Nonlocal Kinetic Equation Research Articles

A Hamilton–Jacobi approach to nonlocal kinetic equations

A Hamilton–Jacobi approach to nonlocal kinetic equations

About the measurement of electric field and electron temperature by the spectroscopic method in a gas mixture

The paper is dedicated to the advance of the spectroscopic method for measuring the electric fields and electron temperatures in a mixture of inert gases with significantly different excitation and ionization potentials. The method based on the measurement of the ratio of the spectral line intensities was applied to the Ne–Kr discharge. Plasma parameters were evaluated based on the solution of the non-local Boltzmann kinetic equation for the Ne–Kr mixture at low pressures and currents. Electric fields and electron temperatures were obtained as functions of the discharge current, total gas pressure, and density of the Kr admixture. The discharge current varied in the range from 1 to 5 mA, the pressure from 0.27 to 1 Torr, and the Kr admixture from 0.5% to 11% from the total gas pressure. The results demonstrate the possibility of using the described technique for diagnostics of dusty plasmas and other plasma objects operating on gas mixtures with significantly different excitation and ionization potentials.

Macroscopic limits of non-local kinetic descriptions of vehicular traffic

We study the derivation of macroscopic traffic models out of optimal speed and follow-the-leader particle dynamics as hydrodynamic limits of non-local Povzner-type kinetic equations. As a first step, we show that optimal speed vehicle dynamics produce a first order macroscopic model with non-local flux. Next, we show that non-local follow-the-leader vehicle dynamics have a universal macroscopic counterpart in the second order Aw-Rascle-Zhang traffic model, at least when the non-locality of the interactions is sufficiently small. Finally, we show that the same qualitative result holds also for a general class of follow-the-leader dynamics based on the headway of the vehicles rather than on their speed. We also investigate the correspondence between the solutions to particle models and their macroscopic limits by means of numerical simulations.

The Kinetics of Sorption-Desorption Phenomena: Local and Non-Local Kinetic Equations.

The kinetics of adsorption phenomena are investigated in terms of local and non-local kinetic equations of the Langmuir type. The sample is assumed in the shape of a slab, limited by two homogeneous planar-parallel surfaces, in such a manner that the problem can be considered one-dimensional. The local kinetic equations in time are analyzed when both saturation and non-saturation regimes are considered. These effects result from an extra dependence of the adsorption coefficient on the density of adsorbed particles, which implies the consideration of nonlinear balance equations. Non-local kinetic equations, arising from the existence of a time delay characterizing a type of reaction occurring between a bulk particle and the surface, are analyzed and show the existence of adsorption effects accompanied by temporal oscillations.

Semiclassical Approach to the Nonlocal Kinetic Model of Metal Vapor Active Media

A semiclassical approach based on the WKB–Maslov method is developed for the kinetic ionization equation in dense plasma with approximations characteristic of metal vapor active media excited by a contracted discharge. We develop the technique for constructing the leading term of the semiclassical asymptotics of the Cauchy problem solution for the kinetic equation under the supposition of weak diffusion. In terms of the approach developed, the local cubic nonlinear term in the original kinetic equation is considered in a nonlocal form. This allows one to transform the nonlinear nonlocal kinetic equation to an associated linear partial differential equation with a given accuracy of the asymptotic parameter using the dynamical system of moments of the desired solution of the equation. The Cauchy problem solution for the nonlinear nonlocal kinetic equation can be obtained from the solution of the associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation. Within the developed approach, the plasma relaxation in metal vapor active media is studied with asymptotic solutions expressed in terms of higher transcendental functions. The qualitative analysis of such the solutions is given.

Probability-free relativistic kinetic theory of classical systems of charged particles

In the complete system of equations of evolution of the classical system of charges and the electromagnetic field generated by them, the field variables are excluded. An exact closed relativistic non-Hamiltonian system of nonlocal kinetic equations, that describes the evolution of a system of charges in terms of their microscopic distribution functions, is obtained. The solutions of this system of equations are non-invariant with respect to time reversal, and also have the property of hereditarity.

Schauder estimates for nonlocal kinetic equations and applications

In this paper we develop a new method based on Littlewood-Paley's decomposition and heat kernel estimates in integral form, to establish Schauder's estimate for the following degenerate nonlocal equation in R2d with Hölder coefficients:∂tu=Lκ;v(α)u+b⋅∇u+f,u0=0, where u=u(t,x,v) and Lκ;v(α) is a nonlocal α-stable-like operator with α∈(1,2) and kernel function κ, which acts on the variable v. As an application, we show the strong well-posedness to the following degenerate stochastic differential equation with Hölder drift b:dZt=b(t,Zt)dt+(0,σ(t,Zt)dLt(α)),Z0=(x,v)∈R2d, where Lt(α) is a d-dimensional rotationally invariant and symmetric α-stable process with α∈(1,2), and b:R+×R2d→R2d is a (γ,β)-order Hölder continuous function in (x,v) with γ∈(2+α2(1+α),1) and β∈(1−α2,1), σ:R+×R2d→Rd⊗Rd is a Lipschitz function. Moreover, we also show that for almost all ω, the following random transport equation has a unique Cb1-solution:∂tu(t,x,ω)+(b(t,x)+Lt(α)(ω))⋅∇xu(t,x,ω)=0,u(0,x)=φ(x), where φ∈Cb1(Rd) and b:R+×Rd→Rd is a bounded continuous function of (t,x) and γ-order Hölder continuous in x uniformly in t with γ∈(2+α2(1+α),1).

Theoretical research on the transport and ionization rate coefficients in glow discharge dusty plasma

The electron kinetic model for investigating the transport and ionization rate coefficients of argon glow discharge dusty plasma is developed from the Boltzmann equation. Both of the electron–neutral and electron–dust collisions are considered as collision terms in the kinetic equation. The kinetic equation is simplified by employing the local approximation and nonlocal approach under the same discharge conditions, and the corresponding simplified kinetic equations are known as local and nonlocal kinetic equations respectively. Then the electron energy distribution function (EEDF) is obtained by numerically solving the local and nonlocal kinetic equations and the dust charging equations simultaneously. Based on the obtained EEDFs, the effective electron temperature, electron mobility, electron diffusion coefficient and ionization rate coefficient are calculated for different discharge conditions. It is shown that the EEDFs calculated from the local kinetic model clearly differ from the nonlocal EEDFs and both the local and nonlocal EEDFs are also clearly different with Maxwellian distributions. The appearance of dust particles results in an obvious decrease of high energy electrons and increase of low energy electrons when axial electric field is low. With the increase of axial electric field, the influence of dust particles on the EEDFs becomes smaller. The electron mobility and diffusion coefficients calculated on the basis of local and nonlocal EEDFs do not differ greatly to the dust-free ones. While, when dust density nd = 106 cm−3, the electron mobility increases obviously compared with the dust-free results at low axial electric field and decreases with the increasing axial electric field until they are close to the dust-free ones. Meanwhile, electron diffusion coefficients for dusty case become smaller and decrease with the increasing axial electric field. The ionization rate coefficients decrease when dust particles are introduced and they approach the dust-free results gradually with the increasing axial electric field.

Mean-field limit of a spatially-extended FitzHugh-Nagumo neural network

We consider a spatially-extended model for a network of interacting FitzHugh-Nagumo neurons without noise, and rigorously establish its mean-field limit towards a nonlocal kinetic equation as the number of neurons goes to infinity. Our approach is based on deterministic methods, and namely on the stability of the solutions of the kinetic equation with respect to their initial data. The main difficulty lies in the adaptation in a deterministic framework of arguments previously introduced for the mean-field limit of stochastic systems of interacting particles with a certain class of locally Lipschitz continuous interaction kernels. This result establishes a rigorous link between the microscopic and mesoscopic scales of observation of the network, which can be further used as an intermediary step to derive macroscopic models. We also propose a numerical scheme for the discretization of the solutions of the kinetic model, based on a particle method, in order to study the dynamics of its solutions, and to compare it with the microscopic model.

Hölder Continuity for a Family of Nonlocal Hypoelliptic Kinetic Equations

In this work, Holder continuity is obtained for solutions to the nonlocal kinetic Fokker--Planck equation and to a family of related equations with general integro-differential operators. These equ...

Ambipolar field role in formation of electron distribution function in gas discharge plasma

It is shown that the local approximation for electron distribution function (EDF) determination at plasma periphery, where the ambipolar field is dominant, is not applicable even at high pressures when the characteristic plasma size exceeds the energy relaxation length of the electrons R > λε. Therefore, consistent results can be obtained only when solving the complete kinetic equation in both energy and spatial variables (i.e. it is necessary to solve nonlocal kinetic equation).

Nonequilibrium thermodynamics with binary quantum correlations.

The balance equations for thermodynamic quantities are derived from the nonlocal quantum kinetic equation. The nonlocal collisions lead to molecular contributions to the observables and currents. The corresponding correlated parts of the observables are found to be given by the rate to form a molecule multiplied with its lifetime which can be considered as collision duration. Explicit expressions of these molecular contributions are given in terms of the scattering phase shifts. The two-particle form of the entropy is derived extending the Landau quasiparticle picture by two-particle molecular contributions. There is a continuous exchange of correlation and kinetic energies condensing into the rate of correlated variables for energy and momentum. For the entropy, an explicit gain remains and Boltzmann's H theorem is proved including the molecular parts of the entropy.

On Rosenau-type approximations to fractional diffusion equations

Owing to Rosenau argument [28], originally proposed to obtain a regularized version of the Chapman-Enskog expansion of hydrodynamics, we introduce a non-local linear kinetic equation which approximates a fractional diffusion equation. We then show that the solution to this approximation, apart of a rapidly vanishing in time perturbation, approaches the fundamental solution of the fractional diffusion (a Levy stable law) at large times.

On an aggregation in birth-and-death stochastic dynamics

We consider birth-and-death stochastic dynamics of particle systems with attractive interaction. The heuristic generator of the dynamics has a constant birth rate and density-dependent decreasing death rate. The corresponding statistical dynamics is constructed. Using the Vlasov-type scaling we derive the limiting mesoscopic evolution and prove that this evolution propagates chaos. We study a nonlinear non-local kinetic equation for the first correlation function (density of population). The existence of uniformly bounded solutions as well as solutions growing inside of a bounded domain and expanding in the space are shown. These solutions describe two regimes in the mesoscopic system: regulation and aggregation.

A nonlocal kinetic model for predator–prey interactions

We extend the aggregation model from Fetecau (2011) by adding a field of vision to individuals and by including a second species. The two species, assumed to have a predator–prey relationship, have dynamics governed by nonlocal kinetic equations that include advection and turning. The latter is the main mechanism for aggregation and orientation, which results from interactions among individuals of the same species as well as predator–prey relationships. We illustrate numerically a diverse set of predator–prey behaviors that can be captured by this model. We show that a prey’s escape outcome depends on the social interactions between its group members, the prey’s field of vision and the sophistication of the predator’s hunting strategies.

Generalized hydrodynamic reductions of the kinetic equation for a soliton gas

We derive generalized multiflow hydrodynamic reductions of the nonlocal kinetic equation for a soliton gas and investigate their structure. These reductions not only provide further insight into the properties of the new kinetic equation but also could prove to be representatives of a novel class of integrable systems of hydrodynamic type beyond the conventional semi-Hamiltonian framework.

Diffuse and constricted modes of a dc discharge in neon: Simulation of the hysteresis transition

Results are presented from theoretical studies of high-pressure (∼100 Torr) dc discharges in neon. The diffuse and constricted discharge modes are studied using a model including the equation of balance for charged and excited particles, heat conduction equations for the neutral gas and plasma electrons, and Poisson’s equation for the radial electric field at a fixed total discharge current. A specific feature of the constricted mode in the investigated range of low fields and high degrees of ionization is that the excitation and ionization rates in the center of the discharge tube and at the periphery differ by several orders of magnitude. This implies that, in the constricted mode, the region where the electron energy distribution function is Maxwellian due to electron-electron collisions may adjoin the region (beyond the constriction zone) where the high-energy part of the distribution function is depleted. The hysteresis transition between the diffuse and constricted modes is analyzed. A transition from the constricted to the diffuse mode can be regarded as a manifestation of the nonlocal character of the formation of the electron distribution function, specifically, the diffusion of high-energy electrons capable of producing gas ionization from the central (constricted) region toward the periphery. The nonlocal formation of the distribution function is described by a nonlocal kinetic equation accounting for electron-electron collisions and electron transport along the radius of the discharge tube. Since only high-energy electrons produce gas ionization, the effect of the nonlocal formation of the electron distribution function is taken into account by introducing the effective temperature of the high-energy part of the distribution function and solving the equation for the radial profile of the high-energy part of the distribution function. This approach allows one to approximately take into account the nonlocal character of the electron distribution function without substantial expenditure of computer resources. The nonlocal model makes it possible to numerically simulate the hysteresis transition between the diffuse and constricted modes, which is impossible in the local approximation.

10.1007/s11823-008-1008-y

The stratification of a positive column of a low-pressure glow discharge in inert gases has been studied with the help of a self-consistent hybrid model. The model is based on the solution of a nonlocal kinetic Boltzmann equation for the electron distribution function, a nonstationary drift-diffusion equation for the ions, and the Poisson equation for the electric field. Spatial electron and ion density distributions and the electric field distribution in the positive column were obtained. The converged solution of the model gives a self-consistent resonant strata length L and the value and the form of the modulated plasma parameters. An unexpected surprising result was obtained: for a given potential drop in the positive column of a low-pressure glow discharge, a self-consistent spatially modulated striation-like electric field does not lead to the resonant increasing of the ionization frequencies in the discharge as compared with a constant electric field with the same potential drop. Usually, it was assumed that, in spatially modulated field distributions, all the parameters in a striated plasma will be more pronounced and have a resonant form.

Spatiotemporal evolution of the distribution function of electrons in sign-changing electric field

A numerical model for solving the Boltzmann unsteady non-local kinetic equation for the distribution function of electrons over energy is constructed. The Boltzmann equation for isotropic part of the distribution function written in natural variables the kinetic energy — the coordinate was solved by the pseudo-unsteady method. The model was applied for describing the spatiotemporal evolution of the distribution function of electrons in a uniform electric field. For a model distribution of the electric field with the “negative” value in the Faraday dark space and the “positive” value in the positive column of the glow discharge, the main macroscopic parameters of electrons are obtained, the diffusion mechanism of the electron current transfer in the negative electric field region is confirmed.

Convergence to equilibrium of a multiscale model for suspensions

We consider a multiscale model describing the flow of a concentrated suspension. The model couples the macroscopic equation of conservation of momentum with a nonlinear nonlocal kinetic equation describing at the microscopic level the rheological behaviour of the fluid. We study the long-time limit of the time-dependent solution. For this purpose, we use the entropy method to prove the convergence to equilibrium of the kinetic equation.