Discrete Kinetic Equations Research Articles

A fresh perspective on solution to fragmentation problems

The population balances offer a continuum framework to model fragmentation of discrete size drops, bubbles, particles, crystals, polymer chains, etc. We have examined the available approaches to solve the resulting population balances and the older discrete kinetic equations. The analytical solutions have enhanced our understanding of breakup processes. We offer in this work new analytical solutions using a fresh and counter-intuitive approach. The approximate numerical solutions approach analytical solutions as the fineness of discretization increases. We looked for scaling patterns in coarse numerical solutions and harnessed the observed scaling to obtain closed-form analytical solutions in the limit of infinitely fine discretization. The approach provides analytical solutions for a family of breakup functions, including evolving populations of infinite number of particles. The success of the new approach is contingent on the discretization step having internal consistency concerning particle number and mass. The approach holds promise for empirical breakage functions.

On Equivalence between Kinetic Equations and Geodesic Equations in Spaces with Affine Connection

Discrete kinetic equations describing binary processes of agglomeration and fragmentation are considered using formal equivalence between the kinetic equations and the geodesic equations of some affinely connected space A associated with the kinetic equation and called the kinetic space of affine connection. The geometric properties of equations are treated locally in some coordinate chart (x;U). The peculiarity of the space A is that in the coordinates (x) of some selected local chart, the Christoffel symbols defining the affine connection of the space A are constant. Examples of the Smoluchowski equation for agglomeration processes without fragmentation and the exchange-driven growth equation are considered for small dimensions in terms of geodesic equations. When fragmentation is taken into account, the kinetic equations can be written as equations of quasigeodesics. Particular cases of spaces with symmetries are discussed.

Diffusion of Biological Organisms: Fickian and Fokker--Planck Type Diffusions

In this paper we derive diffusion equations in a heterogeneous environment. We consider a system of discrete kinetic equations that consists of two phenotypes of different turning frequencies. The ...

Pólya distribution and its asymptotics in nucleation theory

A model of condensation-decay rate constants that are linear with respect to the number of monomers in the nucleus is considered. In a particular case of stable growth, this model leads to an exact solution of discrete kinetic equations of the theory of heterogeneous nucleation in the form of the Polya distribution function. An asymptotic solution in the region of large nucleus sizes that satisfies the normalization condition and provides correct mean nucleus size has been found. It is shown that, in terms of the logarithmic invariant size, the obtained distribution has a universal time-independent form. The obtained solution, being more general than the double-exponent distribution used previously, describes both Gaussian and asymmetric distributions depending on the rate constant of condensation on a bare core. The obtained results are useful for modeling processes in some systems, in particular, the growth of linear chains, two-dimensional clusters, and filamentary nanocrystals.

Solving the discrete S-model kinetic equations with arbitrary order polynomial approximations

A numerical method for solving the model Boltzmann equation using arbitrary order polynomials is presented. The S-model is solved by a discrete ordinate method with velocity space discretized with a truncated Hermite polynomial expansion. Physical space is discretized according to the Conservative Flux Approximation scheme with extension to allow non-uniform grid spacing. This approach, which utilizes Legendre polynomials, allows the spatial representation and flux calculation to be of arbitrary order. High order boundary conditions are implemented. Various results are shown to demonstrate the utility and limitations of the method with comparison to solutions of the Euler and Navier–Stokes equations and from the Direct Simulation Monte Carlo and Unified Gas Kinetic schemes. The effect of both the velocity space discretization and Knudsen number on the convergence properties of the scheme are also investigated.

Three-dimensional finite-difference lattice Boltzmann model and its application to inviscid compressible flows with shock waves

In this paper, a three-dimensional (3D) finite-difference lattice Boltzmann model for simulating compressible flows with shock waves is developed in the framework of the double-distribution-function approach. In the model, a density distribution function is adopted to model the flow field, while a total energy distribution function is adopted to model the temperature field. The discrete equilibrium density and total energy distribution functions are derived from the Hermite expansions of the continuous equilibrium distribution functions. The discrete velocity set is obtained by choosing the abscissae of a suitable Gauss–Hermite quadrature with sufficient accuracy. In order to capture the shock waves in compressible flows and improve the numerical accuracy and stability, an implicit–explicit finite-difference numerical technique based on the total variation diminishing flux limitation is introduced to solve the discrete kinetic equations. The model is tested by numerical simulations of some typical compressible flows with shock waves ranging from 1D to 3D. The numerical results are found to be in good agreement with the analytical solutions and/or other numerical results reported in the literature.

On Nonexistence of Dissipative Estimates for Discrete Kinetic Equations

UDC 517.9, 533.7, 533.723 Our study concerns the existence of a global solution to the discrete kinetic equations in Sobolev spaces. We obtain a decomposition of the solution and clarify the influence of the oscillations generated by the interaction operator. We show that there exists a submanifold Mdiss of initial data for which dissipative solution exit. If initial data u 0 , v 0 , w 0 deviate from the submanifold Mdiss, then the interaction operator generates solitons, the nondissipative part of the solution. The amplitude of solitons is proportional to the distance from u 0 , v 0 , w 0 to the submanifold Mdiss, which means that the solution can stabilize as t →∞ only on compact sets of spatial variables. Bibliography :6 titles.

On discrete kinetic equations

On discrete kinetic equations

The existence of global solutions to the Cauchy problem for discrete kinetic equations (nonperiodic case)

We establish the existence of global solutions to discrete kinetic equations in Sobolev spaces. We also obtain an expansion of the solution and investigate the influence of oscillations generated by the interaction operator. Bibliography: 7 titles.

The existence of global solutions to the Cauchy problem for discrete kinetic equations. II

UDC 517.9 We prove the existence of global solutions to discrete kinetic equations. We also derive an expansion of the solution and investigate the influence of oscillations generated by the interaction operator. Bibliography :9 titles.

The existence of global solutions to the cauchy problem for discrete kinetic equations

UDC 517.9 We prove the existence of global solutions to discrete kinetic equations. We also derive an expansion of the solution and investigate the influence of oscillations generated by the interaction operator. Bibliography :9 titles.

A fast iterative model for discrete velocity calculations on triangular grids

A fast synthetic type iterative model is proposed to speed up the slow convergence of discrete velocity algorithms for solving linear kinetic equations on triangular lattices. The efficiency of the scheme is verified both theoretically by a discrete Fourier stability analysis and computationally by solving a rarefied gas flow problem. The stability analysis of the discrete kinetic equations yields the spectral radius of the typical and the proposed iterative algorithms and reveal the drastically improved performance of the latter one for any grid resolution. This is the first time that stability analysis of the full discrete kinetic equations related to rarefied gas theory is formulated, providing the detailed dependency of the iteration scheme on the discretization parameters in the phase space. The corresponding characteristics of the model deduced by solving numerically the rarefied gas flow through a duct with triangular cross section are in complete agreement with the theoretical findings. The proposed approach may open a way for fast computation of rarefied gas flows on complex geometries in the whole range of gas rarefaction including the hydrodynamic regime.

Existence and stability analysis for the Carleman kinetic system

A global existence theorem for the discrete Carleman system in the Sobolev class W1,2 is proved by the Leray-Schauder topological degree method, which was not previously applied to discrete kinetic equations. The instability of the nonequilibrium steady flow on a bounded interval is established in the linear approximation.

Kinetics of helium bubble formation in nuclear materials

The formation and growth of helium bubbles due to self-irradiation in plutonium has been modelled by discrete kinetic equations for the number densities of bubbles having k atoms. Analysis of these equations shows that the bubble size distribution function can be approximated by a composite of: (i) the solution of partial differential equations describing the continuum limit of the theory but corrected to take into account the effects of discreteness, and (ii) a local expansion about the advancing leading edge of the distribution function in size space. Both approximations contribute to the memory term in a close integrodifferential equation for the monomer concentration of single helium atoms. The present boundary layer theory for discrete equations is compared to the numerical solution of the full kinetic model and to the previous approximation of Schaldach and Wolfer involving a truncated system of moment equations.

Plane Waves Propagating in Gases Composed of Composite Particles

The quantum discrete kinetic equations are solved to study the propagation of plane waves in a system of composite particles with hard-sphere interactions and the filling factor (ν) being 1/2. We compare the dispersion relations thus obtained by the relevant Pauli-blocking parameter B which describes the different-statistics particles for the quantum analog of the discrete Boltzmann system when B is positive (Bose gases), zero (Boltzmann gases), and negative (Fermi Gases). We found, as the effective magnetic field being zero (ν = 1/2 using the composite fermion formulation), the electric field effect will induce anomalous dispersion relations.

Acoustic waves propagating in composite-particle gases under uniform gravitational fields

The quantum discrete kinetic equations are solved to study the propagation of plane waves in a system of composite particles under the magnetic field which is subjected to the Beltrami field condition and particles being also under uniform external gravitational field. We compare the dispersion relations thus obtained by the relevant Pauli-blocking factor $\ensuremath{\theta}$ which describes the different-statistics particles for the quantum analog of the discrete Boltzmann system when $\ensuremath{\theta}$ is positive (say, $\ensuremath{\theta}=1$ for Bose gases), zero (Boltzmann gases), and negative ($\ensuremath{\theta}=\ensuremath{-}1$ for Fermi Gases). We found, as the effect of magnetic field being zero (using the Beltrami field condition), the gravitational field effect will induce anomalous dispersion relations (e.g., negative sound speed).

Dynamical Shannon entropy and information Tsallis entropy in complex systems

In this work, we offer a new approach to the concept of the information entropy for the description of the dynamic behaviour of complex systems. On the basis of the unification of the known information approach to the Shannon entropy, submitted in works of authors [Phys. Rev. E 62 (5) (2000) 6178; Physica A 303 (2002) 427], and generalization of the Boltzman–Gibbs entropy, offered by Tsallis [Braz. J. Phys. 29 (1) (1999) 1], we have received a new representation of the dynamic information entropy. Here we present concrete applications of the received equations to the study of complex systems related to the electric signals of ECGs of healthy people and patients with myocardial infarction (MI) as an example. Various modifications of the non-Markovity parameter are also submitted. They were received with the help of the new approach to the information entropy. We have received detailed information about Markov, quasi-Markov and non-Markov characteristics of the RR-interval fluctuations of the ECGs with the help of the chain of Zwanzig’-Mori's discrete kinetic equations and dynamical Tsallis entropy.

Treatment of laser-induced thermal acoustics in the framework of discrete kinetic theory

The physics behind the laser-induced thermal acoustics technique is dealt with on a microscopic level. A discrete velocity model of the Boltzmann equation for inelastically interacting gas mixtures in the presence of two counterpropagating laser beams is established. The collisional scheme for the model is developed by taking into account elastic and inelastic interactions between the gas particles, on the one hand, and the interactions between monochromatic laser photons and gas particles, on the other hand. The formation and evolution of laser-pulse-driven thermal and density gratings are simulated by numerically solving the discrete kinetic equations based on the fractional step method. Numerical results are provided for a wide scope of Knudsen numbers.

Equilibria for discrete kinetic equations

We develop a systematic method of constructing equilibria for kinetic models with discrete microscopic velocities. The approach is based on a suitable entropy maximum principle. The $H$ theorem is demonstrated in the continuous and discrete space-time realizations. In addition, we discuss an extension of the Lattice Boltzmann method to irregular grids.

Some remarks on the extended discrete kinetic equations

Abstract Extended kinetic equations, taking into account removal and generation interactions, and presence of a background medium, are considered in the discrete velocity formalism, and applied for studying the problem of a gas in a uniform host medium according to a planar 8-velocity model. After deriving the relevant governing equations, equilibria are investigated in detail for varying absorption, generation and scattering properties of the fixed, but in general not isotropic, background. Some stability properties and preliminary results about shock wave structure are also discussed.