interacting gluons, viz. R=2GeV−1 and R=5GeV−1 as well as for different values of Regge intercept λG. Our results are compared with those of most recent global DGLAP fits obtained by various parametrization groups viz. PDF4LHC15, NNPDF3.0, HERAPDF15, CT14 and ABM12.

Anomalous temperature dependence in the structural organization of charge alternation in imidazolium-based ionic liquids of various alkyl chain lengths

We report on trends observed in the X-ray static structure factor obtained for four homologous imidazolium-based ionic liquids. In a series of low-energy (laboratory) and high-energy synchrotron X-ray diffraction experiments, the temperature dependent structure was investigated in 1-alkyl-3-methylimidazolium cations, with chain lengths of n=4, 6, 8, and 10, paired with the trifluoromethylsulfonate (TF) anion. Interestingly, the charge alternation peak in the liquids displays an anomalous temperature evolution, increasing in amplitude as the temperature is raised. Analysis of the pair distribution function (PDF) evolution with temperature for all compositions attributes the observed anomalous behavior to a specific distance in the liquid. Extensive peak modelling of the structure factors confirms the anomaly for all compositions, which combined with the analysis of the temperature of the PDFs, suggests that the entropic tendency to decohere strict charge alternation overcomes enthalpic tendencies in these liquids.

Detailed modeling of CO2 addition effects on the evolution of soot particle size distribution functions in premixed laminar ethylene flames

This study investigates computationally the influence of CO2 addition on the sooting behavior in premixed laminar ethylene/ oxygen/ argon burner stabilized stagnation (BSS) flames at the atmospheric pressure. The discrete sectional aerosol dynamics method combined with a reversible nucleation model and a novel model of reversible polycyclic aromatic hydrocarbon (PAH) condensation were employed to predict the size evolution of the particle size distribution (PSD) function. The predicted temperature profiles and PSD functions are in reasonably good agreement with the experimental data for nascent soot measured in the BSS configuration. The evolution of the PSD functions shows that CO2 addition reduces the soot nucleation and mass growth rates, consequently lowering the soot yield. The addition of CO2 reduces the concentrations of H, C2H2, C6H6, and large PAHs (e.g., pyrene) which all suppress the soot formation process through a chemical effect; the thermal effect of CO2 is not as strong as the chemical effect.

Cloud Structure of Three Galactic Infrared Dark Star-forming Regions from Combining Ground- and Space-based Bolometric Observations

Abstract We have modified the iterative procedure introduced by Lin et al., to systematically combine the submillimeter images taken from ground-based (e.g., CSO, JCMT, APEX) and space (e.g., Herschel, Planck) telescopes. We applied the updated procedure to observations of three well-studied Infrared Dark Clouds (IRDCs): G11.11−0.12, G14.225−0.506, and G28.34+0.06, and then performed single-component, modified blackbody fits to each pixel to derive ∼10″ resolution dust temperature and column density maps. The derived column density maps show that these three IRDCs exhibit complex filamentary structures embedded with rich clumps/cores. We compared the column density probability distribution functions (N-PDFs) and two-point correlation (2PT) functions of the column density field between these IRDCs with several OB-cluster-forming regions. Based on the observed correlation between the luminosity-to-mass ratio and the power-law index of the N-PDF, and complementary hydrodynamical simulations for a 104 molecular cloud, we hypothesize that cloud evolution can be better characterized by the evolution of the (column) density distribution function and the relative power of dense structures as a function of spatial scales, rather than merely based on the presence of star-forming activity. An important component of our approach is to provide a model-independent quantification of cloud evolution. Based on the small analyzed sample, we propose four evolutionary stages, namely, cloud integration, stellar assembly, cloud pre-dispersal, and dispersed cloud. The initial cloud integration stage and the final dispersed cloud stage may be distinguished from the two intermediate stages by a steeper than −4 power-law index of the N-PDF. The cloud integration stage and the subsequent stellar assembly stage are further distinguished from each other by the larger luminosity-to-mass ratio (>40 ) of the latter. A future large survey of molecular clouds with high angular resolution may establish more precise evolutionary tracks in the parameter space of N-PDF, 2PT function, and luminosity-to-mass ratio.

Size evolution of soot particles from gasoline and n-heptane/toluene blend in a burner stabilized stagnation flame

The evolution of soot particle size distribution function (PSDF) in premixed flames of gasoline (34% aromatics by volume) and a n-heptane/toluene blend (66% n-heptane/34% toluene by volume) was investigated in the burner stabilized stagnation (BSS) flame configuration, using the micro-orifice probe sampling and scanning mobility particle sizer (SMPS). The aim of this study is to illustrate the similarities and differences of sooting propensity between the real fuel and the simple hydrocarbon blend in premixed flame conditions. The mole ratio of carbon to oxygen (C/O) in the unburned gas was kept constant at 0.6 and similar maximum flame temperatures and temperature-time histories were kept between the two cases, so that we could focus on the fuel composition effects on sooting propensity. In addition, the size distribution, the total number density, and the volume fraction of soot were also compared to those previously measured for ethylene and propene flames under comparable conditions. It was observed that the particle size distributions of both gasoline and heptane/toluene flames evolve from the unimodal distribution (nucleation mode only) to the bimodal (both nucleation and coagulation mode) distribution. Compared to the heptane/toluene blend, the soot formation in gasoline flame features more persistent nucleation and much faster growth rate.

Extracting the QCD ΛMS¯ parameter in Drell–Yan process using Collins–Soper–Sterman approach

In this work, we directly fit the QCD dimensional transmutation parameter, [Formula: see text], to experimental data of Drell–Yan (DY) observables. For this purpose, we first obtain the evolution of transverse momentum dependent parton distribution functions (TMDPDFs) up to the next-to-next-to-leading logarithm (NNLL) approximation based on Collins–Soper–Sterman (CSS) formalism. As is expecting the TMDPDFs are appearing at larger values of transverse momentum by increasing the energy scales and also the order of approximation. Then we calculate the cross-section related to the TMDPDFs in the DY process. As a consequence of global fitting to the five sets of experimental data at different low center-of-mass energies and one set at high center-of-mass energy, using CETQ06 parametrizations as our boundary condition, we obtain [Formula: see text] MeV corresponding to the renormalized coupling constant [Formula: see text] which is within the acceptable range for this quantity. The goodness of [Formula: see text] shows the results for DY cross-section are in good agreement with different experimental sets, containing E288, E605 and R209 at low center-of-mass energies and [Formula: see text], CDF data at high center-of-mass energy. The repeated calculations, using HERAPDFs parametrizations is yielding us numerical values for fitted parameters very close to what we obtain using CETQ06 PDFs set. This indicates that the obtained results have enough stability by variations in the boundary conditions.

Heavy Flavour Production as Probe of Gluon Sivers Function

Heavy flavour production like $J/\psi$ and $D$- meson production in scattering of electrons/unpolarized protons off polarized proton target offer promising probes to investigate gluon Sivers function. In this talk, I will summarize our recent work on trasverse single spin asymmetry in $J/\psi $ -production and $D$ - meson production in $p p^\uparrow$ scattering using a generalized parton model approach. We compare predictions obtained using different models of gluon Sivers function within this approach and then, taking into account the transverse momentum dependent evolution of the unpolarized parton distribution functions and gluon Sivers function, we study the effect of evolution on asymmetry.

Constraining nuclear data via cosmological observations: Neutrino energy transport and big bang nucleosynthesis

We introduce a new computational capability that moves toward a self-consistent calculation of neutrino transport and nuclear reactions for big bang nucleosynthesis (BBN). Such a self-consistent approach is needed to be able to extract detailed information about nuclear reactions and physics beyond the standard model from precision cosmological observations of primordial nuclides and the cosmic microwave background radiation. We calculate the evolution of the early universe through the epochs of weak decoupling, weak freeze-out and big bang nucleosynthesis (BBN) by simultaneously coupling a full strong, electromagnetic, and weak nuclear reaction network with a multi-energy group Boltzmann neutrino energy transport scheme. The modular structure of our approach allows the dissection of the relative contributions of each process responsible for evolving the dynamics of the early universe. Such an approach allows a detailed account of the evolution of the active neutrino energy distribution functions alongside and self-consistently with the nuclear reactions and entropy/heat generation and 'ow between the neutrino and photon/electron/positron/baryon plasma components. Our calculations reveal nonlinear feedback in the time evolution of neutrino distribution functions and plasma thermodynamic conditions. We discuss the time development of neutrino spectral distortions and concomitant entropy production and extraction from the plasma. These e↑ects result in changes in the computed values of the BBN deuterium and helium-4 yields that are on the order of a half-percent relative to a baseline standard BBN calculation with no neutrino transport. This is an order of magnitude larger e↑ect than in previous estimates. For particular implementations of quantum corrections in plasma thermodynamics, our calculations show a 0.4% increase in deuterium and a 0.6% decrease in 4 He over our baseline. The magnitude of these changes are on the order of uncertainties in the nuclear physics for the case of deuterium and are potentially signi↓cant for the error budget of helium in upcoming cosmological observations.

Evolution equations for the double parton distributions. Initial conditions and transverse momentum dependence

In the first part of this contribution we discuss the problem of the initial conditions for the evolution of double parton distribution functions (PDFs). We show that one can construct a framework based on the expansion in terms of the Dirichlet functions in which both single and double PDFs satisfy momentum sum rules. In the second part, we propose how to include the transverse momentum dependence for the double parton distribution functions using the extension of the Kimber-Martin-Ryskin framework previously applied to the single PDFs.