Exact Kinetic Equation Research Articles

Моделирование электропроводности ионизированного газа на основе численного анализа кинетики электронов

A weakly ionized medium formed during impact ionization of a rarefied gas by fast electrons is considered. The development of a model of radiative conductivity based on approximations of the spatial homogeneity of the distribution function of slow secondary electrons and its symmetry with respect to the direction of the electric field is presented. The equations for the concentration, drift velocity, and specific energy of slow electrons are derived from the kinetic equation. The numerical solution of the exact kinetic equation is used to construct the coefficients of the equations for the moments, as well as to test the radiative conductivity model.

Three dimensional high-order gas-kinetic scheme for supersonic isotropic turbulence II: Coarse-graining analysis of compressible Ksgs budget

The direct numerical simulation (DNS) of compressible isotropic turbulence up to the supersonic regime with Mat=1.2 has been investigated by the high-order gas-kinetic scheme (HGKS) (Cao et al. (2019) [8]). In this study, the DNS on a much higher initial turbulent Mach number up to Mat=2.0 is performed by HGKS, which confirms the super robustness of HGKS. The coarse-graining analysis of subgrid-scale (SGS) turbulent kinetic energy Ksgs budget is fully analyzed for constructing one-equation SGS model in the compressible large eddy simulation (LES). The exact compressible SGS turbulent kinetic energy Ksgs transport equation is derived with density weighted filtering process. Based on the compressible Ksgs transport equation, the coarse-graining processes are implemented on three sets of unresolved grids with the Box filter. The coarse-graining analysis of compressible Ksgs budgets shows that all unresolved source terms are dominant in the current system. Especially, the magnitude of SGS pressure-dilation term is on the order of SGS solenoidal dissipation term within the initial acoustic time scale. Therefore, the SGS pressure-dilation term cannot be neglected as that in the previous work. The delicate coarse-graining analysis of SGS diffusion terms in compressible Ksgs equation also confirms that both the fluctuation velocity triple correlation term and the pressure-velocity correlation term are dominant terms. The current analysis provides an indication on the order of magnitude of all SGS terms in compressible Ksgs budget, which provides a solid basis for compressible LES modeling of high Mach number turbulent flow.

On different kinetic approaches for computing planar gas expansion under pulsed evaporation into vacuum

The numerical study of one-dimensional gas expansion under pulsed evaporation into vacuum is carried out on the basis of the direct simulation Monte Carlo method, the exact Boltzmann kinetic equation, and the S-model kinetic equation. The results are presented for various levels of evaporation intensity, defined by the amount of evaporated material. Special attention has been paid to the calculation of the average axial energy of particles, the velocity vector of which deviates from the axis by no more than a small prescribed angle α. This characteristic of the flow is important for analysis of time-of-flight distributions in pulsed laser ablation. It is found that for intense evaporation, the average axial energy has a maximum as a function of time. The presented results allow us to establish the relative accuracy of the considered kinetic approaches for various flow regimes.

Derivation of lattice Boltzmann equation via analytical characteristic integral**Project supported by the National Science and Technology Major Project, China (Grant No. 2017ZX06002002).

A lattice Boltzmann (LB) theory, the analytical characteristic integral (ACI) LB theory, is proposed in this paper. ACI LB theory takes the Bhatnagar–Gross–Krook (BGK)-Boltzmann equation as the exact kinetic equation behind Navier–Stokes continuum and momentum equations and constructs an LB equation by rigorously integrating the BGK-Boltzmann equation along characteristics. It is a general theory, supporting most existing LB equations including the standard lattice BGK (LBGK) equation inherited from lattice-gas automata, whose theoretical foundation had been questioned. ACI LB theory also indicates that the characteristic parameter of an LB equation is collision number, depicting the particle-interaction intensity in the time span of the LB equation, instead of the traditionally assumed relaxation time, and the over-relaxation time problem is merely a manifestation of the temporal evolution of equilibrium distribution along characteristics under high collision number, irrelevant to particle kinetics. In ACI LB theory, the temporal evolution of equilibrium distribution along characteristics is the determinant of LB method accuracy and we numerically prove this.

Macroscopic Irreversibility and Decay to Kinetic Equilibrium of the 1-Body PDF for Finite Hard-Sphere Systems

The conditions for the occurrence of the so-called macroscopic irreversibility property and the related phenomenon of decay to kinetic equilibrium which may characterize the 1-body probability density function (PDF) associated with hard-sphere systems are investigated. The problem is set in the framework of the axiomatic “ab initio” theory of classical statistical mechanics developed recently and the related establishment of an exact kinetic equation realized by the Master equation for the same kinetic PDF. As shown in the paper the task involves the introduction of a suitable functional of the 1-body PDF, identified here with the Master kinetic information. It is then proved that, provided the same PDF is prescribed in terms of suitably smooth, i.e., stochastic, solution of the Master kinetic equation, the two properties indicated above are indeed realized.

Vibration and stability analysis of a simply-supported Rayleigh beam with spinning and axial motions

The vibration and stability of a simply supported beam are analyzed when the beam has an axially moving motion as well as a spinning motion. When a beam has spinning and axial motions, rotary inertia plays an important role on the lateral vibration. Compared to previous studies, the present study adopts the Rayleigh beam theory and derives more exact kinetic energy and equations of motion. The rotary inertia terms derived by the present study are compared to those of the previous studies. We investigate the natural frequencies between the present and previous studies. In addition, the critical speed and stability boundary for the spinning and moving speeds are also analyzed. It can be observed from the computed natural frequencies and dynamic responses that the present equations of motion are more reliable than the previous equations because the present equations fully consider the rotary inertia terms.

The electrical conductance growth of a metallic granular packing

We report on measurements of the electrical conductivity on a two-dimensional packing of metallic disks when a stable current of ~1 mA flows through the system. At low applied currents, the conductance σ is found to increase by a pattern σ(t) = σ ∞ − Δσ E α [ − (t/τ) α ], where E α denotes the Mittag-Leffler function of order α ∈ (0,1). By changing the inclination angle θ of the granular bed from horizontal, we have studied the impact of the effective gravitational acceleration g eff = gsinθ on the relaxation features of the conductance σ(t). The characteristic timescale τ is found to grow when effective gravity g eff decreases. By changing both the distance between the electrodes and the number of grains in the packing, we have shown that the long term resistance decay observed in the experiment is related to local micro-contacts rearrangements at each disk. By focusing on the electro-mechanical processes that allow both creation and breakdown of micro-contacts between two disks, we present an approach to granular conduction based on subordination of stochastic processes. In order to imitate, in a very simplified way, the conduction dynamics of granular material at low currents, we impose that the micro-contacts at the interface switch stochastically between two possible states, “on” and “off”, characterizing the conductivity of the micro-contact. We assume that the time intervals between the consecutive changes of state are governed by a certain waiting-time distribution. It is demonstrated how the microscopic random dynamics regarding the micro-contacts leads to the macroscopic observation of slow conductance growth, described by an exact fractional kinetic equations.

Asymptotic orderings and approximations of the Master kinetic equation for large hard spheres systems

In this paper the problem is posed of determining the physically-meaningful asymptotic orderings holding for the statistical description of a large N-body system of hard spheres, i.e., formed by N≡1ε≫1 particles, which are allowed to undergo instantaneous and purely elastic unary, binary or multiple collisions. Starting point is the axiomatic treatment recently developed [Tessarotto et al., 2013–2016] and the related discovery of an exact kinetic equation realized by Master equation which advances in time the 1-body probability density function (PDF) for such a system. As shown in the paper the task involves introducing appropriate asymptotic orderings in terms of ε for all the physically-relevant parameters. The goal is that of identifying the relevant physically-meaningful asymptotic approximations applicable for the Master kinetic equation, together with their possible relationships with the Boltzmann and Enskog kinetic equations, and holding in appropriate asymptotic regimes. These correspond either to dilute or dense systems and are formed either by small-size or finite-size identical hard spheres, the distinction between the various cases depending on suitable asymptotic orderings in terms of ε.

Nonlinear vortex-phonon interactions in a Bose–Einstein condensate

We consider the nonlinear coupling between an exact vortex solution in a Bose–Einstein condensate and a spectrum of elementary excitations in the medium. These excitations, or Bogoliubov–de Gennes modes, are indeed a special kind of phonons. We treat the spectrum of elementary excitations in the medium as a gas of quantum particles, sometimes also called bogolons. An exact kinetic equation for the bogolon gas is derived, and an approximate form of this equation, valid in the quasi-classical limit, is also obtained. We study the energy transfer between the vortex and the bogolon gas, and establish conditions for vortex instability and damping.

The Master kinetic equation for the statistical treatment of the Boltzmann-Sinai classical dynamical system

In this investigation, exact particular realizations are sought for the microscopic statistical description which is associated with the classical dynamical system (CDS) formed by N identical smooth hard spheres subject to elastic collisions (S N -CDS). The problem is posed in the framework of the ab initio statistical description of S N -CDS recently developed. It is shown that the Liouville equation associated with SN-CDS admits an exact particular solution for the N-body probability density function (PDF). This is factorized in terms of the i-th particle 1-body PDF (for all i = 1, N) via suitable weighting factors, which are denoted here as particle occupation coefficients. The latter are found to depend functionally only on the 1-body PDFs which are associated with each of the remaining particles belonging to S N -CDS. Furthermore, the 1-body PDF is proved to obey a well-defined statistical equation, referred to here as Master kinetic equation. This is an exact kinetic equation which takes into account the occurrence of configuration-space correlations due to the finite size of the extended particles, while depending functionally on the same 1-body PDF only. The asymptotic approximation of the Master equation, which holds in validity of the Boltzmann-Grad limit, is shown to recover in a suitable asymptotic sense the customary Boltzmann equation. Finally, a critical analysis is presented of the original and modified versions of the Enskog kinetic equation, as well as of some of the non-linear kinetic approaches formulated in the past for dense granular gases. Their conditions of validity and main differences with respect to the present theory are pointed out.

A Method for Esterification Reaction Rate Prediction of Aliphatic Monocarboxylic Acids with Primary Alcohols in 1,4‐Dioxane Based on Two Parametrical Taft Equation

ABSTRACTEsterification reaction rates of aliphatic monocarboxylic acids with primary alcohols in 1,4‐dioxane as inert solvent were investigated. Acids were esterified with 1‐propanol and alcohols with acetic acid as model reactants at a constant temperature of 60°C, at a fixed ionic strength and pH in a batch reactor with a constant volume. For evaluation of reaction rates, an exact kinetic equation for the equilibrium reaction was applied. Under these conditions and for low reactants, concentrations reaction rate depends only on the structure of reactants and, therefore, can be predicted by a correlation equation with two Taft coefficients (inductive and steric effects). From these equations, it is possible to estimate the esterification reaction rate constant for other acid‐alcohol pairs. This methodology may also be suitable for other kinetic systems measured under comparable experimental conditions.

Singularity of Inertial Particle Concentration in the Viscous Sublayer of Wall-bounded Turbulent Flows

The exact kinetic equation for probability density function (PDF) of the velocity and the position of inertial particle transported by turbulent non-Gaussian fluid velocity fields in the viscous sublayer of wall-bounded turbulent flow is analyzed by the method of matched asymptotic expansions. It is shown that the particle concentration near the wall exhibits a power-law singularity giving rise to the phenomenon of particle accumulation. It is shown how the corresponding exponent depends upon the particle Stokes number. The result is in good agreement with previously published results of numerical simulations. A corresponding singularity is found for the standardized higher-order moments of particle velocity.

Fractional kinetic model for granular compaction

We present an approach to granular compaction based on subordination of stochastic processes. In order to imitate, in a very simplified way, the compaction dynamics of granular material under tapping, we impose that particles switch stochastically between the two possible orientational states characterizing the average volumes of the grain in the presence of other grains. The main physical idea of our approach is that the interaction of grains with their environment is taken into account with the aid of the temporal subordination. Accordingly, we assume that the time intervals between the consecutive grain’s reorientations are governed by a certain waiting-time distribution ψ(t). It is demonstrated how the presence of the trapping events leads to the macroscopic observation of slow compaction dynamics, described by an exact fractional kinetic equation. We also perform numerical simulations to examine our analytical result. In addition, we reproduce the memory effects numerically by considering the response of the system to the abrupt change in the external excitation.

Kinetics of Enzyme Action on Surface-Attached Substrates: A Practical Guide to Progress Curve Analysis in Any Kinetic Situation

In the present work, exact kinetic equations describing the action of an enzyme in solution on a substrate attached to a surface have been derived in the framework of the Michaelis-Menten mechanism but without resorting to the often-used steady-state approximation. The here-derived kinetic equations are cast in a workable format, allowing us to introduce a simple and universal procedure for the quantitative analysis of enzyme surface kinetics that is valid for any kinetic situation. The results presented here should allow experimentalists studying the kinetics of enzyme action on immobilized substrates to analyze their data in a perfectly rigorous way.

X-ray bursting neutron star atmosphere models using an exact relativistic kinetic equation for Compton scattering

Theoretical spectra of X-ray bursting neutron star (NS) model atmospheres are widely used to determine the basic NS parameters such as their masses and radii. We construct accurate NS atmosphere models using for the first time an exact treatment of Compton scattering via the integral relativistic kinetic equation. We also compare the results with the previous calculations based on the Kompaneets operator. We solve the radiation transfer equation together with the hydrostatic equilibrium equation accounting exactly for the radiation pressure by electron scattering. We thus construct a new set of plane-parallel atmosphere models in LTE for hot NSs. The models were computed for six chemical compositions (pure H, pure He, solar H/He mix with various heavy elements abundances Z = 1, 0.3, 0.1, and 0.01 Z_sun, and three log g = 14.0, 14.3, and 14.6. For each chemical composition and log g, we compute more than 26 model atmospheres with various luminosities relative to the Eddington luminosity L_Edd computed for the Thomson cross-section. The maximum relative luminosities L/L_Edd reach values of up to 1.1 for high gravity models. The emergent spectra of all models are redshifted and fitted by diluted blackbody spectra in the 3--20 keV energy range appropriate for the RXTE/PCA. We also compute the color correction factors f_c. The radiative acceleration g_rad in our luminous, hot-atmosphere models is significantly smaller than in corresponding models based on the Kompaneets operator, because of the Klein-Nishina reduction of the electron scattering cross-section, and therefore formally "super-Eddington" model atmospheres do exist. The differences between the new and old model atmospheres are small for L / L_Edd < 0.8. For the same g_rad / g, the new f_c are slightly larger (by approximately 1%) than the old values.

Dynamic -equation model for large-eddy simulation of compressible flows

AbstractThis paper presents a dynamic one-equation eddy viscosity model for large-eddy simulation (LES) of compressible flows. The transport equation for subgrid-scale (SGS) kinetic energy is introduced to predict SGS kinetic energy. The exact SGS kinetic energy transport equation for compressible flows is derived formally. Each of the unclosed terms in the SGS kinetic energy equation is modelled separately and dynamically closed, instead of being grouped into production and dissipation terms, as in the Reynolds averaged Navier–Stokes equations. All of the SGS terms in the filtered total energy equation are found to reappear in the SGS kinetic energy equation. Therefore, these terms can be included in the total energy equation without adding extra computational cost.A prioritests using direct numerical simulation (DNS) of decaying isotropic turbulence show that, for a Smagorinsky-type eddy viscosity model, the correlation between the SGS stress and the model is comparable to that from the original model. Also, the suggested model for the pressure dilatation term in the SGS kinetic energy equation is found to have a high correlation with its actual value. Ina posterioritests, the proposed dynamic$k$-equation model is applied to decaying isotropic turbulence and normal shock–isotropic turbulence interaction, and yields good agreement with available experimental and DNS data. Compared with the results of the dynamic Smagorinsky model (DSM), the$k$-equation model predicts better energy spectra at high wavenumbers, similar kinetic energy decay and fluctuations of thermodynamic quantities for decaying isotropic turbulence. For shock–turbulence interaction, the$k$-equation model and the DSM predict similar evolutions of turbulent intensities across shocks, owing to the dominant effect of linear interaction. The proposed$k$-equation model is more robust in that local averaging over neighbouring control volumes is sufficient to regularize the dynamic procedure. The behaviour of pressure dilatation and dilatational dissipation is discussed through the budgets of the SGS kinetic energy equation, and the importance of the dilatational dissipation term is addressed.

Generalized thermodynamic and transport properties. I. Simple liquids

We propose a method by which the generalized transport properties and coefficients at all wavelengths and frequencies can be obtained by inversion of an exact kinetic equation. The necessary data are the density-density, energy-energy, and density-energy time correlation functions, which can be obtained by molecular-dynamics simulation. In addition, also the coupling between viscous stress tensor and energy flux vector can be obtained without approximation. This allows one to check the validity of the Markov assumption in a straightforward way. As a first test case, the theory is applied to liquid argon in two thermodynamic states. For this system, we calculate and discuss generalized thermodynamic (enthalpy, specific heats, and thermal expansion) and transport properties (longitudinal viscosity, thermal conductivity).

Generalized thermodynamic and transport properties. II. Molecular liquids

In the present paper, we extend the method described in paper I [D. Bertolini and A. Tani, preceding paper, Phys. Rev. E 83, 031201 (2011)] to molecular liquids, which allows us to solve the exact kinetic equation proposed by de Schepper et al. [Phys. Rev. A 38, 271 (1988)] without approximations. In particular, generalized thermodynamic properties (enthalpy, specific heat, and thermal expansion coefficient) and transport properties (longitudinal viscosity, thermal conductivity) have been calculated for three liquids of increasing complexity, namely dimethyl sulfoxide, hydrogen fluoride, and SPC/E water. All results have been obtained by the molecular formalism as well as the atomic one, corrected for intramolecular correlations that are due to the models adopted. As done for simple liquids, the coupling between the viscous stress tensor and the energy flux vector has been calculated exactly. We also show that the Markov assumption for the dynamics related to thermal conductivity can only be adopted with caution.

Mapping the reaction dynamics in Liouville space onto a reaction coordinate space: A first-principle- based theory

We have derived here an exact kinetic equation for the time evolution of the probability distribution for a general reaction coordinate space, starting from a multidimensional Liouville equation based on first-principles theory. To make the equation tractable we use two standard approximations, which reduce the exact equation into a Fokker-Planck-type equation with a sink term. As illustrative examples, we consider its application to two important classes of reactions, viz., the electron transfer and diffusion-controlled reactions. The kinetic equations for the reaction coordinates corresponding to these reactions become one dimensional. We also provide a scheme to find the critical value of the reaction coordinate at which the reaction takes place, and also to obtain the initial distribution function in the same space for different experimental situations. In the case of photoinduced electron transfer, we introduce the effect of excitation frequency through the use of suitable initial distribution functions.

A stochastic Keller–Segel model of chemotaxis

We introduce stochastic models of chemotaxis generalizing the deterministic Keller–Segel model. These models include fluctuations which are important in systems with small particle numbers or close to a critical point. Following Dean’s approach, we derive the exact kinetic equation satisfied by the density distribution of cells. In the mean field limit where statistical correlations between cells are neglected, we recover the Keller–Segel model governing the smooth density field. We also consider hydrodynamic and kinetic models of chemotaxis that take into account the inertia of the particles and lead to a delay in the adjustment of the velocity of cells with the chemotactic gradient. We make the connection with the Cattaneo model of chemotaxis and the telegraph equation.